fe_evaluation.h

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//
// Copyright (C) 2011 - 2021 by the deal.II authors
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// This file is part of the deal.II library.
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// it, and/or modify it under the terms of the GNU Lesser General
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#ifndef dealii_matrix_free_fe_evaluation_h
#define dealii_matrix_free_fe_evaluation_h


#include <deal.II/base/config.h>

#include <deal.II/base/array_view.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/smartpointer.h>
#include <deal.II/base/symmetric_tensor.h>
#include <deal.II/base/template_constraints.h>
#include <deal.II/base/vectorization.h>

#include <deal.II/lac/vector_operation.h>

#include <deal.II/matrix_free/evaluation_flags.h>
#include <deal.II/matrix_free/evaluation_kernels.h>
#include <deal.II/matrix_free/evaluation_selector.h>
#include <deal.II/matrix_free/evaluation_template_factory.h>
#include <deal.II/matrix_free/fe_evaluation_data.h>
#include <deal.II/matrix_free/mapping_data_on_the_fly.h>
#include <deal.II/matrix_free/matrix_free.h>
#include <deal.II/matrix_free/shape_info.h>
#include <deal.II/matrix_free/tensor_product_kernels.h>
#include <deal.II/matrix_free/type_traits.h>
#include <deal.II/matrix_free/vector_access_internal.h>

#include <type_traits>


DEAL_II_NAMESPACE_OPEN



/**
 * This is the base class for the FEEvaluation classes.  This class needs
 * usually not be called in user code and does not have any public
 * constructor. The usage is through the class FEEvaluation instead. It
 * implements a reinit method that is used to set pointers so that operations
 * on quadrature points can be performed quickly, access functions to vectors
 * for the FEEvaluationBase::read_dof_values(),
 * FEEvaluationBase::set_dof_values(), and
 * FEEvaluationBase::distribute_local_to_global() functions, as well as
 * methods to access values and gradients of finite element functions. It also
 * inherits the geometry access functions provided by the class
 * FEEvaluationData.
 *
 * This class has five template arguments:
 *
 * @tparam dim Dimension in which this class is to be used
 *
 * @tparam n_components Number of vector components when solving a system of
 * PDEs. If the same operation is applied to several components of a PDE (e.g.
 * a vector Laplace equation), they can be applied simultaneously with one
 * call (and often more efficiently)
 *
 * @tparam Number Number format, usually @p double or @p float
 *
 * @tparam is_face Whether the class is used for a cell integrator (with
 * quadrature dimension the same as the space dimension) or for a face
 * integrator (with quadrature dimension one less)
 *
 * @tparam VectorizedArrayType Type of array to be woked on in a vectorized
 *                             fashion, defaults to VectorizedArray<Number>
 *
 * @note Currently only VectorizedArray<Number, width> is supported as
 *       VectorizedArrayType.
 *
 *
 * @ingroup matrixfree
 */
template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
class FEEvaluationBase
  : public FEEvaluationData<dim, VectorizedArrayType, is_face>
{
public:
  using number_type = Number;
  using value_type  = Tensor<1, n_components_, VectorizedArrayType>;
  using gradient_type =
    Tensor<1, n_components_, Tensor<1, dim, VectorizedArrayType>>;
  using hessian_type =
    Tensor<1, n_components_, Tensor<2, dim, VectorizedArrayType>>;
  static constexpr unsigned int dimension    = dim;
  static constexpr unsigned int n_components = n_components_;

  /**
   * @name 1: Reading from and writing to vectors
   */
  //@{
  /**
   * For the vector @p src, read out the values on the degrees of freedom of
   * the current cell, and store them internally. Similar functionality as the
   * function DoFAccessor::get_interpolated_dof_values when no constraints are
   * present, but it also includes constraints from hanging nodes, so one can
   * see it as a similar function to AffineConstraints::read_dof_values as
   * well. Note that if vectorization is enabled, the DoF values for several
   * cells are set.
   *
   * If some constraints on the vector are inhomogeneous, use the function
   * read_dof_values_plain instead and provide the vector with useful data
   * also in constrained positions by calling AffineConstraints::distribute.
   * When accessing vector entries during the solution of linear systems, the
   * temporary solution should always have homogeneous constraints and this
   * method is the correct one.
   *
   * If the given vector template class is a block vector (determined through
   * the template function 'IsBlockVector<VectorType>::value', which checks
   * for vectors derived from dealii::BlockVectorBase) or an
   * std::vector<VectorType> or std::vector<VectorType *>, this function reads
   * @p n_components blocks from the block vector starting at the index
   * @p first_index. For non-block vectors, @p first_index is ignored.
   *
   * @note If this class was constructed without a MatrixFree object and the
   * information is acquired on the fly through a
   * DoFHandler<dim>::cell_iterator, only one single cell is used by this
   * class and this function extracts the values of the underlying components
   * on the given cell. This call is slower than the ones done through a
   * MatrixFree object and lead to a structure that does not effectively use
   * vectorization in the evaluate routines based on these values (instead,
   * VectorizedArray::size() same copies are worked on).
   */
  template <typename VectorType>
  void
  read_dof_values(const VectorType & src,
                  const unsigned int first_index = 0,
                  const std::bitset<VectorizedArrayType::size()> &mask =
                    std::bitset<VectorizedArrayType::size()>().flip());

  /**
   * For the vector @p src, read out the values on the degrees of freedom of
   * the current cell, and store them internally. Similar functionality as the
   * function DoFAccessor::get_interpolated_dof_values. As opposed to the
   * read_dof_values function, this function reads out the plain entries from
   * vectors, without taking stored constraints into account. This way of
   * access is appropriate when the constraints have been distributed on the
   * vector by a call to AffineConstraints::distribute previously. This
   * function is also necessary when inhomogeneous constraints are to be used,
   * as MatrixFree can only handle homogeneous constraints. Note that if
   * vectorization is enabled, the DoF values for several cells are set.
   *
   * If the given vector template class is a block vector (determined through
   * the template function 'IsBlockVector<VectorType>::value', which checks
   * for vectors derived from dealii::BlockVectorBase) or an
   * std::vector<VectorType> or std::vector<VectorType *>, this function reads
   * @p n_components blocks from the block vector starting at the index
   * @p first_index. For non-block vectors, @p first_index is ignored.
   *
   * @note If this class was constructed without a MatrixFree object and the
   * information is acquired on the fly through a
   * DoFHandler<dim>::cell_iterator, only one single cell is used by this
   * class and this function extracts the values of the underlying components
   * on the given cell. This call is slower than the ones done through a
   * MatrixFree object and lead to a structure that does not effectively use
   * vectorization in the evaluate routines based on these values (instead,
   * VectorizedArray::size() same copies are worked on).
   */
  template <typename VectorType>
  void
  read_dof_values_plain(const VectorType & src,
                        const unsigned int first_index = 0,
                        const std::bitset<VectorizedArrayType::size()> &mask =
                          std::bitset<VectorizedArrayType::size()>().flip());

  /**
   * Takes the values stored internally on dof values of the current cell and
   * sums them into the vector @p dst. The function also applies constraints
   * during the write operation. The functionality is hence similar to the
   * function AffineConstraints::distribute_local_to_global. If vectorization
   * is enabled, the DoF values for several cells are used.
   *
   * If the given vector template class is a block vector (determined through
   * the template function 'IsBlockVector<VectorType>::value', which checks
   * for vectors derived from dealii::BlockVectorBase) or an
   * std::vector<VectorType> or std::vector<VectorType *>, this function
   * writes to @p n_components blocks of the block vector starting at the
   * index @p first_index. For non-block vectors, @p first_index is ignored.
   *
   * The @p mask can be used to suppress the write access for some of the
   * cells contained in the current cell vectorization batch, e.g. in case of
   * local time stepping, where some cells are excluded from a call. A value
   * of `true` in the bitset means that the respective lane index will be
   * processed, whereas a value of `false` skips this index. The default
   * setting is a bitset that contains all ones, which will write the
   * accumulated integrals to all cells in the batch.
   *
   * @note If this class was constructed without a MatrixFree object and the
   * information is acquired on the fly through a
   * DoFHandler<dim>::cell_iterator, only one single cell is used by this
   * class and this function extracts the values of the underlying components
   * on the given cell. This call is slower than the ones done through a
   * MatrixFree object and lead to a structure that does not effectively use
   * vectorization in the evaluate routines based on these values (instead,
   * VectorizedArray::size() same copies are worked on).
   */
  template <typename VectorType>
  void
  distribute_local_to_global(
    VectorType &                                    dst,
    const unsigned int                              first_index = 0,
    const std::bitset<VectorizedArrayType::size()> &mask =
      std::bitset<VectorizedArrayType::size()>().flip()) const;

  /**
   * Takes the values stored internally on dof values of the current cell and
   * writes them into the vector @p dst. The function skips the degrees of
   * freedom which are constrained. As opposed to the
   * distribute_local_to_global method, the old values at the position given
   * by the current cell are overwritten. Thus, if a degree of freedom is
   * associated to more than one cell (as usual in continuous finite
   * elements), the values will be overwritten and only the value written last
   * is retained. Please note that in a parallel context this function might
   * also touch degrees of freedom owned by other MPI processes, so that a
   * subsequent update or accumulation of ghost values as done by
   * MatrixFree::loop() might invalidate the degrees of freedom set by this
   * function.
   *
   * If the given vector template class is a block vector (determined through
   * the template function 'IsBlockVector<VectorType>::value', which checks
   * for vectors derived from dealii::BlockVectorBase) or an
   * std::vector<VectorType> or std::vector<VectorType *>, this function
   * writes to @p n_components blocks of the block vector starting at the
   * index @p first_index. For non-block vectors, @p first_index is ignored.
   *
   * The @p mask can be used to suppress the write access for some
   * of the cells contained in the current cell vectorization batch, e.g. in
   * case of local time stepping, where some  cells are excluded from a call.
   * A value of `true` in the bitset means that the respective lane index will
   * be processed, whereas a value of `false` skips this index. The default
   * setting is a bitset that contains all ones, which will write the
   * accumulated integrals to all cells in the batch.
   *
   * @note If this class was constructed without a MatrixFree object and the
   * information is acquired on the fly through a
   * DoFHandler<dim>::cell_iterator, only one single cell is used by this
   * class and this function extracts the values of the underlying components
   * on the given cell. This call is slower than the ones done through a
   * MatrixFree object and lead to a structure that does not effectively use
   * vectorization in the evaluate routines based on these values (instead,
   * VectorizedArray::size() same copies are worked on).
   */
  template <typename VectorType>
  void
  set_dof_values(VectorType &       dst,
                 const unsigned int first_index = 0,
                 const std::bitset<VectorizedArrayType::size()> &mask =
                   std::bitset<VectorizedArrayType::size()>().flip()) const;

  /**
   * Same as set_dof_values(), but without resolving constraints.
   */
  template <typename VectorType>
  void
  set_dof_values_plain(
    VectorType &                                    dst,
    const unsigned int                              first_index = 0,
    const std::bitset<VectorizedArrayType::size()> &mask =
      std::bitset<VectorizedArrayType::size()>().flip()) const;

  //@}

  /**
   * @name 2: Access to data at quadrature points or the gather vector data
   */
  //@{
  /**
   * Return the value stored for the local degree of freedom with index @p
   * dof. If the object is vector-valued, a vector-valued return argument is
   * given. Thus, the argument @p dof can at most run until @p
   * dofs_per_component rather than @p dofs_per_cell since the different
   * components of a vector-valued FE are return together. Note that when
   * vectorization is enabled, values from several cells are grouped
   * together. If @p set_dof_values was called last, the value corresponds to
   * the one set there. If @p integrate was called last, it instead
   * corresponds to the value of the integrated function with the test
   * function of the given index.
   *
   * Note that the derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  value_type
  get_dof_value(const unsigned int dof) const;

  /**
   * Write a value to the field containing the degrees of freedom with
   * component @p dof. Writes to the same field as is accessed through @p
   * get_dof_value. Therefore, the original data that was read from a vector
   * is overwritten as soon as a value is submitted.
   *
   * Note that the derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  void
  submit_dof_value(const value_type val_in, const unsigned int dof);

  /**
   * Return the value of a finite element function at quadrature point number
   * @p q_point after a call to FEEvaluation::evaluate() with
   * EvaluationFlags::values set, or the value that has been stored there with
   * a call to FEEvaluationBase::submit_value(). If the object is
   * vector-valued, a vector-valued return argument is given. Note that when
   * vectorization is enabled, values from several cells are grouped together.
   *
   * Note that the derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  value_type
  get_value(const unsigned int q_point) const;

  /**
   * Write a value to the field containing the values on quadrature points
   * with component @p q_point. Access to the same field as through
   * get_value(). If applied before the function FEEvaluation::integrate()
   * with EvaluationFlags::values set is called, this specifies the value
   * which is tested by all basis function on the current cell and integrated
   * over.
   *
   * Note that the derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  void
  submit_value(const value_type val_in, const unsigned int q_point);

  /**
   * Return the gradient of a finite element function at quadrature point
   * number @p q_point after a call to FEEvaluation::evaluate() with
   * EvaluationFlags::gradients, or the value that has been stored there with
   * a call to FEEvaluationBase::submit_gradient().
   *
   * Note that the derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  gradient_type
  get_gradient(const unsigned int q_point) const;

  /**
   * Return the derivative of a finite element function at quadrature point
   * number @p q_point after a call to
   * FEEvaluation::evaluate(EvaluationFlags::gradients) the direction normal
   * to the face: $\boldsymbol \nabla u(\mathbf x_q) \cdot \mathbf n(\mathbf
   * x_q)$
   *
   * This call is equivalent to calling get_gradient() * get_normal_vector()
   * but will use a more efficient internal representation of data.
   *
   * Note that the derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  value_type
  get_normal_derivative(const unsigned int q_point) const;

  /**
   * Write a contribution that is tested by the gradient to the field
   * containing the values on quadrature points with component @p q_point.
   * Access to the same field as through get_gradient(). If applied before the
   * function FEEvaluation::integrate(EvaluationFlags::gradients) is called,
   * this specifies what is tested by all basis function gradients on the
   * current cell and integrated over.
   *
   * Note that the derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  void
  submit_gradient(const gradient_type grad_in, const unsigned int q_point);

  /**
   * Write a contribution that is tested by the gradient to the field
   * containing the values on quadrature points with component @p
   * q_point. Access to the same field as through get_gradient() or
   * get_normal_derivative(). If applied before the function
   * FEEvaluation::integrate(EvaluationFlags::gradients) is called, this
   * specifies what is tested by all basis function gradients on the current
   * cell and integrated over.
   *
   * @note This operation writes the data to the same field as
   * submit_gradient(). As a consequence, only one of these two can be
   * used. Usually, the contribution of a potential call to this function must
   * be added into the contribution for submit_gradient().
   *
   * @note The derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  void
  submit_normal_derivative(const value_type   grad_in,
                           const unsigned int q_point);

  /**
   * Write a contribution that is tested by the Hessian to the field
   * containing the values at quadrature points with component @p q_point.
   * Access to the same field as through get_hessian(). If applied before the
   * function FEEvaluation::integrate(EvaluationFlags::hessians) is called,
   * this specifies what is tested by the Hessians of all basis functions on the
   * current cell and integrated over.
   *
   * Note that the derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  void
  submit_hessian(const hessian_type hessian_in, const unsigned int q_point);

  /**
   * Return the Hessian of a finite element function at quadrature point
   * number @p q_point after a call to
   * FEEvaluation::evaluate(EvaluationFlags::hessians). If only the diagonal
   * or even the trace of the Hessian, the Laplacian, is needed, use the other
   * functions below.
   *
   * Note that the derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  Tensor<1, n_components_, Tensor<2, dim, VectorizedArrayType>>
  get_hessian(const unsigned int q_point) const;

  /**
   * Return the diagonal of the Hessian of a finite element function at
   * quadrature point number @p q_point after a call to
   * FEEvaluation::evaluate(EvaluationFlags::hessians).
   *
   * Note that the derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  gradient_type
  get_hessian_diagonal(const unsigned int q_point) const;

  /**
   * Return the Laplacian (i.e., the trace of the Hessian) of a finite element
   * function at quadrature point number @p q_point after a call to
   * FEEvaluation::evaluate(EvaluationFlags::hessians). Compared to the case
   * when computing the full Hessian, some operations can be saved when only
   * the Laplacian is requested.
   *
   * Note that the derived class FEEvaluationAccess overloads this operation
   * with specializations for the scalar case (n_components == 1) and for the
   * vector-valued case (n_components == dim).
   */
  value_type
  get_laplacian(const unsigned int q_point) const;

#ifdef DOXYGEN
  // doxygen does not anyhow mention functions coming from partial template
  // specialization of the base class, in this case FEEvaluationAccess<dim,dim>.
  // For now, hack in those functions manually only to fix documentation:

  /**
   * Return the divergence of a vector-valued finite element at quadrature
   * point number @p q_point after a call to @p evaluate(...,true,...).
   *
   * @note Only available for n_components_==dim.
   */
  VectorizedArrayType
  get_divergence(const unsigned int q_point) const;

  /**
   * Return the symmetric gradient of a vector-valued finite element at
   * quadrature point number @p q_point after a call to @p
   * evaluate(...,true,...). It corresponds to <tt>0.5
   * (grad+grad<sup>T</sup>)</tt>.
   *
   * @note Only available for n_components_==dim.
   */
  SymmetricTensor<2, dim, VectorizedArrayType>
  get_symmetric_gradient(const unsigned int q_point) const;

  /**
   * Return the curl of the vector field, $\nabla \times v$ after a call to @p
   * evaluate(...,true,...).
   *
   * @note Only available for n_components_==dim.
   */
  Tensor<1, (dim == 2 ? 1 : dim), VectorizedArrayType>
  get_curl(const unsigned int q_point) const;

  /**
   * Write a contribution that is tested by the divergence to the field
   * containing the values on quadrature points with component @p q_point.
   * Access to the same field as through @p get_gradient. If applied before
   * the function @p integrate(...,true) is called, this specifies what is
   * tested by all basis function gradients on the current cell and integrated
   * over.
   *
   * @note Only available for n_components_==dim.
   *
   * @note This operation writes the data to the same field as
   * submit_gradient(). As a consequence, only one of these two can be
   * used. Usually, the contribution of a potential call to this function must
   * be added into the diagonal of the contribution for submit_gradient().
   */
  void
  submit_divergence(const VectorizedArrayType div_in,
                    const unsigned int        q_point);

  /**
   * Write a contribution that is tested by the symmetric gradient to the field
   * containing the values on quadrature points with component @p q_point.
   * Access to the same field as through @p get_symmetric_gradient. If applied before
   * the function @p integrate(...,true) is called, this specifies the
   * symmetric gradient which is tested by all basis function symmetric
   * gradients on the current cell and integrated over.
   *
   * @note Only available for n_components_==dim.
   *
   * @note This operation writes the data to the same field as
   * submit_gradient(). As a consequence, only one of these two can be
   * used. Usually, the contribution of a potential call to this function must
   * be added to the respective entries of the rank-2 tensor for
   * submit_gradient().
   */
  void
  submit_symmetric_gradient(
    const SymmetricTensor<2, dim, VectorizedArrayType> grad_in,
    const unsigned int                                 q_point);

  /**
   * Write the components of a curl containing the values on quadrature point
   * @p q_point. Access to the same data field as through @p get_gradient.
   *
   * @note Only available for n_components_==dim.
   *
   * @note This operation writes the data to the same field as
   * submit_gradient(). As a consequence, only one of these two can be
   * used. Usually, the contribution of a potential call to this function must
   * be added to the respective entries of the rank-2 tensor for
   * submit_gradient().
   */
  void
  submit_curl(const Tensor<1, dim == 2 ? 1 : dim, VectorizedArrayType> curl_in,
              const unsigned int                                       q_point);

#endif

  /**
   * Takes values at quadrature points, multiplies by the Jacobian determinant
   * and quadrature weights (JxW) and sums the values for all quadrature
   * points on the cell. The result is a scalar, representing the integral
   * over the function over the cell. If a vector-element is used, the
   * resulting components are still separated. Moreover, if vectorization is
   * enabled, the integral values of several cells are contained in the slots
   * of the returned VectorizedArray field.
   *
   * @note In case the FEEvaluation object is initialized with a batch of
   * cells where not all lanes in the SIMD vector VectorizedArray are
   * representing actual data, this method performs computations on dummy data
   * (that is copied from the last valid lane) that will not make sense. Thus,
   * the user needs to make sure that it is not used in any computation
   * explicitly, like when summing the results of several cells.
   */
  value_type
  integrate_value() const;

  //@}

  /**
   * Provides a unified interface to access data in a vector of
   * VectorizedArray fields of length MatrixFree::n_cell_batches() +
   * MatrixFree::n_ghost_cell_batches() for both cells (plain read) and faces
   * (indirect addressing).
   */
  VectorizedArrayType
  read_cell_data(const AlignedVector<VectorizedArrayType> &array) const;

  /**
   * Provides a unified interface to set data in a vector of
   * VectorizedArray fields of length MatrixFree::n_cell_batches() +
   * MatrixFree::n_ghost_cell_batches() for both cells (plain read) and faces
   * (indirect addressing).
   */
  void
  set_cell_data(AlignedVector<VectorizedArrayType> &array,
                const VectorizedArrayType &         value) const;

  /**
   * The same as above, just for std::array of length of VectorizedArrayType for
   * arbitrary data type.
   */
  template <typename T>
  std::array<T, VectorizedArrayType::size()>
  read_cell_data(const AlignedVector<std::array<T, VectorizedArrayType::size()>>
                   &array) const;

  /**
   * The same as above, just for std::array of length of VectorizedArrayType for
   * arbitrary data type.
   */
  template <typename T>
  void
  set_cell_data(
    AlignedVector<std::array<T, VectorizedArrayType::size()>> &array,
    const std::array<T, VectorizedArrayType::size()> &         value) const;

  /**
   * Return the underlying MatrixFree object.
   */
  const MatrixFree<dim, Number, VectorizedArrayType> &
  get_matrix_free() const;

protected:
  /**
   * Constructor. Made protected to prevent users from directly using this
   * class. Takes all data stored in MatrixFree. If applied to problems with
   * more than one finite element or more than one quadrature formula selected
   * during construction of @p matrix_free, @p dof_no, @p
   * first_selected_component and @p quad_no allow to select the appropriate
   * components.
   */
  FEEvaluationBase(
    const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
    const unsigned int                                  dof_no,
    const unsigned int first_selected_component,
    const unsigned int quad_no,
    const unsigned int fe_degree,
    const unsigned int n_q_points,
    const bool         is_interior_face,
    const unsigned int active_fe_index,
    const unsigned int active_quad_index,
    const unsigned int face_type);

  /**
   * Constructor that comes with reduced functionality and works similar as
   * FEValues. The arguments are similar to the ones passed to the constructor
   * of FEValues, with the notable difference that FEEvaluation expects a
   * one-dimensional
   * quadrature formula, Quadrature<1>, instead of a @p dim
   * dimensional one. The finite element can be both scalar or vector valued,
   * but this method always only selects a scalar base element at a time (with
   * @p n_components copies as specified by the class template argument). For
   * vector-valued elements, the optional argument @p first_selected_component
   * allows to specify the index of the base element to be used for
   * evaluation. Note that the internal data structures always assume that the
   * base element is primitive, non-primitive are not supported currently.
   *
   * As known from FEValues, a call to the reinit method with a
   * Triangulation::cell_iterator is necessary to make the geometry and
   * degrees of freedom of the current class known. If the iterator includes
   * DoFHandler information (i.e., it is a DoFHandler::cell_iterator or
   * similar), the initialization allows to also read from or write to vectors
   * in the standard way for DoFHandler::active_cell_iterator types for one
   * cell at a time. However, this approach is much slower than the path with
   * MatrixFree with MPI since index translation has to be done. As only one
   * cell at a time is used, this method does not vectorize over several
   * elements (which is most efficient for vector operations), but only
   * possibly within the element if the evaluate/integrate routines are
   * combined inside user code (e.g. for computing cell matrices).
   *
   * The optional FEEvaluationData object allows several
   * FEEvaluation objects to share the geometry evaluation, i.e., the
   * underlying mapping and quadrature points do only need to be evaluated
   * once. This only works if the quadrature formulas are the same. Otherwise,
   * a new evaluation object is created. Make sure to not pass an optional
   * object around when you intend to use the FEEvaluation object in %parallel
   * with another one because otherwise the intended sharing may create race
   * conditions.
   */
  FEEvaluationBase(
    const Mapping<dim> &      mapping,
    const FiniteElement<dim> &fe,
    const Quadrature<1> &     quadrature,
    const UpdateFlags         update_flags,
    const unsigned int        first_selected_component,
    const FEEvaluationData<dim, VectorizedArrayType, is_face> *other);

  /**
   * Copy constructor. If FEEvaluationBase was constructed from a mapping, fe,
   * quadrature, and update flags, the underlying geometry evaluation based on
   * FEValues will be deep-copied in order to allow for using in parallel with
   * threads.
   */
  FEEvaluationBase(const FEEvaluationBase &other);

  /**
   * Copy assignment operator. If FEEvaluationBase was constructed from a
   * mapping, fe, quadrature, and update flags, the underlying geometry
   * evaluation based on FEValues will be deep-copied in order to allow for
   * using in parallel with threads.
   */
  FEEvaluationBase &
  operator=(const FEEvaluationBase &other);

  /**
   * Destructor.
   */
  ~FEEvaluationBase();

  /**
   * A unified function to read from and write into vectors based on the given
   * template operation. It can perform the operation for @p read_dof_values,
   * @p distribute_local_to_global, and @p set_dof_values. It performs the
   * operation for several vectors at a time.
   */
  template <typename VectorType, typename VectorOperation>
  void
  read_write_operation(
    const VectorOperation &                        operation,
    const std::array<VectorType *, n_components_> &vectors,
    const std::array<
      const std::vector<ArrayView<const typename VectorType::value_type>> *,
      n_components_> &                              vectors_sm,
    const std::bitset<VectorizedArrayType::size()> &mask,
    const bool apply_constraints = true) const;

  /**
   * A unified function to read from and write into vectors based on the given
   * template operation for DG-type schemes where all degrees of freedom on
   * cells are contiguous. It can perform the operation for read_dof_values(),
   * distribute_local_to_global(), and set_dof_values() for several vectors at
   * a time, depending on n_components.
   */
  template <typename VectorType, typename VectorOperation>
  void
  read_write_operation_contiguous(
    const VectorOperation &                        operation,
    const std::array<VectorType *, n_components_> &vectors,
    const std::array<
      const std::vector<ArrayView<const typename VectorType::value_type>> *,
      n_components_> &                              vectors_sm,
    const std::bitset<VectorizedArrayType::size()> &mask) const;

  /**
   * A unified function to read from and write into vectors based on the given
   * template operation for the case when we do not have an underlying
   * MatrixFree object. It can perform the operation for @p read_dof_values,
   * @p distribute_local_to_global, and @p set_dof_values. It performs the
   * operation for several vectors at a time, depending on n_components.
   */
  template <typename VectorType, typename VectorOperation>
  void
  read_write_operation_global(
    const VectorOperation &                        operation,
    const std::array<VectorType *, n_components_> &vectors) const;

  /**
   * Apply hanging-node constraints.
   */
  void
  apply_hanging_node_constraints(const bool transpose) const;

  /**
   * This is the general array for all data fields.
   */
  AlignedVector<VectorizedArrayType> *scratch_data_array;

  /**
   * A pointer to the underlying data.
   */
  const MatrixFree<dim, Number, VectorizedArrayType> *matrix_free;

  /**
   * A temporary data structure necessary to read degrees of freedom when no
   * MatrixFree object was given at initialization.
   */
  mutable std::vector<types::global_dof_index> local_dof_indices;
};



/**
 * This class provides access to the data fields of the FEEvaluation classes.
 * Generic access is achieved through the base class, and specializations for
 * scalar and vector-valued elements are defined separately.
 *
 * @ingroup matrixfree
 */
template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType = VectorizedArray<Number>>
class FEEvaluationAccess : public FEEvaluationBase<dim,
                                                   n_components_,
                                                   Number,
                                                   is_face,
                                                   VectorizedArrayType>
{
  static_assert(
    std::is_same<Number, typename VectorizedArrayType::value_type>::value,
    "Type of Number and of VectorizedArrayType do not match.");

public:
  using number_type = Number;
  using value_type  = Tensor<1, n_components_, VectorizedArrayType>;
  using gradient_type =
    Tensor<1, n_components_, Tensor<1, dim, VectorizedArrayType>>;
  static constexpr unsigned int dimension    = dim;
  static constexpr unsigned int n_components = n_components_;
  using BaseClass =
    FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>;

protected:
  /**
   * Constructor. Made protected to prevent initialization in user code. Takes
   * all data stored in MatrixFree. If applied to problems with more than one
   * finite element or more than one quadrature formula selected during
   * construction of @p matrix_free, @p first_selected_component and @p
   * quad_no allow to select the appropriate components.
   */
  FEEvaluationAccess(
    const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
    const unsigned int                                  dof_no,
    const unsigned int first_selected_component,
    const unsigned int quad_no,
    const unsigned int fe_degree,
    const unsigned int n_q_points,
    const bool         is_interior_face  = true,
    const unsigned int active_fe_index   = numbers::invalid_unsigned_int,
    const unsigned int active_quad_index = numbers::invalid_unsigned_int,
    const unsigned int face_type         = numbers::invalid_unsigned_int);

  /**
   * Constructor with reduced functionality for similar usage of FEEvaluation
   * as FEValues, including matrix assembly.
   */
  FEEvaluationAccess(
    const Mapping<dim> &      mapping,
    const FiniteElement<dim> &fe,
    const Quadrature<1> &     quadrature,
    const UpdateFlags         update_flags,
    const unsigned int        first_selected_component,
    const FEEvaluationData<dim, VectorizedArrayType, is_face> *other);

  /**
   * Copy constructor
   */
  FEEvaluationAccess(const FEEvaluationAccess &other);

  /**
   * Copy assignment operator
   */
  FEEvaluationAccess &
  operator=(const FEEvaluationAccess &other);
};



/**
 * This class provides access to the data fields of the FEEvaluation classes.
 * Partial specialization for scalar fields that defines access with simple
 * data fields, i.e., scalars for the values and Tensor<1,dim> for the
 * gradients.
 *
 * @ingroup matrixfree
 */
template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
class FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>
  : public FEEvaluationBase<dim, 1, Number, is_face, VectorizedArrayType>
{
  static_assert(
    std::is_same<Number, typename VectorizedArrayType::value_type>::value,
    "Type of Number and of VectorizedArrayType do not match.");

public:
  using number_type                       = Number;
  using value_type                        = VectorizedArrayType;
  using gradient_type                     = Tensor<1, dim, VectorizedArrayType>;
  using hessian_type                      = Tensor<2, dim, VectorizedArrayType>;
  static constexpr unsigned int dimension = dim;
  using BaseClass =
    FEEvaluationBase<dim, 1, Number, is_face, VectorizedArrayType>;

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::get_dof_value()
   */
  value_type
  get_dof_value(const unsigned int dof) const;

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::submit_dof_value()
   */
  void
  submit_dof_value(const value_type val_in, const unsigned int dof);

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::get_value()
   */
  value_type
  get_value(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::submit_value()
   */
  void
  submit_value(const value_type val_in, const unsigned int q_point);

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::submit_value()
   */
  void
  submit_value(const Tensor<1, 1, VectorizedArrayType> val_in,
               const unsigned int                      q_point);

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::get_gradient()
   */
  gradient_type
  get_gradient(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::get_normal_derivative()
   */
  value_type
  get_normal_derivative(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::submit_gradient()
   */
  void
  submit_gradient(const gradient_type grad_in, const unsigned int q_point);

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::submit_normal_derivative()
   */
  void
  submit_normal_derivative(const value_type   grad_in,
                           const unsigned int q_point);

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::get_hessian()
   */
  hessian_type
  get_hessian(unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::get_hessian_diagonal()
   */
  gradient_type
  get_hessian_diagonal(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::submit_hessian()
   */
  void
  submit_hessian(const hessian_type hessian_in, const unsigned int q_point);

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::get_laplacian()
   */
  value_type
  get_laplacian(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::integrate_value()
   */
  value_type
  integrate_value() const;

protected:
  /**
   * Constructor. Made protected to avoid initialization in user code. Takes
   * all data stored in MatrixFree. If applied to problems with more than one
   * finite element or more than one quadrature formula selected during
   * construction of @p matrix_free, @p first_selected_component and @p
   * quad_no allow to select the appropriate components.
   */
  FEEvaluationAccess(
    const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
    const unsigned int                                  dof_no,
    const unsigned int first_selected_component,
    const unsigned int quad_no,
    const unsigned int fe_degree,
    const unsigned int n_q_points,
    const bool         is_interior_face  = true,
    const unsigned int active_fe_index   = numbers::invalid_unsigned_int,
    const unsigned int active_quad_index = numbers::invalid_unsigned_int,
    const unsigned int face_type         = numbers::invalid_unsigned_int);

  /**
   * Constructor with reduced functionality for similar usage of FEEvaluation
   * as FEValues, including matrix assembly.
   */
  FEEvaluationAccess(
    const Mapping<dim> &      mapping,
    const FiniteElement<dim> &fe,
    const Quadrature<1> &     quadrature,
    const UpdateFlags         update_flags,
    const unsigned int        first_selected_component,
    const FEEvaluationData<dim, VectorizedArrayType, is_face> *other);

  /**
   * Copy constructor
   */
  FEEvaluationAccess(const FEEvaluationAccess &other);

  /**
   * Copy assignment operator
   */
  FEEvaluationAccess &
  operator=(const FEEvaluationAccess &other);
};



/**
 * This class provides access to the data fields of the FEEvaluation classes.
 * Partial specialization for fields with as many components as the underlying
 * space dimension, i.e., values are of type Tensor<1,dim> and gradients of
 * type Tensor<2,dim>. Provides some additional functions for access, like the
 * symmetric gradient and divergence.
 *
 * @ingroup matrixfree
 */
template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
class FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>
  : public FEEvaluationBase<dim, dim, Number, is_face, VectorizedArrayType>
{
  static_assert(
    std::is_same<Number, typename VectorizedArrayType::value_type>::value,
    "Type of Number and of VectorizedArrayType do not match.");

public:
  using number_type                       = Number;
  using value_type                        = Tensor<1, dim, VectorizedArrayType>;
  using gradient_type                     = Tensor<2, dim, VectorizedArrayType>;
  static constexpr unsigned int dimension = dim;
  static constexpr unsigned int n_components = dim;
  using BaseClass =
    FEEvaluationBase<dim, dim, Number, is_face, VectorizedArrayType>;

  /**
   * @copydoc FEEvaluationBase<dim,dim,Number,is_face>::get_gradient()
   */
  gradient_type
  get_gradient(const unsigned int q_point) const;

  /**
   * Return the divergence of a vector-valued finite element at quadrature
   * point number @p q_point after a call to @p evaluate(...,true,...).
   */
  VectorizedArrayType
  get_divergence(const unsigned int q_point) const;

  /**
   * Return the symmetric gradient of a vector-valued finite element at
   * quadrature point number @p q_point after a call to @p
   * evaluate(...,true,...). It corresponds to <tt>0.5
   * (grad+grad<sup>T</sup>)</tt>.
   */
  SymmetricTensor<2, dim, VectorizedArrayType>
  get_symmetric_gradient(const unsigned int q_point) const;

  /**
   * Return the curl of the vector field, $\nabla \times v$ after a call to @p
   * evaluate(...,true,...).
   */
  Tensor<1, (dim == 2 ? 1 : dim), VectorizedArrayType>
  get_curl(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<dim,dim,Number,is_face>::get_hessian()
   */
  Tensor<3, dim, VectorizedArrayType>
  get_hessian(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<dim,dim,Number,is_face>::get_hessian_diagonal()
   */
  gradient_type
  get_hessian_diagonal(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<dim,dim,Number,is_face>::submit_gradient()
   */
  void
  submit_gradient(const gradient_type grad_in, const unsigned int q_point);

  /**
   * Write a contribution that is tested by the gradient to the field
   * containing the values on quadrature points with component @p q_point.
   * This function is an alternative to the other submit_gradient function
   * when using a system of fixed number of equations which happens to
   * coincide with the dimension for some dimensions, but not all. To allow
   * for dimension-independent programming, this function can be used instead.
   */
  void
  submit_gradient(
    const Tensor<1, dim, Tensor<1, dim, VectorizedArrayType>> grad_in,
    const unsigned int                                        q_point);

  /**
   * Write a contribution that is tested by the divergence to the field
   * containing the values on quadrature points with component @p q_point.
   * Access to the same field as through @p get_gradient. If applied before
   * the function @p integrate(...,true) is called, this specifies what is
   * tested by all basis function gradients on the current cell and integrated
   * over.
   */
  void
  submit_divergence(const VectorizedArrayType div_in,
                    const unsigned int        q_point);

  /**
   * Write a contribution that is tested by the symmetric gradient to the field
   * containing the values on quadrature points with component @p q_point.
   * Access to the same field as through @p get_symmetric_gradient. If applied before
   * the function @p integrate(...,true) is called, this specifies the
   * symmetric gradient which is tested by all basis function symmetric
   * gradients on the current cell and integrated over.
   */
  void
  submit_symmetric_gradient(
    const SymmetricTensor<2, dim, VectorizedArrayType> grad_in,
    const unsigned int                                 q_point);

  /**
   * Write the components of a curl containing the values on quadrature point
   * @p q_point. Access to the same data field as through @p get_gradient.
   */
  void
  submit_curl(const Tensor<1, dim == 2 ? 1 : dim, VectorizedArrayType> curl_in,
              const unsigned int                                       q_point);

protected:
  /**
   * Constructor. Made protected to avoid initialization in user code. Takes
   * all data stored in MatrixFree. If applied to problems with more than one
   * finite element or more than one quadrature formula selected during
   * construction of @p matrix_free, @p first_selected_component and @p
   * quad_no allow to select the appropriate components.
   */
  FEEvaluationAccess(
    const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
    const unsigned int                                  dof_no,
    const unsigned int first_selected_component,
    const unsigned int quad_no,
    const unsigned int dofs_per_cell,
    const unsigned int n_q_points,
    const bool         is_interior_face  = true,
    const unsigned int active_fe_index   = numbers::invalid_unsigned_int,
    const unsigned int active_quad_index = numbers::invalid_unsigned_int,
    const unsigned int face_type         = numbers::invalid_unsigned_int);

  /**
   * Constructor with reduced functionality for similar usage of FEEvaluation
   * as FEValues, including matrix assembly.
   */
  FEEvaluationAccess(
    const Mapping<dim> &      mapping,
    const FiniteElement<dim> &fe,
    const Quadrature<1> &     quadrature,
    const UpdateFlags         update_flags,
    const unsigned int        first_selected_component,
    const FEEvaluationData<dim, VectorizedArrayType, is_face> *other);

  /**
   * Copy constructor
   */
  FEEvaluationAccess(const FEEvaluationAccess &other);

  /**
   * Copy assignment operator
   */
  FEEvaluationAccess &
  operator=(const FEEvaluationAccess &other);
};


/**
 * This class provides access to the data fields of the FEEvaluation classes.
 * Partial specialization for scalar fields in 1d that defines access with
 * simple data fields, i.e., scalars for the values and Tensor<1,1> for the
 * gradients.
 *
 * @ingroup matrixfree
 */
template <typename Number, bool is_face, typename VectorizedArrayType>
class FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>
  : public FEEvaluationBase<1, 1, Number, is_face, VectorizedArrayType>
{
  static_assert(
    std::is_same<Number, typename VectorizedArrayType::value_type>::value,
    "Type of Number and of VectorizedArrayType do not match.");

public:
  using number_type                       = Number;
  using value_type                        = VectorizedArrayType;
  using gradient_type                     = Tensor<1, 1, VectorizedArrayType>;
  using hessian_type                      = Tensor<2, 1, VectorizedArrayType>;
  static constexpr unsigned int dimension = 1;
  using BaseClass =
    FEEvaluationBase<1, 1, Number, is_face, VectorizedArrayType>;

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::get_dof_value()
   */
  value_type
  get_dof_value(const unsigned int dof) const;

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::submit_dof_value()
   */
  void
  submit_dof_value(const value_type val_in, const unsigned int dof);

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::get_value()
   */
  value_type
  get_value(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::submit_value()
   */
  void
  submit_value(const value_type val_in, const unsigned int q_point);

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::submit_value()
   */
  void
  submit_value(const gradient_type val_in, const unsigned int q_point);

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::get_gradient()
   */
  gradient_type
  get_gradient(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::get_divergence()
   */
  value_type
  get_divergence(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<dim,1,Number,is_face>::get_normal_derivative()
   */
  value_type
  get_normal_derivative(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::submit_gradient()
   */
  void
  submit_gradient(const gradient_type grad_in, const unsigned int q_point);

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::submit_gradient()
   */
  void
  submit_gradient(const value_type grad_in, const unsigned int q_point);

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::submit_gradient()
   */
  void
  submit_gradient(const Tensor<2, 1, VectorizedArrayType> grad_in,
                  const unsigned int                      q_point);

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::submit_normal_derivative()
   */
  void
  submit_normal_derivative(const value_type   grad_in,
                           const unsigned int q_point);

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::submit_normal_derivative()
   */
  void
  submit_normal_derivative(const gradient_type grad_in,
                           const unsigned int  q_point);

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::get_hessian()
   */
  hessian_type
  get_hessian(unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::get_hessian_diagonal()
   */
  gradient_type
  get_hessian_diagonal(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::submit_hessian()
   */
  void
  submit_hessian(const hessian_type hessian_in, const unsigned int q_point);

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::get_laplacian()
   */
  value_type
  get_laplacian(const unsigned int q_point) const;

  /**
   * @copydoc FEEvaluationBase<1,1,Number,is_face>::integrate_value()
   */
  value_type
  integrate_value() const;

protected:
  /**
   * Constructor. Made protected to avoid initialization in user code. Takes
   * all data stored in MatrixFree. If applied to problems with more than one
   * finite element or more than one quadrature formula selected during
   * construction of @p matrix_free, @p first_selected_component and @p
   * quad_no allow to select the appropriate components.
   */
  FEEvaluationAccess(
    const MatrixFree<1, Number, VectorizedArrayType> &matrix_free,
    const unsigned int                                dof_no,
    const unsigned int                                first_selected_component,
    const unsigned int                                quad_no,
    const unsigned int                                fe_degree,
    const unsigned int                                n_q_points,
    const bool                                        is_interior_face = true,
    const unsigned int active_fe_index   = numbers::invalid_unsigned_int,
    const unsigned int active_quad_index = numbers::invalid_unsigned_int,
    const unsigned int face_type         = numbers::invalid_unsigned_int);

  /**
   * Constructor with reduced functionality for similar usage of FEEvaluation
   * as FEValues, including matrix assembly.
   */
  FEEvaluationAccess(
    const Mapping<1> &      mapping,
    const FiniteElement<1> &fe,
    const Quadrature<1> &   quadrature,
    const UpdateFlags       update_flags,
    const unsigned int      first_selected_component,
    const FEEvaluationData<1, VectorizedArrayType, is_face> *other);

  /**
   * Copy constructor
   */
  FEEvaluationAccess(const FEEvaluationAccess &other);

  /**
   * Copy assignment operator
   */
  FEEvaluationAccess &
  operator=(const FEEvaluationAccess &other);
};



/**
 * The class that provides all functions necessary to evaluate functions at
 * quadrature points and cell integrations. In functionality, this class is
 * similar to FEValues, however, it includes a lot of specialized functions
 * that make it much faster (between 5 and 500, depending on the polynomial
 * degree). For evaluation of face terms in DG, see the class
 * FEFaceEvaluation.
 *
 * <h3>Usage and initialization</h3>
 *
 * <h4>Fast usage in combination with MatrixFree</h4>
 *
 * The first and foremost way of usage is to initialize this class from a
 * MatrixFree object that caches everything related to the degrees of freedom
 * and the mapping information. This way, it is possible to use vectorization
 * for applying a differential operator for several cells at once.
 *
 * The capabilities of FEEvaluation span a large spectrum of integration tasks
 * for weak forms. In general, there are two classes of tasks that get
 * done. One is the @p evaluate path that interpolates from a solution vector
 * to quadrature points:
 *
 * @code
 * FEEvaluation<dim,fe_degree> fe_eval(matrix_free);
 * for (unsigned int cell_index = cell_range.first;
 *      cell_index < cell_range.second; ++cell_index)
 *   {
 *     fe_eval.reinit(cell_index);
 *     fe_eval.read_dof_values(vector);
 *     fe_eval.evaluate(EvaluationFlags::values);   // interpolate values only
 *     for (unsigned int q=0; q<fe_eval.n_q_points; ++q)
 *       {
 *         VectorizedArray<double> val = fe_eval.get_value(q);
 *         // do something with val
 *       }
 *   }
 * @endcode
 *
 * Likewise, a gradient of the finite element solution represented by
 * `vector` can be interpolated to the quadrature points by
 * `fe_eval.get_gradient(q)`. The combination of read_dof_values(), evaluate()
 * and get_value() is similar to what FEValues::get_function_values or
 * FEValues::get_function_gradients does, but it is in general much faster
 * because it makes use of the tensor product, see the description of the
 * evaluation routines below, and can do this operation for several cells at
 * once through vectorization.
 *
 * The second class of tasks done by FEEvaluation are integration tasks for
 * right hand sides. In finite element computations, these typically consist
 * of multiplying a quantity on quadrature points (a function value, or a
 * field interpolated by the finite element space itself) by a set of test
 * functions and integrating over the cell through summation of the values in
 * each quadrature point, multiplied by the quadrature weight and the Jacobian
 * determinant of the transformation. If a generic Function object is given
 * and we want to compute $v_i = \int_\Omega \varphi_i f dx$, this is done by
 * the following cell-wise integration:
 *
 * @code
 * FEEvaluation<dim,fe_degree> fe_eval(matrix_free);
 * Function<dim> &function = ...;
 * for (unsigned int cell_index = cell_range.first;
 *      cell_index < cell_range.second; ++cell_index)
 *   {
 *     fe_eval.reinit(cell_index);
 *     for (unsigned int q=0; q<fe_eval.n_q_points; ++q)
 *       {
 *         const Point<dim,VectorizedArray<double> > p_vect =
 *           fe_eval.quadrature_point(q);
 *         // Need to evaluate function for each component in VectorizedArray
 *         VectorizedArray<double> f_value = 0.0;
 *         for (unsigned int v=0; v<VectorizedArray<double>::size(); ++v)
 *           {
 *             Point<dim> p;
 *             for (unsigned int d=0; d<dim; ++d)
 *               p[d] = p_vect[d][v];
 *             f_value[v] = function.value(p);
 *           }
 *         fe_eval.submit_value(f_value, q);
 *       }
 *     fe_eval.integrate(EvaluationFlags::values);
 *     fe_eval.distribute_local_to_global(dst);
 *   }
 * @endcode
 *
 * In this code, the call to @p fe_eval.submit_value() prepares for the
 * multiplication by the test function prior to the actual integration (inside
 * the submit call, the value to be tested is also multiplied by the
 * determinant of the Jacobian and the quadrature weight). In the
 * @p integrate() call, an integral contribution tested by each basis function
 * underlying the FEEvaluation object (e.g. the four linear shape functions of
 * FE_Q@<2@>(1) in 2D) is computed, which gives the vector entries to be
 * summed into the @p dst vector. Note that the above code needs to explicitly
 * loop over the components in the vectorized array for evaluating the
 * function, which is necessary for interfacing with a generic Function object
 * with double arguments. Simple functions can also be implemented in
 * VectorizedArray form directly as VectorizedArray provides the basic math
 * operations.
 *
 * For evaluating a bilinear form, the evaluation on a source vector is
 * combined with the integration involving test functions that get written
 * into a result vector. This setting is the context of matrix-free operator
 * evaluation and explained in the step-37 and step-48 tutorial programs.
 *
 * Note that the two vector accesses through FEEvaluation::read_dof_values and
 * FEEvaluation::distribute_local_to_global resolve constraints on the fly,
 * based on the AffineConstraints object specified at the MatrixFree::reinit()
 * call. In case the values in the degrees of freedom are of interest (usually
 * only the values in quadrature points are necessary), these can be accessed
 * through FEEvaluation::get_dof_value(i), where i is the index of the basis
 * function. Note that the numbering of the degrees of freedom for continuous
 * elements in FEEvaluation is different from the ordering in FE_Q (or
 * FEValues) because FEEvaluation needs to access them in lexicographic order,
 * which is the ordering used in FE_DGQ, for instance. Re-indexing would be
 * too expensive because the access inside evaluate() and integrate() is on
 * the critical path in the tensorial evaluation parts. An alternative to
 * filling the DoF values by read_dof_values() before an evaluate() call is to
 * manually assign a value by a set_dof_value() call. Likewise, if the local
 * result of integration should be further processed rather than scattered
 * into a vector by distribute_local_to_global(), one can access it by
 * get_dof_value() after an integrate() call. An example for using the values
 * of an integral in a different context is fast assembly of matrices as shown
 * in the next subsection.
 *
 * For most operator evaluation tasks that repeatedly go through the mesh, the
 * realization by MatrixFree that combines pre-computed data for the mapping
 * (Jacobian transformations for the geometry description) with on-the-fly
 * evaluation of basis functions is the most efficient way of doing things. In
 * other words, the framework selects a trade-off between memory usage and
 * initialization of objects that is suitable for replacement of matrix-vector
 * products or explicit time integration in a matrix-free way.
 *
 * <h4>Usage without pre-initialized MatrixFree object</h4>
 *
 * The second form of usage is to initialize FEEvaluation from geometry
 * information generated by FEValues. This allows to apply the integration
 * loops on the fly without prior initialization of MatrixFree objects. This
 * can be useful when the memory and initialization cost of MatrixFree is not
 * acceptable, e.g. when a different number of quadrature points should be
 * used for one single evaluation in error computation. Also, when using the
 * routines of this class to assemble matrices the trade-off implied by the
 * MatrixFree class may not be desired. In such a case, the cost to initialize
 * the necessary geometry data on the fly is comparably low and thus avoiding
 * a global object MatrixFree can be useful. When used in this way, reinit
 * methods reminiscent from FEValues with a cell iterator are used. However,
 * note that this model results in working on a single cell at a time, with
 * geometry data duplicated in all components of the vectorized array. Thus,
 * vectorization is only useful when it can apply the same operation on
 * different data, e.g. when performing matrix assembly.
 *
 * As an example, consider the following code to assemble the contributions to
 * the Laplace matrix:
 *
 * @code
 * FEEvaluation<dim,fe_degree> fe_eval (mapping, finite_element,
 *                                      QGauss<1>(fe_degree+1), flags);
 * for (const auto &cell : dof_handler.active_cell_iterators())
 *   {
 *     fe_eval.reinit(cell);
 *     for (unsigned int i=0; i<dofs_per_cell;
 *          i += VectorizedArray<double>::size())
 *       {
 *         const unsigned int n_items =
 *           i+VectorizedArray<double>::size() > dofs_per_cell ?
 *           (dofs_per_cell - i) :
 *           VectorizedArray<double>::size();
 *
 *         // Set n_items unit vectors
 *         for (unsigned int j=0; j<dofs_per_cell; ++j)
 *           fe_eval.set_dof_value(VectorizedArray<double>(), j);
 *         for (unsigned int v=0; v<n_items; ++v)
 *           {
 *             VectorizedArray<double> one_value = VectorizedArray<double>();
 *             one_value[v] = 1.;
 *             fe_eval.set_dof_value(one_value, i+v);
 *           }
 *
 *         // Apply operator on unit vector to generate the next few matrix
 *         // columns
 *         fe_eval.evaluate(EvaluationFlags::values|EvaluationFlags::gradients);
 *         for (unsigned int q=0; q<n_q_points; ++q)
 *           {
 *             fe_eval.submit_value(10.*fe_eval.get_value(q), q);
 *             fe_eval.submit_gradient(fe_eval.get_gradient(q), q);
 *           }
 *         fe_eval.integrate(EvaluationFlags::values|EvaluationFlags::gradients);
 *
 *         // Insert computed entries in matrix
 *         for (unsigned int v=0; v<n_items; ++v)
 *           for (unsigned int j=0; j<dofs_per_cell; ++j)
 *             cell_matrix(fe_eval.get_internal_dof_numbering()[j],
 *                         fe_eval.get_internal_dof_numbering()[i+v])
 *               = fe_eval.get_dof_value(j)[v];
 *       }
 *     ...
 *   }
 * @endcode
 *
 * This code generates the columns of the cell matrix with the loop over @p i
 * above. The way this is done is the following: FEEvaluation's routines focus
 * on the evaluation of finite element operators, so for computing a cell
 * matrix out of an operator evaluation it is applied to all the unit vectors
 * on the cell. Applying the operator on a unit vector might seem inefficient
 * but the evaluation routines used here are so quick that they still work
 * much faster than what is possible with FEValues. In particular, the
 * complexity is <code>(fe_degree+1)<sup>2*dim+1</sup> </code> rather than
 * <code>(fe_degree+1)<sup>3*dim</sup> </code>.
 *
 * Due to vectorization, we can generate matrix columns for several unit
 * vectors at a time (e.g. 4). The variable @p n_items make sure that we do
 * the last iteration where the number of cell dofs is not divisible by the
 * vectorization length correctly. Also note that we need to get the internal
 * dof numbering applied by fe_eval because FEEvaluation internally uses a
 * lexicographic numbering of degrees of freedom as explained above.
 *
 * <h4>Internal data organization</h4>
 *
 * The temporary data for holding the solution values on the local degrees of
 * freedom as well as the interpolated values, gradients, and Hessians on
 * quadrature points is a scratch array provided by
 * MatrixFree::acquire_scratch_data() that is re-used between different calls
 * to FEEvaluation. Therefore, constructing an FEEvaluation object is
 * typically cheap and does not involve any expensive operation. Only a few
 * dozen pointers to the actual data fields are set during
 * construction. Therefore, no negative performance impact arises when
 * creating an FEEvaluation several times per loop, such as at the top of a
 * `local_cell_operation` operation that is split in small chunks for a parallel
 * for loop, obviating a separate scratch data field for parallel loops as
 * necessary in the loop of @p WorkStream.
 *
 * When using the FEEvaluation class in multithreaded mode, the thread local
 * storage of the scratch data in MatrixFree automatically makes sure that
 * each thread gets it private data array. Note, however, that deal.II must be
 * compiled with thread support also when all the thread parallelization is
 * provided externally and not done via deal.II's routines, such as
 * OpenMP. This is because deal.II needs to know the notation of thread local
 * storage. The FEEvaluation kernels have been verified to work within OpenMP
 * loops.
 *
 * <h4>Vectorization scheme through VectorizedArray</h4>
 *
 * This class is designed to perform all arithmetics on single-instruction
 * multiple-data (SIMD) instructions present on modern CPUs by explicit
 * vectorization, which are made available in deal.II through the class
 * VectorizedArray, using the widest vector width available at
 * configure/compile time. In order to keep programs flexible, FEEvaluation
 * always applies vectorization over several elements. This is often the best
 * compromise because computations on different elements are usually
 * independent in the finite element method (except of course the process of
 * adding an integral contribution to a global residual vector), also in more
 * complicated scenarios: Stabilization parameter can e.g. be defined as the
 * maximum of some quantities on all quadrature points of a cell divided by
 * the cell's volume, but without locally mixing the results with
 * neighbors. Using the terminology from computer architecture, the design of
 * FEEvaluation relies on not doing any cross-lane data exchange when
 * operating on the cell in typical integration scenarios.
 *
 * When the number of cells in the problem is not a multiple of the number of
 * array elements in the SIMD vector, the implementation of FEEvaluation fills
 * in some dummy entries in the unused SIMD lanes and carries them around
 * nonetheless, a choice made necessary since the length of VectorizedArray is
 * fixed at compile time. Yet, this approach most often results in superior
 * code as compared to an auto-vectorization setup where an alternative
 * unvectorized code path would be necessary next to the vectorized version to
 * be used on fully populated lanes, together with a dispatch mechanism. In
 * @p read_dof_values, the empty lanes resulting from a reinit() call to an
 * incomplete batch of cells are set to zero, whereas
 * @p distribute_local_to_global or @p set_dof_values simply ignores the
 * content in the empty lanes. The number of actually filled SIMD lanes can by
 * queried by MatrixFree::n_components_filled().
 *
 * Obviously, the computations performed on the artificial lanes (without real
 * data) should never be mixed with valid results. The contract in using this
 * class is that the user makes sure that lanes are not crossed in user code,
 * in particular since it is not clear a priori which cells are going to be
 * put together in vectorization. For example, results on an element should
 * not be added to results on other elements except through the global vector
 * access methods or by access that is masked by
 * MatrixFree::n_components_filled(). No guarantee can be made that results on
 * artificial lanes will always be zero that can safely be added to other
 * results: The data on JxW or Jacobians is copied from the last valid lane in
 * order to avoid division by zero that could trigger floating point
 * exceptions or trouble in other situations.
 *
 * <h3>Description of evaluation routines</h3>
 *
 * This class contains specialized evaluation routines for elements based on
 * tensor-product quadrature formulas and tensor-product-like shape functions,
 * including standard FE_Q or FE_DGQ elements and quadrature points symmetric
 * around 0.5 (like Gauss quadrature), FE_DGP elements based on truncated
 * tensor products as well as the faster case of Gauss-Lobatto elements with
 * Gauss-Lobatto quadrature which give diagonal mass matrices and quicker
 * evaluation internally. The main benefit of this class is the evaluation of
 * all shape functions in all quadrature or integration over all shape
 * functions in <code>dim (fe_degree+1)<sup>dim+1</sup> </code> operations
 * instead of the slower <code> (fe_degree+1)<sup>2*dim</sup></code>
 * complexity in the evaluation routines of FEValues. This is done by an
 * algorithm called sum factorization which factors out constant factors
 * during the evaluation along a coordinate direction. This algorithm is the
 * basis of many spectral element algorithms.
 *
 * Note that many of the operations available through this class are inherited
 * from the base class FEEvaluationBase, in particular reading from and
 * writing to vectors. Also, the class inherits from FEEvaluationAccess that
 * implements access to values, gradients and Hessians of the finite element
 * function on quadrature points.
 *
 * This class assumes that the shape functions of the FiniteElement under
 * consideration do <em>not</em> depend on the geometry of the cells in real
 * space. Currently, other finite elements cannot be treated with the
 * matrix-free concept.
 *
 * <h4>Degree of finite element as a compile-time parameter</h4>
 *
 * The class FEEvaluation as two usage models. The first usage model is to
 * specify the polynomial degree as a template parameter. This guarantees
 * maximum
 * efficiency: The evaluation with sum factorization performs a number of nested
 * short 1D loops of length equal to the polynomial degree plus one. If the
 * loop bounds are known at compile time, the compiler can unroll loops as
 * deemed most efficient by its heuristics. At least the innermost loop is
 * almost always completely unrolled, avoiding the loop overhead.
 *
 * However, carrying the polynomial degree (and the number of quadrature
 * points) as a template parameter makes things more complicated in codes
 * where different polynomial degrees should be considered, e.g. in
 * application codes where the polynomial degree is given through an input
 * file. The second usage model is to rely on pre-compiled code for polynomial
 * degrees. While a user code can use different functions for the cells (that
 * get e.g. invoked by some dynamic dispatch mechanism for the various degree
 * templates), deal.II also supports usage of this class based on the
 * information in the element passed to the initialization. For this usage
 * model, set the template parameter for the polynomial degree to -1 and
 * choose an arbitrary number for the number of quadrature points. That code
 * part contains pre-compiled templated code for polynomial degrees between 1
 * and 6 and common quadrature formulas, which runs almost as fast as the
 * templated version. In case the chosen degree is not pre-compiled, an
 * evaluator object with template specialization for -1 is invoked that runs
 * according to run-time bounds.
 *
 * An overview of the performance of FEEvaluation is given in the following
 * figure. It considers the time spent per degree of freedom for evaluating
 * the Laplacian with continuous finite elements using a code similar to the
 * step-37 tutorial program for single-precision arithmetics. The time is
 * based on an experiment on a single core of an Intel Xeon E5-2687W v4,
 * running at 3.4 GHz and measured at problem sizes of around 10 million. The
 * plot lists the computational time (around 0.1 seconds) divided by the
 * number of degrees freedom.
 *
 * @image html fe_evaluation_laplacian_time_per_dof.png
 *
 * The figure shows that the templated computational kernels are between 2.5
 * and 3 times faster than the non-templated ones. The fastest turnaround on
 * this setup is for polynomial degree 5 at 7.4e-9 seconds per degree of
 * freedom or 134 million degrees of freedom per second - on a single
 * core. The non-templated version is also fastest at polynomial degree 5 with
 * 2.1e-9 seconds per degree of freedom or 48 million degrees of freedom per
 * second. Note that using FEEvaluation with template `degree=-1` selects the
 * fast path for degrees between one and six, and the slow path for other
 * degrees.
 *
 * <h4>Pre-compiling code for more polynomial degrees</h4>
 *
 * It is also possible to pre-compile the code in FEEvaluation for a different
 * maximal polynomial degree. This is controlled by the class
 * internal::FEEvaluationFactory and the implementation in
 * `include/deal.II/matrix_free/evaluation_template_factory.templates.h`. By
 * setting the macro `FE_EVAL_FACTORY_DEGREE_MAX` to the desired integer and
 * instantiating the classes FEEvaluationFactory and FEFaceEvaluationFactory
 * (the latter for FEFaceEvaluation) creates paths to templated functions for
 * a possibly larger set of degrees. You can check if fast
 * evaluation/integration for a given degree/n_quadrature_points pair by calling
 * FEEvaluation::fast_evaluation_supported() or
 * FEFaceEvaluation::fast_evaluation_supported().
 *
 * <h3>Handling multi-component systems</h3>
 *
 * FEEvaluation also allows for treating vector-valued problems through a
 * template parameter on the number of components:
 *
 * @code
 * FEEvaluation<dim,fe_degree,n_q_points_1d,n_components> fe_eval(matrix_free);
 * @endcode
 *
 * If used this way, the components can be gathered from several components of
 * an @p std::vector<VectorType> through the call
 *
 * @code
 * fe_eval.read_dof_values(src, 0);
 * @endcode
 *
 * where the 0 means that the vectors starting from the zeroth vector in the
 * @p std::vector should be used, <code>src[0], src[1], ...,
 * src[n_components-1]</code>.
 *
 * An alternative way for reading multi-component systems is possible if the
 * DoFHandler underlying the MatrixFree data is based on an FESystem of @p
 * n_components entries. In that case, a single vector is provided for the
 * read_dof_values() and distribute_local_to_global() calls.
 *
 * An important property of FEEvaluation in multi-component systems is the
 * layout of multiple components in the get_value(), get_gradient(), or
 * get_dof_value() calls. In this case, instead of a scalar return field
 * VectorizedArray@<double@> a tensor is returned,
 *
 * @code
 * get_value -> Tensor<1,n_components,VectorizedArray<double>>
 * get_gradient -> Tensor<1,n_components,Tensor<1,dim,VectorizedArray<double>>
 * @endcode
 *
 * In a similar vein, the submit_value() and submit_gradient() calls take
 * tensors of values. Note that there exist specializations for @p
 * n_components=1 and @p n_components=dim, which are provided through the base
 * class FEEvaluationAccess. In the scalar case, these provide the scalar
 * return types described above. In the vector-valued case, the gradient is
 * converted from <code>Tensor@<1,dim,Tensor@<1,dim,VectorizedArray@<double@>
 * @> @></code> to <code>Tensor@<2,dim,VectorizedArray@<double@>
 * @></code>. Furthermore, additional operations such as the diveregence or
 * curl are available.
 *
 * In case different shape functions are combined, for example mixed finite
 * element formulations in Stokes flow, two FEEvaluation objects are created,
 * one for the velocity and one for the pressure. Those are then combined on
 * quadrature points:
 *
 * @code
 * FEEvaluation<dim,degree_p+1,degree_p+2,dim> velocity (data, 0);
 * FEEvaluation<dim,degree_p,  degree_p+2,1, > pressure (data, 1);
 *
 * for (unsigned int cell=cell_range.first; cell<cell_range.second; ++cell)
 *   {
 *     velocity.reinit (cell);
 *     velocity.read_dof_values (src.block(0));
 *     velocity.evaluate (EvaluationFlags::gradients);
 *     pressure.reinit (cell);
 *     pressure.read_dof_values (src.block(1));
 *     pressure.evaluate (EvaluationFlags::values);
 *
 *     for (unsigned int q=0; q<velocity.n_q_points; ++q)
 *       {
 *         SymmetricTensor<2,dim,VectorizedArray<double> > sym_grad_u =
 *           velocity.get_symmetric_gradient (q);
 *         VectorizedArray<double> pres = pressure.get_value(q);
 *         VectorizedArray<double> div = -trace(sym_grad_u);
 *         pressure.submit_value (div, q);
 *
 *         // subtract p * I
 *         for (unsigned int d=0; d<dim; ++d)
 *           sym_grad_u[d][d] -= pres;
 *
 *         velocity.submit_symmetric_gradient(sym_grad_u, q);
 *      }
 *
 *     velocity.integrate (EvaluationFlags::gradients);
 *     velocity.distribute_local_to_global (dst.block(0));
 *     pressure.integrate (EvaluationFlags::values);
 *     pressure.distribute_local_to_global (dst.block(1));
 *   }
 * @endcode
 *
 * This code assumes that a BlockVector of two components describes the
 * velocity and pressure components, respectively. For identifying the
 * different DoFHandler objects for velocity and pressure, the second argument
 * to the FEEvaluation objects specify the respective component 0 for velocity
 * and 1 for pressure. For further examples of vector-valued problems, the
 * deal.II test suite includes a few additional examples as well, e.g. the
 * Stokes operator described above is found at
 * https://github.com/dealii/dealii/blob/master/tests/matrix_free/matrix_vector_stokes_noflux.cc
 *
 * <h3>Handling several integration tasks and data storage in quadrature
 * points</h3>
 *
 * The design of FEEvaluation and MatrixFree separates the geometry from the
 * basis functions. Therefore, several DoFHandler objects (or the same
 * DoFHandler equipped with different constraint objects) can share the same
 * geometry information like in the Stokes example above. All geometry is
 * cached once in MatrixFree, so FEEvaluation does not need to do expensive
 * initialization calls and rather sets a few pointers. This realization is
 * based on the idea that the geometry information is needed only once also
 * when several fields are evaluated, in a departure from FEValues which sets
 * up the internal mapping data for each field. If for example a
 * multi-component PDE involves the shape values on one component and the
 * shape gradient on the other, no efficiency is lost if both are based on the
 * same MatrixFree object where the update flags specify that both @p
 * update_values , @p update_gradients , and @p update_jxw_values are
 * given. The selection of desired quantities of shape values is through the
 * flags in the evaluate() or integrate calls and the access at quadrature
 * points:
 *
 * @code
 * fe_eval1.evaluate(EvaluationFlags::values);
 * fe_eval2.evaluate(EvaluationFlags::gradients);
 * for (unsigned int q=0; q<fe_eval1.n_q_points; ++q)
 *   {
 *     VectorizedArray<double> val1 = fe_eval1.get_value(q);
 *     Tensor<1,dim,VectorizedArray<double> > grad2 = fe_eval2.get_gradient(q);
 *     Point<dim,VectorizedArray<double> > point = fe_eval1.quadrature_point(q);
 *     // ... some complicated formula combining those three...
 *   }
 * @endcode
 *
 * In the loop over quadrature points, one can ask any of the two FEEvaluation
 * objects &mdash; it does not really matter which one because they only keep
 * pointers to the quadrature point data &mdash; to provide the quadrature
 * point location.
 *
 * This observation also translates to the case when different differential
 * operators are implemented in a program, for example the action of a mass
 * matrix for one phase of the algorithm and the action of a stiffness matrix
 * in another one. Only a single MatrixFree object is necessary, maintaining
 * full efficiency by using different local functions with the respective
 * implementation in separate FEEvaluation objects. In other words, a user
 * does not need to bother about being conservative when providing
 * update_flags to the initialization of MatrixFree for efficiency reasons -
 * no overhead incurs inside FEEvaluation, except for at most one or two more
 * @p if statements inside the FEEvaluation::reinit() call. Rather, the
 * largest set of flags necessary among all calls is perfectly fine from an
 * efficiency point of view.
 *
 * For the combination of different fields, including different solution
 * vectors that come from different time steps, it is mandatory that all
 * FEEvaluation objects share the same MatrixFree object. This is because the
 * way cells are looped by MatrixFree::cell_loop() can be different for
 * different DoFHandler or AffineConstraints arguments. More precisely, even
 * though the layout is going to be the same in serial, there is no guarantee
 * about the ordering for different DoFHandler/AffineConstraints in the MPI
 * case. The reason is that the algorithm detects cells that need data
 * exchange with MPI and those can change for different elements &mdash; FE_Q
 * with hanging node constraints connects to more neighbors than a FE_DGQ
 * element, for instance, and cells which need data exchange are put in
 * different positions inside the cell loop. Of course, if the exact same
 * DoFHandler, AffineConstraints, and options (such as the setting for thread
 * parallelism) are set, then the order is going to be the same because the
 * algorithm is deterministic.
 *
 * @tparam dim Dimension in which this class is to be used
 *
 * @tparam fe_degree Degree of the tensor product finite element with
 * fe_degree+1 degrees of freedom per coordinate direction. Can be set to -1
 * if the degree is not known at compile time, but performance will usually be
 * worse by a factor of 2-3.
 *
 * @tparam n_q_points_1d Number of points in the quadrature formula in 1D,
 * defaults to fe_degree+1
 *
 * @tparam n_components Number of vector components when solving a system of
 * PDEs. If the same operation is applied to several components of a PDE (e.g.
 * a vector Laplace equation), they can be applied simultaneously with one
 * call (and often more efficiently). Defaults to 1.
 *
 * @tparam Number Number format, usually @p double or @p float. Defaults to @p
 * double
 *
 * @ingroup matrixfree
 */
template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
class FEEvaluation : public FEEvaluationAccess<dim,
                                               n_components_,
                                               Number,
                                               false,
                                               VectorizedArrayType>
{
  static_assert(
    std::is_same<Number, typename VectorizedArrayType::value_type>::value,
    "Type of Number and of VectorizedArrayType do not match.");

public:
  /**
   * An alias to the base class.
   */
  using BaseClass =
    FEEvaluationAccess<dim, n_components_, Number, false, VectorizedArrayType>;

  /**
   * An underlying number type specified as template argument.
   */
  using number_type = Number;

  /**
   * The type of function values, e.g. `VectorizedArrayType` for
   * `n_components=1` or `Tensor<1,dim,VectorizedArrayType >` for
   * `n_components=dim`.
   */
  using value_type = typename BaseClass::value_type;

  /**
   * The type of gradients, e.g. `Tensor<1,dim,VectorizedArrayType>` for
   * `n_components=1` or `Tensor<2,dim,VectorizedArrayType >` for
   * `n_components=dim`.
   */
  using gradient_type = typename BaseClass::gradient_type;

  /**
   * The dimension given as template argument.
   */
  static constexpr unsigned int dimension = dim;

  /**
   * The number of solution components of the evaluator given as template
   * argument.
   */
  static constexpr unsigned int n_components = n_components_;

  /**
   * The static number of quadrature points determined from the given template
   * argument `n_q_points_1d`. Note that the actual number of quadrature
   * points, `n_q_points`, can be different if `fe_degree=-1` is given and
   * run-time loop lengths are used rather than compile time ones.
   */
  static constexpr unsigned int static_n_q_points =
    Utilities::pow(n_q_points_1d, dim);

  /**
   * The static number of degrees of freedom of a scalar component determined
   * from the given template argument `fe_degree`. Note that the actual number
   * of degrees of freedom `dofs_per_component` can be different if
   * `fe_degree=-1` is given or if the underlying is of more complicated type
   * than the usual FE_Q or FE_DGQ ones, such as FE_DGP.
   */
  static constexpr unsigned int static_dofs_per_component =
    Utilities::pow(fe_degree + 1, dim);

  /**
   * The static number of degrees of freedom of all components determined from
   * the given template argument `fe_degree`. Note that the actual number of
   * degrees of freedom `dofs_per_cell` can be different if `fe_degree=-1` is
   * given or if the underlying is of more complicated type than the usual
   * FE_Q or FE_DGQ ones, such as FE_DGP.
   */
  static constexpr unsigned int tensor_dofs_per_cell =
    static_dofs_per_component * n_components;

  /**
   * The static number of degrees of freedom of all components determined from
   * the given template argument `fe_degree`. Note that the actual number of
   * degrees of freedom `dofs_per_cell` can be different if `fe_degree=-1` is
   * given or if the underlying is of more complicated type than the usual
   * FE_Q or FE_DGQ ones, such as FE_DGP.
   */
  static constexpr unsigned int static_dofs_per_cell =
    static_dofs_per_component * n_components;

  /**
   * Constructor. Takes all data stored in MatrixFree. If applied to problems
   * with more than one finite element or more than one quadrature formula
   * selected during construction of @p matrix_free, the appropriate component
   * can be selected by the optional arguments.
   *
   * @param matrix_free Data object that contains all data
   *
   * @param dof_no If matrix_free was set up with multiple DoFHandler
   * objects, this parameter selects to which DoFHandler/AffineConstraints pair
   * the given evaluator should be attached to.
   *
   * @param quad_no If matrix_free was set up with multiple Quadrature
   * objects, this parameter selects the appropriate number of the quadrature
   * formula.
   *
   * @param first_selected_component If the dof_handler selected by dof_no
   * uses an FESystem consisting of more than one component, this parameter
   * allows for selecting the component where the current evaluation routine
   * should start. Note that one evaluator does not support combining
   * different shape functions in different components. In other words, the
   * same base element of a FESystem needs to be set for the components
   * between @p first_selected_component and
   * <code>first_selected_component+n_components_</code>.
   *
   * @param active_fe_index If matrix_free was set up with DoFHandler
   * objects with hp::FECollections, this parameter selects to which
   * DoFHandler/AffineConstraints pair the given evaluator should be attached
   * to.
   *
   * @param active_quad_index If matrix_free was set up with hp::Collection
   * objects, this parameter selects the appropriate number of the quadrature
   * formula.
   */
  FEEvaluation(
    const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
    const unsigned int                                  dof_no  = 0,
    const unsigned int                                  quad_no = 0,
    const unsigned int first_selected_component                 = 0,
    const unsigned int active_fe_index   = numbers::invalid_unsigned_int,
    const unsigned int active_quad_index = numbers::invalid_unsigned_int);

  /**
   * Constructor. Takes all data stored in MatrixFree for a given cell range,
   * which allows to automatically identify the active_fe_index and
   * active_quad_index in case of a p-adaptive strategy.
   *
   * The rest of the arguments are the same as in the constructor above.
   */
  FEEvaluation(const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
               const std::pair<unsigned int, unsigned int> &       range,
               const unsigned int                                  dof_no  = 0,
               const unsigned int                                  quad_no = 0,
               const unsigned int first_selected_component                 = 0);

  /**
   * Constructor that comes with reduced functionality and works similar as
   * FEValues. The arguments are similar to the ones passed to the constructor
   * of FEValues, with the notable difference that FEEvaluation expects a
   * one-dimensional
   * quadrature formula, Quadrature<1>, instead of a @p dim
   * dimensional one. The finite element can be both scalar or vector valued,
   * but this method always only selects a scalar base element at a time (with
   * @p n_components copies as specified by the class template). For
   * vector-valued
   * elements, the optional argument @p first_selected_component allows
   * to specify the index of the base element to be used for evaluation. Note
   * that the internal data structures always assume that the base element is
   * primitive, non-primitive are not supported currently.
   *
   * As known from FEValues, a call to the reinit method with a
   * Triangulation<dim>::cell_iterator is necessary to make the geometry and
   * degrees of freedom of the current class known. If the iterator includes
   * DoFHandler information (i.e., it is a DoFHandler<dim>::cell_iterator or
   * similar), the initialization allows to also read from or write to vectors
   * in the standard way for DoFHandler<dim>::active_cell_iterator types for
   * one cell at a time. However, this approach is much slower than the path
   * with MatrixFree with MPI since index translation has to be done. As only
   * one cell at a time is used, this method does not vectorize over several
   * elements (which is most efficient for vector operations), but only
   * possibly within the element if the evaluate/integrate routines are
   * combined inside user code (e.g. for computing cell matrices).
   */
  FEEvaluation(const Mapping<dim> &      mapping,
               const FiniteElement<dim> &fe,
               const Quadrature<1> &     quadrature,
               const UpdateFlags         update_flags,
               const unsigned int        first_selected_component = 0);

  /**
   * Constructor for the reduced functionality. This constructor is equivalent
   * to the other one except that it makes the object use a $Q_1$ mapping
   * (i.e., an object of type MappingQ(1)) implicitly.
   */
  FEEvaluation(const FiniteElement<dim> &fe,
               const Quadrature<1> &     quadrature,
               const UpdateFlags         update_flags,
               const unsigned int        first_selected_component = 0);

  /**
   * Constructor for the reduced functionality. Similar to the other
   * constructor with FiniteElement argument but using another
   * FEEvaluationBase object to provide information about the geometry. This
   * allows several FEEvaluation objects to share the geometry evaluation, i.e.,
   * the underlying mapping and quadrature points do only need to be evaluated
   * once. Make sure to not pass an optional object around when you intend to
   * use the FEEvaluation object in %parallel to the given one because
   * otherwise the intended sharing may create race conditions.
   */
  FEEvaluation(const FiniteElement<dim> &                               fe,
               const FEEvaluationData<dim, VectorizedArrayType, false> &other,
               const unsigned int first_selected_component = 0);

  /**
   * Copy constructor. If FEEvaluationBase was constructed from a mapping, fe,
   * quadrature, and update flags, the underlying geometry evaluation based on
   * FEValues will be deep-copied in order to allow for using in parallel with
   * threads.
   */
  FEEvaluation(const FEEvaluation &other);

  /**
   * Copy assignment operator. If FEEvaluationBase was constructed from a
   * mapping, fe, quadrature, and update flags, the underlying geometry
   * evaluation based on FEValues will be deep-copied in order to allow for
   * using in parallel with threads.
   */
  FEEvaluation &
  operator=(const FEEvaluation &other);

  /**
   * Initialize the operation pointer to the current cell batch index. Unlike
   * the reinit functions taking a cell iterator as argument below and the
   * FEValues::reinit() methods, where the information related to a particular
   * cell is generated in the reinit call, this function is very cheap since
   * all data is pre-computed in @p matrix_free, and only a few indices have
   * to be set appropriately.
   */
  void
  reinit(const unsigned int cell_batch_index);

  /**
   * Initialize the data to the current cell using a TriaIterator object as
   * usual in FEValues. The argument is either of type
   * DoFHandler::active_cell_iterator or DoFHandler::level_cell_iterator. This
   * option is only available if the FEEvaluation object was created with a
   * finite element, quadrature formula and correct update flags and
   * <b>without</b> a MatrixFree object. This initialization method loses the
   * ability to use vectorization, see also the description of the
   * FEEvaluation class. When this reinit method is used, FEEvaluation can
   * also read from vectors (but less efficient than with data coming from
   * MatrixFree).
   */
  template <bool level_dof_access>
  void
  reinit(const TriaIterator<DoFCellAccessor<dim, dim, level_dof_access>> &cell);

  /**
   * Initialize the data to the current cell using a TriaIterator object as
   * usual in FEValues. This option is only available if the FEEvaluation
   * object was created with a finite element, quadrature formula and correct
   * update flags and <b>without</b> a MatrixFree object. This initialization
   * method loses the ability to use vectorization, see also the description
   * of the FEEvaluation class. When this reinit method is used, FEEvaluation
   * can <b>not</b> read from vectors because no DoFHandler information is
   * available.
   */
  void
  reinit(const typename Triangulation<dim>::cell_iterator &cell);

  /**
   * Check if face evaluation/integration is supported.
   */
  static bool
  fast_evaluation_supported(const unsigned int given_degree,
                            const unsigned int give_n_q_points_1d);

  /**
   * Evaluate the function values, the gradients, and the Hessians of the
   * polynomial interpolation from the DoF values in the input vector to the
   * quadrature points on the unit cell.  The function arguments specify which
   * parts shall actually be computed. This function has to be called first so
   * that the access functions @p get_value(), @p get_gradient() or @p
   * get_laplacian give useful information (unless these values have been set
   * manually).
   */
  void
  evaluate(const EvaluationFlags::EvaluationFlags evaluation_flag);

  /**
   * Like above but with separate bool flags.
   * @deprecated use evaluate() with the EvaluationFlags argument.
   */
  DEAL_II_DEPRECATED_EARLY void
  evaluate(const bool evaluate_values,
           const bool evaluate_gradients,
           const bool evaluate_hessians = false);

  /**
   * Evaluate the function values, the gradients, and the Hessians of the
   * polynomial interpolation from the DoF values in the input array @p
   * values_array to the quadrature points on the unit cell. If multiple
   * components are involved in the current FEEvaluation object, the sorting
   * in @p values_array is such that all degrees of freedom for the first
   * component come first, then all degrees of freedom for the second, and so
   * on. The function arguments specify which parts shall actually be
   * computed. This function has to be called first so that the access
   * functions @p get_value(), @p get_gradient() or @p get_laplacian give
   * useful information (unless these values have been set manually).
   */
  void
  evaluate(const VectorizedArrayType *            values_array,
           const EvaluationFlags::EvaluationFlags evaluation_flag);

  /**
   * Like above but using separate bool flags.
   * @deprecated use evaluate() with the EvaluationFlags argument.
   */
  DEAL_II_DEPRECATED_EARLY void
  evaluate(const VectorizedArrayType *values_array,
           const bool                 evaluate_values,
           const bool                 evaluate_gradients,
           const bool                 evaluate_hessians = false);

  /**
   * Read from the input vector and evaluates the function values, the
   * gradients, and the Hessians of the polynomial interpolation of the vector
   * entries from @p input_vector associated with the current cell to the
   * quadrature points on the unit cell. The function arguments specify which
   * parts shall actually be computed. This function has to be called first so
   * that the access functions @p get_value(), @p get_gradient() or @p
   * get_laplacian give useful information (unless these values have been set
   * manually).
   *
   * This call is equivalent to calling read_dof_values() followed by
   * evaluate(), but might internally use some additional optimizations.
   */
  template <typename VectorType>
  void
  gather_evaluate(const VectorType &                     input_vector,
                  const EvaluationFlags::EvaluationFlags evaluation_flag);

  /**
   * @deprecated Please use the gather_evaluate() function with the EvaluationFlags argument.
   */
  template <typename VectorType>
  DEAL_II_DEPRECATED_EARLY void
  gather_evaluate(const VectorType &input_vector,
                  const bool        evaluate_values,
                  const bool        evaluate_gradients,
                  const bool        evaluate_hessians = false);

  /**
   * This function takes the values and/or gradients that are stored on
   * quadrature points, tests them by all the basis functions/gradients on the
   * cell and performs the cell integration. The two function arguments
   * @p integrate_values and @p integrate_gradients are used to enable/disable
   * summation of the contributions submitted to the values or gradients slots,
   * respectively. The result is written into the internal data field
   * @p dof_values (that is usually written into the result vector by the
   * distribute_local_to_global() or set_dof_values() methods).
   */
  void
  integrate(const EvaluationFlags::EvaluationFlags integration_flag);

  /**
   * @deprecated Please use the integrate() function with the EvaluationFlags argument.
   */
  DEAL_II_DEPRECATED_EARLY void
  integrate(const bool integrate_values, const bool integrate_gradients);

  /**
   * This function takes the values and/or gradients that are stored on
   * quadrature points, tests them by all the basis functions/gradients on the
   * cell and performs the cell integration. The two function arguments @p
   * integrate_values and @p integrate_gradients are used to enable/disable
   * summation of the contributions submitted to the values or gradients
   * slots, respectively. As opposed to the other integrate() method, this
   * call stores the result of the testing in the given array @p values_array,
   * whose previous results is overwritten, rather than writing it on the
   * internal data structures behind begin_dof_values().
   */
  void
  integrate(const EvaluationFlags::EvaluationFlags integration_flag,
            VectorizedArrayType *                  values_array,
            const bool                             sum_into_values = false);

  /**
   * @deprecated Please use the integrate() function with the EvaluationFlags argument.
   */
  DEAL_II_DEPRECATED_EARLY void
  integrate(const bool           integrate_values,
            const bool           integrate_gradients,
            VectorizedArrayType *values_array);

  /**
   * This function takes the values and/or gradients that are stored on
   * quadrature points, tests them by all the basis functions/gradients on the
   * cell, performs the cell integration, and adds the result into the global
   * vector @p output_vector on the degrees of freedom associated with the
   * present cell index. The two function arguments @p integrate_values and
   * @p integrate_gradients are used to enable/disable summation of the
   * contributions submitted to the values or gradients slots, respectively.
   *
   * This call is equivalent to calling integrate() followed by
   * distribute_local_to_global(), but might internally use
   * some additional optimizations.
   */
  template <typename VectorType>
  void
  integrate_scatter(const EvaluationFlags::EvaluationFlags integration_flag,
                    VectorType &                           output_vector);

  /**
   * @deprecated Please use the integrate_scatter() function with the EvaluationFlags argument.
   */
  template <typename VectorType>
  DEAL_II_DEPRECATED_EARLY void
  integrate_scatter(const bool  integrate_values,
                    const bool  integrate_gradients,
                    VectorType &output_vector);

  /**
   * Return the q-th quadrature point in real coordinates stored in
   * MappingInfo.
   */
  Point<dim, VectorizedArrayType>
  quadrature_point(const unsigned int q_point) const;

  /**
   * The number of degrees of freedom of a single component on the cell for
   * the underlying evaluation object. Usually close to
   * static_dofs_per_component, but the number depends on the actual element
   * selected and is thus not static.
   */
  const unsigned int dofs_per_component;

  /**
   * The number of degrees of freedom on the cell accumulated over all
   * components in the current evaluation object. Usually close to
   * static_dofs_per_cell = static_dofs_per_component*n_components, but the
   * number depends on the actual element selected and is thus not static.
   */
  const unsigned int dofs_per_cell;

  /**
   * The number of quadrature points in use. If the number of quadrature
   * points in 1d is given as a template, this number is simply the
   * <tt>dim</tt>-th power of that value. If the element degree is set to -1
   * (dynamic selection of element degree), the static value of quadrature
   * points is inaccurate and this value must be used instead.
   */
  const unsigned int n_q_points;

private:
  /**
   * Checks if the template arguments regarding degree of the element
   * corresponds to the actual element used at initialization.
   */
  void
  check_template_arguments(const unsigned int fe_no,
                           const unsigned int first_selected_component);
};



/**
 * The class that provides all functions necessary to evaluate functions at
 * quadrature points and face integrations. The design of the class is similar
 * to FEEvaluation and most of the interfaces are shared with that class, in
 * particular most access functions that come from the common base classes
 * FEEvaluationAccess and FEEvaluationBase. Furthermore, the relation of this
 * class to FEEvaluation is similar to the relation between FEValues and
 * FEFaceValues.
 *
 * @tparam dim Dimension in which this class is to be used
 *
 * @tparam fe_degree Degree of the tensor product finite element with
 *                  fe_degree+1 degrees of freedom per coordinate
 *                  direction. If set to -1, the degree of the underlying
 *                  element will be used, which acts as a run time constant
 *                  rather than a compile time constant that slows down the
 *                  execution.
 *
 * @tparam n_q_points_1d Number of points in the quadrature formula in 1D,
 *                  usually chosen as fe_degree+1
 *
 * @tparam n_components Number of vector components when solving a system of
 *                  PDEs. If the same operation is applied to several
 *                  components of a PDE (e.g. a vector Laplace equation), they
 *                  can be applied simultaneously with one call (and often
 *                  more efficiently)
 *
 * @tparam Number Number format, usually @p double or @p float
 *
 * @tparam VectorizedArrayType Type of array to be woked on in a vectorized
 *                             fashion, defaults to VectorizedArray<Number>
 *
 * @note Currently only VectorizedArray<Number, width> is supported as
 *       VectorizedArrayType.
 */
template <int dim,
          int fe_degree,
          int n_q_points_1d            = fe_degree + 1,
          int n_components_            = 1,
          typename Number              = double,
          typename VectorizedArrayType = VectorizedArray<Number>>
class FEFaceEvaluation : public FEEvaluationAccess<dim,
                                                   n_components_,
                                                   Number,
                                                   true,
                                                   VectorizedArrayType>
{
  static_assert(
    std::is_same<Number, typename VectorizedArrayType::value_type>::value,
    "Type of Number and of VectorizedArrayType do not match.");

public:
  /**
   * An alias to the base class.
   */
  using BaseClass =
    FEEvaluationAccess<dim, n_components_, Number, true, VectorizedArrayType>;

  /**
   * A underlying number type specified as template argument.
   */
  using number_type = Number;

  /**
   * The type of function values, e.g. `VectorizedArrayType` for
   * `n_components=1` or `Tensor<1,dim,VectorizedArrayType >` for
   * `n_components=dim`.
   */
  using value_type = typename BaseClass::value_type;

  /**
   * The type of gradients, e.g. `Tensor<1,dim,VectorizedArrayType>` for
   * `n_components=1` or `Tensor<2,dim,VectorizedArrayType >` for
   * `n_components=dim`.
   */
  using gradient_type = typename BaseClass::gradient_type;

  /**
   * The dimension given as template argument.
   */
  static constexpr unsigned int dimension = dim;

  /**
   * The number of solution components of the evaluator given as template
   * argument.
   */
  static constexpr unsigned int n_components = n_components_;

  /**
   * The static number of quadrature points determined from the given template
   * argument `n_q_points_1d` taken to the power of dim-1. Note that the actual
   * number of quadrature points, `n_q_points`, can be different if
   * `fe_degree=-1` is given and run-time loop lengths are used rather than
   * compile time ones.
   */
  static constexpr unsigned int static_n_q_points =
    Utilities::pow(n_q_points_1d, dim - 1);

  /**
   * The static number of quadrature points on a cell with the same quadrature
   * formula. Note that this value is only present for simpler comparison with
   * the cell quadrature, as the actual number of points is given to a face by
   * the `static_n_q_points` variable.
   */
  static constexpr unsigned int static_n_q_points_cell =
    Utilities::pow(n_q_points_1d, dim);

  /**
   * The static number of degrees of freedom of a scalar component determined
   * from the given template argument `fe_degree`. Note that the actual number
   * of degrees of freedom `dofs_per_component` can be different if
   * `fe_degree=-1` is given.
   */
  static constexpr unsigned int static_dofs_per_component =
    Utilities::pow(fe_degree + 1, dim);

  /**
   * The static number of degrees of freedom of all components determined from
   * the given template argument `fe_degree`. Note that the actual number of
   * degrees of freedom `dofs_per_cell` can be different if `fe_degree=-1` is
   * given.
   */
  static constexpr unsigned int tensor_dofs_per_cell =
    static_dofs_per_component * n_components;

  /**
   * The static number of degrees of freedom of all components determined from
   * the given template argument `fe_degree`. Note that the actual number of
   * degrees of freedom `dofs_per_cell` can be different if `fe_degree=-1` is
   * given.
   */
  static constexpr unsigned int static_dofs_per_cell =
    static_dofs_per_component * n_components;

  /**
   * Constructor. Takes all data stored in MatrixFree. If applied to problems
   * with more than one finite element or more than one quadrature formula
   * selected during construction of @p matrix_free, the appropriate component
   * can be selected by the optional arguments.
   *
   * @param matrix_free Data object that contains all data
   *
   * @param is_interior_face This selects which of the two cells of an
   * internal face the current evaluator will be based upon. The interior face
   * is the main face along which the normal vectors are oriented. The
   * exterior face coming from the other side provides the same normal vector
   * as the interior side, so if the outer normal vector to that side is
   * desired, it must be multiplied by -1.
   *
   * @param dof_no If matrix_free was set up with multiple DoFHandler
   * objects, this parameter selects to which DoFHandler/AffineConstraints pair
   * the given evaluator should be attached to.
   *
   * @param quad_no If matrix_free was set up with multiple Quadrature
   * objects, this parameter selects the appropriate number of the quadrature
   * formula.
   *
   * @param first_selected_component If the dof_handler selected by dof_no
   * uses an FESystem consisting of more than one base element, this parameter
   * selects the number of the base element in FESystem. Note that this does
   * not directly relate to the component of the respective element due to the
   * possibility for a multiplicity in the element.
   *
   * @param active_fe_index If matrix_free was set up with DoFHandler
   * objects with hp::FECollections, this parameter selects to which
   * DoFHandler/AffineConstraints pair the given evaluator should be attached
   * to.
   *
   * @param face_type In the case of a face, indicate its reference-cell type
   * (0 for line or quadrilateral 1 for triangle).
   *
   * @param active_quad_index If matrix_free was set up with hp::Collection
   * objects, this parameter selects the appropriate number of the quadrature
   * formula.
   */
  FEFaceEvaluation(
    const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
    const bool                                          is_interior_face = true,
    const unsigned int                                  dof_no           = 0,
    const unsigned int                                  quad_no          = 0,
    const unsigned int first_selected_component                          = 0,
    const unsigned int active_fe_index   = numbers::invalid_unsigned_int,
    const unsigned int active_quad_index = numbers::invalid_unsigned_int,
    const unsigned int face_type         = numbers::invalid_unsigned_int);

  /**
   * Constructor. Takes all data stored in MatrixFree for a given face range,
   * which allows to automatically identify the active_fe_index and
   * active_quad_index in case of a p-adaptive strategy.
   *
   * The rest of the arguments are the same as in the constructor above.
   */
  FEFaceEvaluation(
    const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
    const std::pair<unsigned int, unsigned int> &       range,
    const bool                                          is_interior_face = true,
    const unsigned int                                  dof_no           = 0,
    const unsigned int                                  quad_no          = 0,
    const unsigned int first_selected_component                          = 0);

  /**
   * Initializes the operation pointer to the current face. This method is the
   * default choice for face integration as the data stored in MappingInfo is
   * stored according to this numbering. Unlike the reinit functions taking a
   * cell iterator as argument below and the FEValues::reinit() methods, where
   * the information related to a particular cell is generated in the reinit
   * call, this function is very cheap since all data is pre-computed in
   * @p matrix_free, and only a few indices and pointers have to be set
   * appropriately.
   */
  void
  reinit(const unsigned int face_batch_number);

  /**
   * As opposed to the reinit() method from the base class, this reinit()
   * method initializes for a given number of cells and a face number. This
   * method is less efficient than the other reinit() method taking a
   * numbering of the faces because it needs to copy the data associated with
   * the faces to the cells in this call.
   */
  void
  reinit(const unsigned int cell_batch_number, const unsigned int face_number);

  /**
   * Check if face evaluation/integration is supported.
   */
  static bool
  fast_evaluation_supported(const unsigned int given_degree,
                            const unsigned int give_n_q_points_1d);

  /**
   * Evaluates the function values, the gradients, and the Laplacians of the
   * FE function given at the DoF values stored in the internal data field
   * `dof_values` (that is usually filled by the read_dof_values() method) at
   * the quadrature points on the unit cell.  The function arguments specify
   * which parts shall actually be computed. Needs to be called before the
   * functions get_value(), get_gradient() or get_normal_derivative() give
   * useful information (unless these values have been set manually by
   * accessing the internal data pointers).
   */
  void
  evaluate(const EvaluationFlags::EvaluationFlags evaluation_flag);

  /**
   * @deprecated Please use the evaluate() function with the EvaluationFlags argument.
   */
  DEAL_II_DEPRECATED_EARLY void
  evaluate(const bool evaluate_values, const bool evaluate_gradients);

  /**
   * Evaluates the function values, the gradients, and the Laplacians of the
   * FE function given at the DoF values in the input array `values_array` at
   * the quadrature points on the unit cell. If multiple components are
   * involved in the current FEEvaluation object, the sorting in values_array
   * is such that all degrees of freedom for the first component come first,
   * then all degrees of freedom for the second, and so on. The function
   * arguments specify which parts shall actually be computed. Needs to be
   * called before the functions get_value(), get_gradient(), or
   * get_normal_derivative() give useful information (unless these values have
   * been set manually).
   */
  void
  evaluate(const VectorizedArrayType *            values_array,
           const EvaluationFlags::EvaluationFlags evaluation_flag);

  /**
   * @deprecated Please use the evaluate() function with the EvaluationFlags argument.
   */
  DEAL_II_DEPRECATED_EARLY void
  evaluate(const VectorizedArrayType *values_array,
           const bool                 evaluate_values,
           const bool                 evaluate_gradients);

  /**
   * Reads from the input vector and evaluates the function values, the
   * gradients, and the Laplacians of the FE function at the quadrature points
   * on the unit cell. The function arguments specify which parts shall
   * actually be computed. Needs to be called before the functions
   * get_value(), get_gradient(), or get_normal_derivative() give useful
   * information.
   *
   * This call is equivalent to calling read_dof_values() followed by
   * evaluate(), but might internally use some additional optimizations.
   */
  template <typename VectorType>
  void
  gather_evaluate(const VectorType &                     input_vector,
                  const EvaluationFlags::EvaluationFlags evaluation_flag);

  /**
   * @deprecated Please use the gather_evaluate() function with the EvaluationFlags argument.
   */
  template <typename VectorType>
  DEAL_II_DEPRECATED_EARLY void
  gather_evaluate(const VectorType &input_vector,
                  const bool        evaluate_values,
                  const bool        evaluate_gradients);

  /**
   * This function takes the values and/or gradients that are stored on
   * quadrature points, tests them by all the basis functions/gradients on the
   * cell and performs the cell integration. The two function arguments
   * `integrate_val` and `integrate_grad` are used to enable/disable some of
   * values or gradients. The result is written into the internal data field
   * `dof_values` (that is usually written into the result vector by the
   * distribute_local_to_global() or set_dof_values() methods).
   */
  void
  integrate(const EvaluationFlags::EvaluationFlags integration_flag);

  /**
   * @deprecated Please use the integrate() function with the EvaluationFlags argument.
   */
  DEAL_II_DEPRECATED_EARLY void
  integrate(const bool integrate_values, const bool integrate_gradients);

  /**
   * This function takes the values and/or gradients that are stored on
   * quadrature points, tests them by all the basis functions/gradients on the
   * cell and performs the cell integration. The two function arguments
   * `integrate_val` and `integrate_grad` are used to enable/disable some of
   * values or gradients. As opposed to the other integrate() method, this
   * call stores the result of the testing in the given array `values_array`.
   */
  void
  integrate(const EvaluationFlags::EvaluationFlags integration_flag,
            VectorizedArrayType *                  values_array);

  /**
   * @deprecated Please use the integrate() function with the EvaluationFlags argument.
   */
  DEAL_II_DEPRECATED_EARLY void
  integrate(const bool           integrate_values,
            const bool           integrate_gradients,
            VectorizedArrayType *values_array);

  /**
   * This function takes the values and/or gradients that are stored on
   * quadrature points, tests them by all the basis functions/gradients on the
   * cell and performs the cell integration. The two function arguments
   * `integrate_val` and `integrate_grad` are used to enable/disable some of
   * values or gradients.
   *
   * This call is equivalent to calling integrate() followed by
   * distribute_local_to_global(), but might internally use some additional
   * optimizations.
   */
  template <typename VectorType>
  void
  integrate_scatter(const EvaluationFlags::EvaluationFlags integration_flag,
                    VectorType &                           output_vector);

  /**
   * @deprecated Please use the integrate_scatter() function with the EvaluationFlags argument.
   */
  template <typename VectorType>
  void
  integrate_scatter(const bool  integrate_values,
                    const bool  integrate_gradients,
                    VectorType &output_vector);

  /**
   * Returns the q-th quadrature point on the face in real coordinates stored
   * in MappingInfo.
   */
  Point<dim, VectorizedArrayType>
  quadrature_point(const unsigned int q_point) const;

  /**
   * The number of degrees of freedom of a single component on the cell for
   * the underlying evaluation object. Usually close to
   * static_dofs_per_component, but the number depends on the actual element
   * selected and is thus not static.
   */
  const unsigned int dofs_per_component;

  /**
   * The number of degrees of freedom on the cell accumulated over all
   * components in the current evaluation object. Usually close to
   * static_dofs_per_cell = static_dofs_per_component*n_components, but the
   * number depends on the actual element selected and is thus not static.
   */
  const unsigned int dofs_per_cell;

  /**
   * The number of quadrature points in use. If the number of quadrature
   * points in 1d is given as a template, this number is simply the
   * <tt>dim-1</tt>-th power of that value. If the element degree is set to -1
   * (dynamic selection of element degree), the static value of quadrature
   * points is inaccurate and this value must be used instead.
   */
  const unsigned int n_q_points;
};



namespace internal
{
  namespace MatrixFreeFunctions
  {
    // a helper function to compute the number of DoFs of a DGP element at
    // compile time, depending on the degree
    template <int dim, int degree>
    struct DGP_dofs_per_component
    {
      // this division is always without remainder
      static constexpr unsigned int value =
        (DGP_dofs_per_component<dim - 1, degree>::value * (degree + dim)) / dim;
    };

    // base specialization: 1d elements have 'degree+1' degrees of freedom
    template <int degree>
    struct DGP_dofs_per_component<1, degree>
    {
      static constexpr unsigned int value = degree + 1;
    };
  } // namespace MatrixFreeFunctions
} // namespace internal


/*----------------------- Inline functions ----------------------------------*/

#ifndef DOXYGEN


namespace internal
{
  // Extract all internal data pointers and indices in a single function that
  // get passed on to the constructor of FEEvaluationData, avoiding to look
  // things up multiple times
  template <bool is_face,
            int  dim,
            typename Number,
            typename VectorizedArrayType>
  inline typename FEEvaluationData<dim, VectorizedArrayType, is_face>::
    InitializationData
    extract_initialization_data(
      const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
      const unsigned int                                  dof_no,
      const unsigned int first_selected_component,
      const unsigned int quad_no,
      const unsigned int fe_degree,
      const unsigned int n_q_points,
      const unsigned int active_fe_index_given,
      const unsigned int active_quad_index_given,
      const unsigned int face_type)
  {
    typename FEEvaluationData<dim, VectorizedArrayType, is_face>::
      InitializationData init_data;

    init_data.dof_info = &matrix_free.get_dof_info(dof_no);
    init_data.mapping_data =
      &internal::MatrixFreeFunctions::
        MappingInfoCellsOrFaces<dim, Number, is_face, VectorizedArrayType>::get(
          matrix_free.get_mapping_info(), quad_no);

    init_data.active_fe_index =
      fe_degree != numbers::invalid_unsigned_int ?
        init_data.dof_info->fe_index_from_degree(first_selected_component,
                                                 fe_degree) :
        (active_fe_index_given != numbers::invalid_unsigned_int ?
           active_fe_index_given :
           0);
    init_data.active_quad_index =
      fe_degree == numbers::invalid_unsigned_int ?
        (active_quad_index_given != numbers::invalid_unsigned_int ?
           active_quad_index_given :
           std::min<unsigned int>(init_data.active_fe_index,
                                  init_data.mapping_data->descriptor.size() -
                                    1)) :
        init_data.mapping_data->quad_index_from_n_q_points(n_q_points);

    init_data.shape_info = &matrix_free.get_shape_info(
      dof_no,
      quad_no,
      init_data.dof_info->component_to_base_index[first_selected_component],
      init_data.active_fe_index,
      init_data.active_quad_index);
    init_data.descriptor =
      &init_data.mapping_data->descriptor
         [is_face ?
            (init_data.active_quad_index * std::max<unsigned int>(1, dim - 1) +
             (face_type == numbers::invalid_unsigned_int ? 0 : face_type)) :
            init_data.active_quad_index];

    return init_data;
  }
} // namespace internal



/*----------------------- FEEvaluationBase ----------------------------------*/

template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline FEEvaluationBase<dim,
                        n_components_,
                        Number,
                        is_face,
                        VectorizedArrayType>::
  FEEvaluationBase(const MatrixFree<dim, Number, VectorizedArrayType> &data_in,
                   const unsigned int                                  dof_no,
                   const unsigned int first_selected_component,
                   const unsigned int quad_no,
                   const unsigned int fe_degree,
                   const unsigned int n_q_points,
                   const bool         is_interior_face,
                   const unsigned int active_fe_index,
                   const unsigned int active_quad_index,
                   const unsigned int face_type)
  : FEEvaluationData<dim, VectorizedArrayType, is_face>(
      internal::extract_initialization_data<is_face>(data_in,
                                                     dof_no,
                                                     first_selected_component,
                                                     quad_no,
                                                     fe_degree,
                                                     n_q_points,
                                                     active_fe_index,
                                                     active_quad_index,
                                                     face_type),
      is_interior_face,
      quad_no,
      first_selected_component)
  , scratch_data_array(data_in.acquire_scratch_data())
  , matrix_free(&data_in)
{
  this->set_data_pointers(scratch_data_array, n_components_);
  Assert(
    this->dof_info->start_components.back() == 1 ||
      static_cast<int>(n_components_) <=
        static_cast<int>(
          this->dof_info->start_components
            [this->dof_info->component_to_base_index[first_selected_component] +
             1]) -
          first_selected_component,
    ExcMessage(
      "You tried to construct a vector-valued evaluator with " +
      std::to_string(n_components) +
      " components. However, "
      "the current base element has only " +
      std::to_string(
        this->dof_info->start_components
          [this->dof_info->component_to_base_index[first_selected_component] +
           1] -
        first_selected_component) +
      " components left when starting from local element index " +
      std::to_string(
        first_selected_component -
        this->dof_info->start_components
          [this->dof_info->component_to_base_index[first_selected_component]]) +
      " (global index " + std::to_string(first_selected_component) + ")"));

  // do not check for correct dimensions of data fields here, should be done
  // in derived classes
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline FEEvaluationBase<dim,
                        n_components_,
                        Number,
                        is_face,
                        VectorizedArrayType>::
  FEEvaluationBase(
    const Mapping<dim> &      mapping,
    const FiniteElement<dim> &fe,
    const Quadrature<1> &     quadrature,
    const UpdateFlags         update_flags,
    const unsigned int        first_selected_component,
    const FEEvaluationData<dim, VectorizedArrayType, is_face> *other)
  : FEEvaluationData<dim, VectorizedArrayType, is_face>(
      other != nullptr &&
          other->mapped_geometry->get_quadrature() == quadrature ?
        other->mapped_geometry :
        std::make_shared<internal::MatrixFreeFunctions::
                           MappingDataOnTheFly<dim, VectorizedArrayType>>(
          mapping,
          quadrature,
          update_flags),
      n_components_,
      first_selected_component)
  , scratch_data_array(new AlignedVector<VectorizedArrayType>())
  , matrix_free(nullptr)
{
  const unsigned int base_element_number =
    fe.component_to_base_index(first_selected_component).first;
  Assert(fe.element_multiplicity(base_element_number) == 1 ||
           fe.element_multiplicity(base_element_number) -
               first_selected_component >=
             n_components_,
         ExcMessage("The underlying element must at least contain as many "
                    "components as requested by this class"));
  (void)base_element_number;

  Assert(this->data == nullptr, ExcInternalError());
  this->data =
    new internal::MatrixFreeFunctions::ShapeInfo<VectorizedArrayType>(
      Quadrature<dim - is_face>(quadrature),
      fe,
      fe.component_to_base_index(first_selected_component).first);

  this->set_data_pointers(scratch_data_array, n_components_);
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline FEEvaluationBase<dim,
                        n_components_,
                        Number,
                        is_face,
                        VectorizedArrayType>::
  FEEvaluationBase(const FEEvaluationBase<dim,
                                          n_components_,
                                          Number,
                                          is_face,
                                          VectorizedArrayType> &other)
  : FEEvaluationData<dim, VectorizedArrayType, is_face>(other)
  , scratch_data_array(other.matrix_free == nullptr ?
                         new AlignedVector<VectorizedArrayType>() :
                         other.matrix_free->acquire_scratch_data())
  , matrix_free(other.matrix_free)
{
  if (other.matrix_free == nullptr)
    {
      Assert(other.mapped_geometry.get() != nullptr, ExcInternalError());
      this->data =
        new internal::MatrixFreeFunctions::ShapeInfo<VectorizedArrayType>(
          *other.data);

      // Create deep copy of mapped geometry for use in parallel
      this->mapped_geometry =
        std::make_shared<internal::MatrixFreeFunctions::
                           MappingDataOnTheFly<dim, VectorizedArrayType>>(
          other.mapped_geometry->get_fe_values().get_mapping(),
          other.mapped_geometry->get_quadrature(),
          other.mapped_geometry->get_fe_values().get_update_flags());
      this->mapping_data = &this->mapped_geometry->get_data_storage();
      this->cell         = 0;

      this->jacobian =
        this->mapped_geometry->get_data_storage().jacobians[0].begin();
      this->J_value =
        this->mapped_geometry->get_data_storage().JxW_values.begin();
    }

  this->set_data_pointers(scratch_data_array, n_components_);
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline FEEvaluationBase<dim,
                        n_components_,
                        Number,
                        is_face,
                        VectorizedArrayType> &
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
operator=(const FEEvaluationBase<dim,
                                 n_components_,
                                 Number,
                                 is_face,
                                 VectorizedArrayType> &other)
{
  // release old memory
  if (matrix_free == nullptr)
    {
      delete this->data;
      delete scratch_data_array;
    }
  else
    {
      matrix_free->release_scratch_data(scratch_data_array);
    }

  this->FEEvaluationData<dim, VectorizedArrayType, is_face>::operator=(other);

  matrix_free = other.matrix_free;

  if (other.matrix_free == nullptr)
    {
      Assert(other.mapped_geometry.get() != nullptr, ExcInternalError());
      this->data =
        new internal::MatrixFreeFunctions::ShapeInfo<VectorizedArrayType>(
          *other.data);
      scratch_data_array = new AlignedVector<VectorizedArrayType>();

      // Create deep copy of mapped geometry for use in parallel
      this->mapped_geometry =
        std::make_shared<internal::MatrixFreeFunctions::
                           MappingDataOnTheFly<dim, VectorizedArrayType>>(
          other.mapped_geometry->get_fe_values().get_mapping(),
          other.mapped_geometry->get_quadrature(),
          other.mapped_geometry->get_fe_values().get_update_flags());
      this->cell         = 0;
      this->mapping_data = &this->mapped_geometry->get_data_storage();
      this->jacobian =
        this->mapped_geometry->get_data_storage().jacobians[0].begin();
      this->J_value =
        this->mapped_geometry->get_data_storage().JxW_values.begin();
    }
  else
    {
      scratch_data_array = matrix_free->acquire_scratch_data();
    }

  this->set_data_pointers(scratch_data_array, n_components_);

  return *this;
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline FEEvaluationBase<dim,
                        n_components_,
                        Number,
                        is_face,
                        VectorizedArrayType>::~FEEvaluationBase()
{
  if (matrix_free != nullptr)
    {
      try
        {
          matrix_free->release_scratch_data(scratch_data_array);
        }
      catch (...)
        {}
    }
  else
    {
      delete scratch_data_array;
      delete this->data;
    }
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline const MatrixFree<dim, Number, VectorizedArrayType> &
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  get_matrix_free() const
{
  Assert(matrix_free != nullptr,
         ExcMessage(
           "FEEvaluation was not initialized with a MatrixFree object!"));
  return *matrix_free;
}


namespace internal
{
  template <int dim,
            int n_components,
            typename Number,
            bool is_face,
            typename VectorizedArrayType,
            typename VectorizedArrayType2,
            typename GlobalVectorType,
            typename FU>
  inline void
  process_cell_data(
    const FEEvaluationBase<dim,
                           n_components,
                           Number,
                           is_face,
                           VectorizedArrayType> &       fe_eval,
    const MatrixFree<dim, Number, VectorizedArrayType> *matrix_free,
    GlobalVectorType &                                  array,
    VectorizedArrayType2 &                              out,
    const FU &                                          fu)
  {
    (void)matrix_free;
    Assert(matrix_free != nullptr, ExcNotImplemented());
    AssertDimension(array.size(),
                    matrix_free->get_task_info().cell_partition_data.back());

    // 1) collect ids of cell
    const auto cells = fe_eval.get_cell_ids();

    // 2) actually gather values
    for (unsigned int i = 0; i < VectorizedArrayType::size(); ++i)
      if (cells[i] != numbers::invalid_unsigned_int)
        fu(out[i],
           array[cells[i] / VectorizedArrayType::size()]
                [cells[i] % VectorizedArrayType::size()]);
  }
} // namespace internal



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline VectorizedArrayType
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  read_cell_data(const AlignedVector<VectorizedArrayType> &array) const
{
  VectorizedArrayType out = Number(1.);
  internal::process_cell_data(
    *this, this->matrix_free, array, out, [](auto &local, const auto &global) {
      local = global;
    });
  return out;
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  set_cell_data(AlignedVector<VectorizedArrayType> &array,
                const VectorizedArrayType &         in) const
{
  internal::process_cell_data(
    *this, this->matrix_free, array, in, [](const auto &local, auto &global) {
      global = local;
    });
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
template <typename T>
inline std::array<T, VectorizedArrayType::size()>
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  read_cell_data(const AlignedVector<std::array<T, VectorizedArrayType::size()>>
                   &array) const
{
  std::array<T, VectorizedArrayType::size()> out;
  internal::process_cell_data(
    *this, this->matrix_free, array, out, [](auto &local, const auto &global) {
      local = global;
    });
  return out;
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
template <typename T>
inline void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  set_cell_data(
    AlignedVector<std::array<T, VectorizedArrayType::size()>> &array,
    const std::array<T, VectorizedArrayType::size()> &         in) const
{
  internal::process_cell_data(
    *this, this->matrix_free, array, in, [](const auto &local, auto &global) {
      global = local;
    });
}



namespace internal
{
  // allows to select between block vectors and non-block vectors, which
  // allows to use a unified interface for extracting blocks on block vectors
  // and doing nothing on usual vectors
  template <typename VectorType, bool>
  struct BlockVectorSelector
  {};

  template <typename VectorType>
  struct BlockVectorSelector<VectorType, true>
  {
    using BaseVectorType = typename VectorType::BlockType;

    static BaseVectorType *
    get_vector_component(VectorType &vec, const unsigned int component)
    {
      AssertIndexRange(component, vec.n_blocks());
      return &vec.block(component);
    }
  };

  template <typename VectorType>
  struct BlockVectorSelector<VectorType, false>
  {
    using BaseVectorType = VectorType;

    static BaseVectorType *
    get_vector_component(VectorType &vec, const unsigned int component)
    {
      // FEEvaluation allows to combine several vectors from a scalar
      // FiniteElement into a "vector-valued" FEEvaluation object with
      // multiple components. These components can be extracted with the other
      // get_vector_component functions. If we do not get a vector of vectors
      // (std::vector<VectorType>, std::vector<VectorType*>, BlockVector), we
      // must make sure that we do not duplicate the components in input
      // and/or duplicate the resulting integrals. In such a case, we should
      // only get the zeroth component in the vector contained set nullptr for
      // the others which allows us to catch unintended use in
      // read_write_operation.
      if (component == 0)
        return &vec;
      else
        return nullptr;
    }
  };

  template <typename VectorType>
  struct BlockVectorSelector<std::vector<VectorType>, false>
  {
    using BaseVectorType = VectorType;

    static BaseVectorType *
    get_vector_component(std::vector<VectorType> &vec,
                         const unsigned int       component)
    {
      AssertIndexRange(component, vec.size());
      return &vec[component];
    }
  };

  template <typename VectorType>
  struct BlockVectorSelector<std::vector<VectorType *>, false>
  {
    using BaseVectorType = VectorType;

    static BaseVectorType *
    get_vector_component(std::vector<VectorType *> &vec,
                         const unsigned int         component)
    {
      AssertIndexRange(component, vec.size());
      return vec[component];
    }
  };
} // namespace internal



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
template <typename VectorType, typename VectorOperation>
inline void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  read_write_operation(
    const VectorOperation &                        operation,
    const std::array<VectorType *, n_components_> &src,
    const std::array<
      const std::vector<ArrayView<const typename VectorType::value_type>> *,
      n_components_> &                              src_sm,
    const std::bitset<VectorizedArrayType::size()> &mask,
    const bool                                      apply_constraints) const
{
  // Case 1: No MatrixFree object given, simple case because we do not need to
  // process constraints and need not care about vectorization -> go to
  // separate function
  if (this->matrix_free == nullptr)
    {
      read_write_operation_global(operation, src);
      return;
    }

  Assert(this->dof_info != nullptr, ExcNotInitialized());
  Assert(this->matrix_free->indices_initialized() == true, ExcNotInitialized());
  if (this->n_fe_components == 1)
    for (unsigned int comp = 0; comp < n_components; ++comp)
      {
        Assert(src[comp] != nullptr,
               ExcMessage("The finite element underlying this FEEvaluation "
                          "object is scalar, but you requested " +
                          std::to_string(n_components) +
                          " components via the template argument in "
                          "FEEvaluation. In that case, you must pass an "
                          "std::vector<VectorType> or a BlockVector to " +
                          "read_dof_values and distribute_local_to_global."));
        internal::check_vector_compatibility(*src[comp], *this->dof_info);
      }
  else
    {
      internal::check_vector_compatibility(*src[0], *this->dof_info);
    }

  // Case 2: contiguous indices which use reduced storage of indices and can
  // use vectorized load/store operations -> go to separate function
  AssertIndexRange(
    this->cell,
    this->dof_info->index_storage_variants[this->dof_access_index].size());
  if (this->dof_info->index_storage_variants
        [is_face ? this->dof_access_index :
                   internal::MatrixFreeFunctions::DoFInfo::dof_access_cell]
        [this->cell] >=
      internal::MatrixFreeFunctions::DoFInfo::IndexStorageVariants::contiguous)
    {
      read_write_operation_contiguous(operation, src, src_sm, mask);
      return;
    }

  // Case 3: standard operation with one index per degree of freedom -> go on
  // here
  constexpr unsigned int n_lanes = VectorizedArrayType::size();
  Assert(mask.count() == n_lanes,
         ExcNotImplemented("Masking currently not implemented for "
                           "non-contiguous DoF storage"));

  unsigned int n_vectorization_actual =
    this->dof_info
      ->n_vectorization_lanes_filled[this->dof_access_index][this->cell];

  bool has_hn_constraints = false;

  if (is_face == false)
    {
      for (unsigned int v = 0; v < n_vectorization_actual; ++v)
        if (this->dof_info->hanging_node_constraint_masks.size() > 0 &&
            this->dof_info->hanging_node_constraint_masks
                [(this->cell * n_lanes + v) * this->n_fe_components +
                 this->first_selected_component] !=
              internal::MatrixFreeFunctions::ConstraintKinds::unconstrained)
          has_hn_constraints = true;
    }

  std::integral_constant<bool,
                         internal::is_vectorizable<VectorType, Number>::value>
    vector_selector;

  const std::size_t dofs_per_component = this->data->dofs_per_component_on_cell;
  std::array<VectorizedArrayType *, n_components> values_dofs;
  for (unsigned int c = 0; c < n_components; ++c)
    values_dofs[c] = const_cast<VectorizedArrayType *>(this->values_dofs) +
                     c * dofs_per_component;

  if (this->dof_info->index_storage_variants
          [is_face ? this->dof_access_index :
                     internal::MatrixFreeFunctions::DoFInfo::dof_access_cell]
          [this->cell] == internal::MatrixFreeFunctions::DoFInfo::
                            IndexStorageVariants::interleaved &&
      (has_hn_constraints == false))
    {
      const unsigned int *dof_indices =
        this->dof_info->dof_indices_interleaved.data() +
        this->dof_info->row_starts[this->cell * this->n_fe_components * n_lanes]
          .first +
        this->dof_info
            ->component_dof_indices_offset[this->active_fe_index]
                                          [this->first_selected_component] *
          n_lanes;
      if (n_components == 1 || this->n_fe_components == 1)
        for (unsigned int i = 0; i < dofs_per_component;
             ++i, dof_indices += n_lanes)
          for (unsigned int comp = 0; comp < n_components; ++comp)
            operation.process_dof_gather(dof_indices,
                                         *src[comp],
                                         0,
                                         values_dofs[comp][i],
                                         vector_selector);
      else
        for (unsigned int comp = 0; comp < n_components; ++comp)
          for (unsigned int i = 0; i < dofs_per_component;
               ++i, dof_indices += n_lanes)
            operation.process_dof_gather(
              dof_indices, *src[0], 0, values_dofs[comp][i], vector_selector);
      return;
    }

  const unsigned int *dof_indices[n_lanes];

  // Assign the appropriate cell ids for face/cell case and get the pointers
  // to the dof indices of the cells on all lanes
  unsigned int        cells_copied[n_lanes];
  const unsigned int *cells;
  bool                has_constraints = false;
  const unsigned int  n_components_read =
    this->n_fe_components > 1 ? n_components : 1;

  if (is_face)
    {
      if (this->dof_access_index ==
          internal::MatrixFreeFunctions::DoFInfo::dof_access_cell)
        for (unsigned int v = 0; v < n_vectorization_actual; ++v)
          cells_copied[v] = this->cell * VectorizedArrayType::size() + v;
      cells =
        this->dof_access_index ==
            internal::MatrixFreeFunctions::DoFInfo::dof_access_cell ?
          &cells_copied[0] :
          (this->is_interior_face ?
             &this->matrix_free->get_face_info(this->cell).cells_interior[0] :
             &this->matrix_free->get_face_info(this->cell).cells_exterior[0]);
      for (unsigned int v = 0; v < n_vectorization_actual; ++v)
        {
          Assert(cells[v] < this->dof_info->row_starts.size() - 1,
                 ExcInternalError());
          const std::pair<unsigned int, unsigned int> *my_index_start =
            &this->dof_info->row_starts[cells[v] * this->n_fe_components +
                                        this->first_selected_component];

          // check whether any of the SIMD lanes has constraints, i.e., the
          // constraint indicator which is the second entry of row_starts
          // increments on this cell
          if (my_index_start[n_components_read].second !=
              my_index_start[0].second)
            has_constraints = true;

          dof_indices[v] =
            this->dof_info->dof_indices.data() + my_index_start[0].first;
        }
      for (unsigned int v = n_vectorization_actual; v < n_lanes; ++v)
        dof_indices[v] = nullptr;
    }
  else
    {
      AssertIndexRange((this->cell + 1) * n_lanes * this->n_fe_components,
                       this->dof_info->row_starts.size());
      for (unsigned int v = 0; v < n_vectorization_actual; ++v)
        {
          const std::pair<unsigned int, unsigned int> *my_index_start =
            &this->dof_info
               ->row_starts[(this->cell * n_lanes + v) * this->n_fe_components +
                            this->first_selected_component];
          if (my_index_start[n_components_read].second !=
              my_index_start[0].second)
            has_constraints = true;

          if (this->dof_info->hanging_node_constraint_masks.size() > 0 &&
              this->dof_info->hanging_node_constraint_masks
                  [(this->cell * n_lanes + v) * this->n_fe_components +
                   this->first_selected_component] !=
                internal::MatrixFreeFunctions::ConstraintKinds::unconstrained)
            has_hn_constraints = true;

          Assert(my_index_start[n_components_read].first ==
                     my_index_start[0].first ||
                   my_index_start[0].first < this->dof_info->dof_indices.size(),
                 ExcIndexRange(0,
                               my_index_start[0].first,
                               this->dof_info->dof_indices.size()));
          dof_indices[v] =
            this->dof_info->dof_indices.data() + my_index_start[0].first;
        }
      for (unsigned int v = n_vectorization_actual; v < n_lanes; ++v)
        dof_indices[v] = nullptr;
    }

  // Case where we have no constraints throughout the whole cell: Can go
  // through the list of DoFs directly
  if (!has_constraints && apply_constraints)
    {
      if (n_vectorization_actual < n_lanes)
        for (unsigned int comp = 0; comp < n_components; ++comp)
          for (unsigned int i = 0; i < dofs_per_component; ++i)
            operation.process_empty(values_dofs[comp][i]);
      if (n_components == 1 || this->n_fe_components == 1)
        {
          for (unsigned int v = 0; v < n_vectorization_actual; ++v)
            for (unsigned int i = 0; i < dofs_per_component; ++i)
              for (unsigned int comp = 0; comp < n_components; ++comp)
                operation.process_dof(dof_indices[v][i],
                                      *src[comp],
                                      values_dofs[comp][i][v]);
        }
      else
        {
          for (unsigned int comp = 0; comp < n_components; ++comp)
            for (unsigned int v = 0; v < n_vectorization_actual; ++v)
              for (unsigned int i = 0; i < dofs_per_component; ++i)
                operation.process_dof(
                  dof_indices[v][comp * dofs_per_component + i],
                  *src[0],
                  values_dofs[comp][i][v]);
        }
      return;
    }

  // In the case where there are some constraints to be resolved, loop over
  // all vector components that are filled and then over local dofs. ind_local
  // holds local number on cell, index iterates over the elements of
  // index_local_to_global and dof_indices points to the global indices stored
  // in index_local_to_global
  if (n_vectorization_actual < n_lanes)
    for (unsigned int comp = 0; comp < n_components; ++comp)
      for (unsigned int i = 0; i < dofs_per_component; ++i)
        operation.process_empty(values_dofs[comp][i]);
  for (unsigned int v = 0; v < n_vectorization_actual; ++v)
    {
      const unsigned int cell_index =
        is_face ? cells[v] : this->cell * n_lanes + v;
      const unsigned int cell_dof_index =
        cell_index * this->n_fe_components + this->first_selected_component;
      const unsigned int n_components_read =
        this->n_fe_components > 1 ? n_components : 1;
      unsigned int index_indicators =
        this->dof_info->row_starts[cell_dof_index].second;
      unsigned int next_index_indicators =
        this->dof_info->row_starts[cell_dof_index + 1].second;

      // For read_dof_values_plain, redirect the dof_indices field to the
      // unconstrained indices
      if (apply_constraints == false &&
          (this->dof_info->row_starts[cell_dof_index].second !=
             this->dof_info->row_starts[cell_dof_index + n_components_read]
               .second ||
           (this->dof_info->hanging_node_constraint_masks.size() > 0 &&
            this->dof_info->hanging_node_constraint_masks[cell_dof_index] !=
              internal::MatrixFreeFunctions::ConstraintKinds::unconstrained)))
        {
          Assert(this->dof_info->row_starts_plain_indices[cell_index] !=
                   numbers::invalid_unsigned_int,
                 ExcNotInitialized());
          dof_indices[v] =
            this->dof_info->plain_dof_indices.data() +
            this->dof_info
              ->component_dof_indices_offset[this->active_fe_index]
                                            [this->first_selected_component] +
            this->dof_info->row_starts_plain_indices[cell_index];
          next_index_indicators = index_indicators;
        }

      if (n_components == 1 || this->n_fe_components == 1)
        {
          unsigned int ind_local = 0;
          for (; index_indicators != next_index_indicators; ++index_indicators)
            {
              const std::pair<unsigned short, unsigned short> indicator =
                this->dof_info->constraint_indicator[index_indicators];
              // run through values up to next constraint
              for (unsigned int j = 0; j < indicator.first; ++j)
                for (unsigned int comp = 0; comp < n_components; ++comp)
                  operation.process_dof(dof_indices[v][j],
                                        *src[comp],
                                        values_dofs[comp][ind_local + j][v]);

              ind_local += indicator.first;
              dof_indices[v] += indicator.first;

              // constrained case: build the local value as a linear
              // combination of the global value according to constraints
              Number value[n_components];
              for (unsigned int comp = 0; comp < n_components; ++comp)
                operation.pre_constraints(values_dofs[comp][ind_local][v],
                                          value[comp]);

              const Number *data_val =
                this->matrix_free->constraint_pool_begin(indicator.second);
              const Number *end_pool =
                this->matrix_free->constraint_pool_end(indicator.second);
              for (; data_val != end_pool; ++data_val, ++dof_indices[v])
                for (unsigned int comp = 0; comp < n_components; ++comp)
                  operation.process_constraint(*dof_indices[v],
                                               *data_val,
                                               *src[comp],
                                               value[comp]);

              for (unsigned int comp = 0; comp < n_components; ++comp)
                operation.post_constraints(value[comp],
                                           values_dofs[comp][ind_local][v]);
              ind_local++;
            }

          AssertIndexRange(ind_local, dofs_per_component + 1);

          for (; ind_local < dofs_per_component; ++dof_indices[v], ++ind_local)
            for (unsigned int comp = 0; comp < n_components; ++comp)
              operation.process_dof(*dof_indices[v],
                                    *src[comp],
                                    values_dofs[comp][ind_local][v]);
        }
      else
        {
          // case with vector-valued finite elements where all components are
          // included in one single vector. Assumption: first come all entries
          // to the first component, then all entries to the second one, and
          // so on. This is ensured by the way MatrixFree reads out the
          // indices.
          for (unsigned int comp = 0; comp < n_components; ++comp)
            {
              unsigned int ind_local = 0;

              // check whether there is any constraint on the current cell
              for (; index_indicators != next_index_indicators;
                   ++index_indicators)
                {
                  const std::pair<unsigned short, unsigned short> indicator =
                    this->dof_info->constraint_indicator[index_indicators];

                  // run through values up to next constraint
                  for (unsigned int j = 0; j < indicator.first; ++j)
                    operation.process_dof(dof_indices[v][j],
                                          *src[0],
                                          values_dofs[comp][ind_local + j][v]);
                  ind_local += indicator.first;
                  dof_indices[v] += indicator.first;

                  // constrained case: build the local value as a linear
                  // combination of the global value according to constraints
                  Number value;
                  operation.pre_constraints(values_dofs[comp][ind_local][v],
                                            value);

                  const Number *data_val =
                    this->matrix_free->constraint_pool_begin(indicator.second);
                  const Number *end_pool =
                    this->matrix_free->constraint_pool_end(indicator.second);

                  for (; data_val != end_pool; ++data_val, ++dof_indices[v])
                    operation.process_constraint(*dof_indices[v],
                                                 *data_val,
                                                 *src[0],
                                                 value);

                  operation.post_constraints(value,
                                             values_dofs[comp][ind_local][v]);
                  ind_local++;
                }

              AssertIndexRange(ind_local, dofs_per_component + 1);

              // get the dof values past the last constraint
              for (; ind_local < dofs_per_component;
                   ++dof_indices[v], ++ind_local)
                {
                  AssertIndexRange(*dof_indices[v], src[0]->size());
                  operation.process_dof(*dof_indices[v],
                                        *src[0],
                                        values_dofs[comp][ind_local][v]);
                }

              if (apply_constraints == true && comp + 1 < n_components)
                next_index_indicators =
                  this->dof_info->row_starts[cell_dof_index + comp + 2].second;
            }
        }
    }
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
template <typename VectorType, typename VectorOperation>
inline void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  read_write_operation_global(
    const VectorOperation &                        operation,
    const std::array<VectorType *, n_components_> &src) const
{
  Assert(!local_dof_indices.empty(), ExcNotInitialized());

  const std::size_t dofs_per_component = this->data->dofs_per_component_on_cell;
  unsigned int      index = this->first_selected_component * dofs_per_component;
  for (unsigned int comp = 0; comp < n_components; ++comp)
    {
      for (unsigned int i = 0; i < dofs_per_component; ++i, ++index)
        {
          operation.process_empty(
            this->values_dofs[comp * dofs_per_component + i]);
          operation.process_dof_global(
            local_dof_indices[this->data->lexicographic_numbering[index]],
            *src[0],
            this->values_dofs[comp * dofs_per_component + i][0]);
        }
    }
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
template <typename VectorType, typename VectorOperation>
inline void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  read_write_operation_contiguous(
    const VectorOperation &                        operation,
    const std::array<VectorType *, n_components_> &src,
    const std::array<
      const std::vector<ArrayView<const typename VectorType::value_type>> *,
      n_components_> &                              vectors_sm,
    const std::bitset<VectorizedArrayType::size()> &mask) const
{
  // This functions processes the functions read_dof_values,
  // distribute_local_to_global, and set_dof_values with the same code for
  // contiguous cell indices (DG case). The distinction between these three
  // cases is made by the input VectorOperation that either reads values from
  // a vector and puts the data into the local data field or write local data
  // into the vector. Certain operations are no-ops for the given use case.

  std::integral_constant<bool,
                         internal::is_vectorizable<VectorType, Number>::value>
                                                               vector_selector;
  const internal::MatrixFreeFunctions::DoFInfo::DoFAccessIndex ind =
    is_face ? this->dof_access_index :
              internal::MatrixFreeFunctions::DoFInfo::dof_access_cell;
  const unsigned int n_lanes = mask.count();

  const std::vector<unsigned int> &dof_indices_cont =
    this->dof_info->dof_indices_contiguous[ind];

  const std::size_t dofs_per_component = this->data->dofs_per_component_on_cell;
  std::array<VectorizedArrayType *, n_components> values_dofs;
  for (unsigned int c = 0; c < n_components; ++c)
    values_dofs[c] = const_cast<VectorizedArrayType *>(this->values_dofs) +
                     c * dofs_per_component;

  // Simple case: We have contiguous storage, so we can simply copy out the
  // data
  if ((this->dof_info->index_storage_variants[ind][this->cell] ==
         internal::MatrixFreeFunctions::DoFInfo::IndexStorageVariants::
           interleaved_contiguous &&
       n_lanes == VectorizedArrayType::size()) &&
      !(is_face &&
        this->dof_access_index ==
          internal::MatrixFreeFunctions::DoFInfo::dof_access_cell &&
        this->is_interior_face == false) &&
      !(!is_face && !this->is_interior_face))
    {
      const unsigned int dof_index =
        dof_indices_cont[this->cell * VectorizedArrayType::size()] +
        this->dof_info
            ->component_dof_indices_offset[this->active_fe_index]
                                          [this->first_selected_component] *
          VectorizedArrayType::size();
      if (n_components == 1 || this->n_fe_components == 1)
        for (unsigned int comp = 0; comp < n_components; ++comp)
          operation.process_dofs_vectorized(dofs_per_component,
                                            dof_index,
                                            *src[comp],
                                            values_dofs[comp],
                                            vector_selector);
      else
        operation.process_dofs_vectorized(dofs_per_component * n_components,
                                          dof_index,
                                          *src[0],
                                          values_dofs[0],
                                          vector_selector);
      return;
    }

  const std::array<unsigned int, VectorizedArrayType::size()> &cells =
    this->get_cell_or_face_ids();

  // More general case: Must go through the components one by one and apply
  // some transformations
  const unsigned int n_filled_lanes =
    this->dof_info->n_vectorization_lanes_filled[ind][this->cell];

  const bool is_ecl =
    (this->dof_access_index ==
       internal::MatrixFreeFunctions::DoFInfo::dof_access_cell &&
     this->is_interior_face == false) ||
    (!is_face && !this->is_interior_face);

  if (vectors_sm[0] != nullptr)
    {
      const auto compute_vector_ptrs = [&](const unsigned int comp) {
        std::array<typename VectorType::value_type *,
                   VectorizedArrayType::size()>
          vector_ptrs = {};

        for (unsigned int v = 0; v < n_filled_lanes; ++v)
          {
            if (mask[v] == false)
              {
                vector_ptrs[v] = nullptr;
                continue;
              }

            Assert(cells[v] != numbers::invalid_unsigned_int,
                   ExcNotImplemented());
            Assert(ind < this->dof_info->dof_indices_contiguous_sm.size(),
                   ExcIndexRange(
                     ind, 0, this->dof_info->dof_indices_contiguous_sm.size()));
            Assert(cells[v] <
                     this->dof_info->dof_indices_contiguous_sm[ind].size(),
                   ExcIndexRange(
                     cells[v],
                     0,
                     this->dof_info->dof_indices_contiguous_sm[ind].size()));

            const auto &temp =
              this->dof_info->dof_indices_contiguous_sm[ind][cells[v]];

            if (temp.first != numbers::invalid_unsigned_int)
              vector_ptrs[v] = const_cast<typename VectorType::value_type *>(
                vectors_sm[comp]->operator[](temp.first).data() + temp.second +
                this->dof_info->component_dof_indices_offset
                  [this->active_fe_index][this->first_selected_component]);
            else
              vector_ptrs[v] = nullptr;
          }
        for (unsigned int v = n_filled_lanes; v < VectorizedArrayType::size();
             ++v)
          vector_ptrs[v] = nullptr;

        return vector_ptrs;
      };

      if (n_filled_lanes == VectorizedArrayType::size() &&
          n_lanes == VectorizedArrayType::size() && !is_ecl)
        {
          if (n_components == 1 || this->n_fe_components == 1)
            {
              for (unsigned int comp = 0; comp < n_components; ++comp)
                {
                  auto vector_ptrs = compute_vector_ptrs(comp);
                  operation.process_dofs_vectorized_transpose(
                    dofs_per_component,
                    vector_ptrs,
                    values_dofs[comp],
                    vector_selector);
                }
            }
          else
            {
              auto vector_ptrs = compute_vector_ptrs(0);
              operation.process_dofs_vectorized_transpose(dofs_per_component *
                                                            n_components,
                                                          vector_ptrs,
                                                          &values_dofs[0][0],
                                                          vector_selector);
            }
        }
      else
        for (unsigned int comp = 0; comp < n_components; ++comp)
          {
            auto vector_ptrs = compute_vector_ptrs(
              (n_components == 1 || this->n_fe_components == 1) ? comp : 0);

            for (unsigned int i = 0; i < dofs_per_component; ++i)
              operation.process_empty(values_dofs[comp][i]);

            if (n_components == 1 || this->n_fe_components == 1)
              {
                for (unsigned int v = 0; v < n_filled_lanes; ++v)
                  if (mask[v] == true)
                    for (unsigned int i = 0; i < dofs_per_component; ++i)
                      operation.process_dof(vector_ptrs[v][i],
                                            values_dofs[comp][i][v]);
              }
            else
              {
                for (unsigned int v = 0; v < n_filled_lanes; ++v)
                  if (mask[v] == true)
                    for (unsigned int i = 0; i < dofs_per_component; ++i)
                      operation.process_dof(
                        vector_ptrs[v][i + comp * dofs_per_component],
                        values_dofs[comp][i][v]);
              }
          }
      return;
    }

  unsigned int dof_indices[VectorizedArrayType::size()];

  for (unsigned int v = 0; v < n_filled_lanes; ++v)
    {
      Assert(cells[v] != numbers::invalid_unsigned_int, ExcNotImplemented());
      dof_indices[v] =
        dof_indices_cont[cells[v]] +
        this->dof_info
            ->component_dof_indices_offset[this->active_fe_index]
                                          [this->first_selected_component] *
          this->dof_info->dof_indices_interleave_strides[ind][cells[v]];
    }

  for (unsigned int v = n_filled_lanes; v < VectorizedArrayType::size(); ++v)
    dof_indices[v] = numbers::invalid_unsigned_int;

  // In the case with contiguous cell indices, we know that there are no
  // constraints and that the indices within each element are contiguous
  if (n_filled_lanes == VectorizedArrayType::size() &&
      n_lanes == VectorizedArrayType::size() && !is_ecl)
    {
      if (this->dof_info->index_storage_variants[ind][this->cell] ==
          internal::MatrixFreeFunctions::DoFInfo::IndexStorageVariants::
            contiguous)
        {
          if (n_components == 1 || this->n_fe_components == 1)
            for (unsigned int comp = 0; comp < n_components; ++comp)
              operation.process_dofs_vectorized_transpose(dofs_per_component,
                                                          dof_indices,
                                                          *src[comp],
                                                          values_dofs[comp],
                                                          vector_selector);
          else
            operation.process_dofs_vectorized_transpose(dofs_per_component *
                                                          n_components,
                                                        dof_indices,
                                                        *src[0],
                                                        &values_dofs[0][0],
                                                        vector_selector);
        }
      else if (this->dof_info->index_storage_variants[ind][this->cell] ==
               internal::MatrixFreeFunctions::DoFInfo::IndexStorageVariants::
                 interleaved_contiguous_strided)
        {
          if (n_components == 1 || this->n_fe_components == 1)
            for (unsigned int i = 0; i < dofs_per_component; ++i)
              {
                for (unsigned int comp = 0; comp < n_components; ++comp)
                  operation.process_dof_gather(dof_indices,
                                               *src[comp],
                                               i * VectorizedArrayType::size(),
                                               values_dofs[comp][i],
                                               vector_selector);
              }
          else
            for (unsigned int comp = 0; comp < n_components; ++comp)
              for (unsigned int i = 0; i < dofs_per_component; ++i)
                {
                  operation.process_dof_gather(dof_indices,
                                               *src[0],
                                               (comp * dofs_per_component + i) *
                                                 VectorizedArrayType::size(),
                                               values_dofs[comp][i],
                                               vector_selector);
                }
        }
      else
        {
          Assert(this->dof_info->index_storage_variants[ind][this->cell] ==
                   internal::MatrixFreeFunctions::DoFInfo::
                     IndexStorageVariants::interleaved_contiguous_mixed_strides,
                 ExcNotImplemented());
          const unsigned int *offsets =
            &this->dof_info->dof_indices_interleave_strides
               [ind][VectorizedArrayType::size() * this->cell];
          if (n_components == 1 || this->n_fe_components == 1)
            for (unsigned int i = 0; i < dofs_per_component; ++i)
              {
                for (unsigned int comp = 0; comp < n_components; ++comp)
                  operation.process_dof_gather(dof_indices,
                                               *src[comp],
                                               0,
                                               values_dofs[comp][i],
                                               vector_selector);
                DEAL_II_OPENMP_SIMD_PRAGMA
                for (unsigned int v = 0; v < VectorizedArrayType::size(); ++v)
                  dof_indices[v] += offsets[v];
              }
          else
            for (unsigned int comp = 0; comp < n_components; ++comp)
              for (unsigned int i = 0; i < dofs_per_component; ++i)
                {
                  operation.process_dof_gather(dof_indices,
                                               *src[0],
                                               0,
                                               values_dofs[comp][i],
                                               vector_selector);
                  DEAL_II_OPENMP_SIMD_PRAGMA
                  for (unsigned int v = 0; v < VectorizedArrayType::size(); ++v)
                    dof_indices[v] += offsets[v];
                }
        }
    }
  else
    for (unsigned int comp = 0; comp < n_components; ++comp)
      {
        for (unsigned int i = 0; i < dofs_per_component; ++i)
          operation.process_empty(values_dofs[comp][i]);
        if (this->dof_info->index_storage_variants[ind][this->cell] ==
            internal::MatrixFreeFunctions::DoFInfo::IndexStorageVariants::
              contiguous)
          {
            if (n_components == 1 || this->n_fe_components == 1)
              {
                for (unsigned int v = 0; v < n_filled_lanes; ++v)
                  if (mask[v] == true)
                    for (unsigned int i = 0; i < dofs_per_component; ++i)
                      operation.process_dof(dof_indices[v] + i,
                                            *src[comp],
                                            values_dofs[comp][i][v]);
              }
            else
              {
                for (unsigned int v = 0; v < n_filled_lanes; ++v)
                  if (mask[v] == true)
                    for (unsigned int i = 0; i < dofs_per_component; ++i)
                      operation.process_dof(dof_indices[v] + i +
                                              comp * dofs_per_component,
                                            *src[0],
                                            values_dofs[comp][i][v]);
              }
          }
        else
          {
            const unsigned int *offsets =
              &this->dof_info->dof_indices_interleave_strides
                 [ind][VectorizedArrayType::size() * this->cell];
            for (unsigned int v = 0; v < n_filled_lanes; ++v)
              AssertIndexRange(offsets[v], VectorizedArrayType::size() + 1);
            if (n_components == 1 || this->n_fe_components == 1)
              for (unsigned int v = 0; v < n_filled_lanes; ++v)
                {
                  if (mask[v] == true)
                    for (unsigned int i = 0; i < dofs_per_component; ++i)
                      operation.process_dof(dof_indices[v] + i * offsets[v],
                                            *src[comp],
                                            values_dofs[comp][i][v]);
                }
            else
              {
                for (unsigned int v = 0; v < n_filled_lanes; ++v)
                  if (mask[v] == true)
                    for (unsigned int i = 0; i < dofs_per_component; ++i)
                      operation.process_dof(dof_indices[v] +
                                              (i + comp * dofs_per_component) *
                                                offsets[v],
                                            *src[0],
                                            values_dofs[comp][i][v]);
              }
          }
      }
}

namespace internal
{
  template <typename Number,
            typename VectorType,
            typename std::enable_if<!IsBlockVector<VectorType>::value,
                                    VectorType>::type * = nullptr>
  decltype(std::declval<VectorType>().begin())
  get_beginning(VectorType &vec)
  {
    return vec.begin();
  }

  template <typename Number,
            typename VectorType,
            typename std::enable_if<IsBlockVector<VectorType>::value,
                                    VectorType>::type * = nullptr>
  typename VectorType::value_type *
  get_beginning(VectorType &)
  {
    return nullptr;
  }

  template <typename VectorType,
            typename std::enable_if<has_shared_vector_data<VectorType>::value,
                                    VectorType>::type * = nullptr>
  const std::vector<ArrayView<const typename VectorType::value_type>> *
  get_shared_vector_data(VectorType &       vec,
                         const bool         is_valid_mode_for_sm,
                         const unsigned int active_fe_index,
                         const internal::MatrixFreeFunctions::DoFInfo *dof_info)
  {
    // note: no hp is supported
    if (is_valid_mode_for_sm &&
        dof_info->dof_indices_contiguous_sm[0 /*any index (<3) should work*/]
            .size() > 0 &&
        active_fe_index == 0)
      return &vec.shared_vector_data();
    else
      return nullptr;
  }

  template <typename VectorType,
            typename std::enable_if<!has_shared_vector_data<VectorType>::value,
                                    VectorType>::type * = nullptr>
  const std::vector<ArrayView<const typename VectorType::value_type>> *
  get_shared_vector_data(VectorType &,
                         const bool,
                         const unsigned int,
                         const internal::MatrixFreeFunctions::DoFInfo *)
  {
    return nullptr;
  }

  template <int n_components, typename VectorType>
  std::pair<
    std::array<typename internal::BlockVectorSelector<
                 typename std::remove_const<VectorType>::type,
                 IsBlockVector<typename std::remove_const<VectorType>::type>::
                   value>::BaseVectorType *,
               n_components>,
    std::array<
      const std::vector<ArrayView<const typename internal::BlockVectorSelector<
        typename std::remove_const<VectorType>::type,
        IsBlockVector<typename std::remove_const<VectorType>::type>::value>::
                                    BaseVectorType::value_type>> *,
      n_components>>
  get_vector_data(VectorType &       src,
                  const unsigned int first_index,
                  const bool         is_valid_mode_for_sm,
                  const unsigned int active_fe_index,
                  const internal::MatrixFreeFunctions::DoFInfo *dof_info)
  {
    // select between block vectors and non-block vectors. Note that the number
    // of components is checked in the internal data
    std::pair<
      std::array<typename internal::BlockVectorSelector<
                   typename std::remove_const<VectorType>::type,
                   IsBlockVector<typename std::remove_const<VectorType>::type>::
                     value>::BaseVectorType *,
                 n_components>,
      std::array<
        const std::vector<
          ArrayView<const typename internal::BlockVectorSelector<
            typename std::remove_const<VectorType>::type,
            IsBlockVector<typename std::remove_const<VectorType>::type>::
              value>::BaseVectorType::value_type>> *,
        n_components>>
      src_data;

    for (unsigned int d = 0; d < n_components; ++d)
      src_data.first[d] = internal::BlockVectorSelector<
        typename std::remove_const<VectorType>::type,
        IsBlockVector<typename std::remove_const<VectorType>::type>::value>::
        get_vector_component(
          const_cast<typename std::remove_const<VectorType>::type &>(src),
          d + first_index);

    for (unsigned int d = 0; d < n_components; ++d)
      src_data.second[d] = get_shared_vector_data(*src_data.first[d],
                                                  is_valid_mode_for_sm,
                                                  active_fe_index,
                                                  dof_info);

    return src_data;
  }
} // namespace internal



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  apply_hanging_node_constraints(const bool transpose) const
{
  if (this->dof_info == nullptr ||
      this->dof_info->hanging_node_constraint_masks.size() == 0)
    return; // nothing to do with faces

  const unsigned int n_vectorization_actual =
    this->dof_info
      ->n_vectorization_lanes_filled[this->dof_access_index][this->cell];

  constexpr unsigned int n_lanes = VectorizedArrayType::size();
  std::array<internal::MatrixFreeFunctions::ConstraintKinds, n_lanes>
    constraint_mask;

  bool hn_available = false;

  unsigned int        cells_copied[n_lanes];
  const unsigned int *cells;

  if (is_face)
    {
      if (this->dof_access_index ==
          internal::MatrixFreeFunctions::DoFInfo::dof_access_cell)
        for (unsigned int v = 0; v < n_vectorization_actual; ++v)
          cells_copied[v] = this->cell * VectorizedArrayType::size() + v;
      cells =
        this->dof_access_index ==
            internal::MatrixFreeFunctions::DoFInfo::dof_access_cell ?
          &cells_copied[0] :
          (this->is_interior_face ?
             &this->matrix_free->get_face_info(this->cell).cells_interior[0] :
             &this->matrix_free->get_face_info(this->cell).cells_exterior[0]);
    }

  for (unsigned int v = 0; v < n_vectorization_actual; ++v)
    {
      const unsigned int cell_index =
        is_face ? cells[v] : this->cell * n_lanes + v;
      const unsigned int cell_dof_index =
        cell_index * this->n_fe_components + this->first_selected_component;

      const auto mask =
        this->dof_info->hanging_node_constraint_masks[cell_dof_index];
      constraint_mask[v] = mask;

      hn_available |=
        (mask != internal::MatrixFreeFunctions::ConstraintKinds::unconstrained);
    }

  if (hn_available == false)
    return; // no hanging node on cell batch -> nothing to do

  for (unsigned int v = n_vectorization_actual; v < n_lanes; ++v)
    constraint_mask[v] =
      internal::MatrixFreeFunctions::ConstraintKinds::unconstrained;

  internal::FEEvaluationHangingNodesFactory<dim, Number, VectorizedArrayType>::
    apply(n_components,
          this->data->data.front().fe_degree,
          this->get_shape_info(),
          transpose,
          constraint_mask,
          this->values_dofs);
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  read_dof_values(const VectorType &                              src,
                  const unsigned int                              first_index,
                  const std::bitset<VectorizedArrayType::size()> &mask)
{
  const auto src_data = internal::get_vector_data<n_components_>(
    src,
    first_index,
    this->dof_access_index ==
      internal::MatrixFreeFunctions::DoFInfo::dof_access_cell,
    this->active_fe_index,
    this->dof_info);

  internal::VectorReader<Number, VectorizedArrayType> reader;
  read_write_operation(reader, src_data.first, src_data.second, mask, true);

  apply_hanging_node_constraints(false);

#  ifdef DEBUG
  this->dof_values_initialized = true;
#  endif
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  read_dof_values_plain(const VectorType & src,
                        const unsigned int first_index,
                        const std::bitset<VectorizedArrayType::size()> &mask)
{
  const auto src_data = internal::get_vector_data<n_components_>(
    src,
    first_index,
    this->dof_access_index ==
      internal::MatrixFreeFunctions::DoFInfo::dof_access_cell,
    this->active_fe_index,
    this->dof_info);

  internal::VectorReader<Number, VectorizedArrayType> reader;
  read_write_operation(reader, src_data.first, src_data.second, mask, false);

#  ifdef DEBUG
  this->dof_values_initialized = true;
#  endif
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  distribute_local_to_global(
    VectorType &                                    dst,
    const unsigned int                              first_index,
    const std::bitset<VectorizedArrayType::size()> &mask) const
{
#  ifdef DEBUG
  Assert(this->dof_values_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif

  apply_hanging_node_constraints(true);

  const auto dst_data = internal::get_vector_data<n_components_>(
    dst,
    first_index,
    this->dof_access_index ==
      internal::MatrixFreeFunctions::DoFInfo::dof_access_cell,
    this->active_fe_index,
    this->dof_info);

  internal::VectorDistributorLocalToGlobal<Number, VectorizedArrayType>
    distributor;
  read_write_operation(distributor, dst_data.first, dst_data.second, mask);
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  set_dof_values(VectorType &                                    dst,
                 const unsigned int                              first_index,
                 const std::bitset<VectorizedArrayType::size()> &mask) const
{
#  ifdef DEBUG
  Assert(this->dof_values_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif

  const auto dst_data = internal::get_vector_data<n_components_>(
    dst,
    first_index,
    this->dof_access_index ==
      internal::MatrixFreeFunctions::DoFInfo::dof_access_cell,
    this->active_fe_index,
    this->dof_info);

  internal::VectorSetter<Number, VectorizedArrayType> setter;
  read_write_operation(setter, dst_data.first, dst_data.second, mask);
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  set_dof_values_plain(
    VectorType &                                    dst,
    const unsigned int                              first_index,
    const std::bitset<VectorizedArrayType::size()> &mask) const
{
#  ifdef DEBUG
  Assert(this->dof_values_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif

  const auto dst_data = internal::get_vector_data<n_components_>(
    dst,
    first_index,
    this->dof_access_index ==
      internal::MatrixFreeFunctions::DoFInfo::dof_access_cell,
    this->active_fe_index,
    this->dof_info);

  internal::VectorSetter<Number, VectorizedArrayType> setter;
  read_write_operation(setter, dst_data.first, dst_data.second, mask, false);
}



/*------------------------------ access to data fields ----------------------*/



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE Tensor<1, n_components_, VectorizedArrayType>
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  get_dof_value(const unsigned int dof) const
{
  AssertIndexRange(dof, this->data->dofs_per_component_on_cell);
  const std::size_t dofs = this->data->dofs_per_component_on_cell;
  Tensor<1, n_components_, VectorizedArrayType> return_value;
  for (unsigned int comp = 0; comp < n_components; ++comp)
    return_value[comp] = this->values_dofs[comp * dofs + dof];
  return return_value;
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE Tensor<1, n_components_, VectorizedArrayType>
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  get_value(const unsigned int q_point) const
{
#  ifdef DEBUG
  Assert(this->values_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif

  AssertIndexRange(q_point, this->n_quadrature_points);
  const std::size_t                             nqp = this->n_quadrature_points;
  Tensor<1, n_components_, VectorizedArrayType> return_value;
  for (unsigned int comp = 0; comp < n_components; ++comp)
    return_value[comp] = this->values_quad[comp * nqp + q_point];
  return return_value;
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE
  Tensor<1, n_components_, Tensor<1, dim, VectorizedArrayType>>
  FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
    get_gradient(const unsigned int q_point) const
{
#  ifdef DEBUG
  Assert(this->gradients_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif

  AssertIndexRange(q_point, this->n_quadrature_points);
  Assert(this->jacobian != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));
  const std::size_t nqp = this->n_quadrature_points;
  Tensor<1, n_components_, Tensor<1, dim, VectorizedArrayType>> grad_out;

  // Cartesian cell
  if (!is_face && this->cell_type == internal::MatrixFreeFunctions::cartesian)
    {
      for (unsigned int d = 0; d < dim; ++d)
        for (unsigned int comp = 0; comp < n_components; ++comp)
          grad_out[comp][d] =
            this->gradients_quad[(comp * dim + d) * nqp + q_point] *
            this->jacobian[0][d][d];
    }
  // cell with general/affine Jacobian
  else
    {
      const Tensor<2, dim, VectorizedArrayType> &jac =
        this->jacobian[this->cell_type > internal::MatrixFreeFunctions::affine ?
                         q_point :
                         0];
      for (unsigned int comp = 0; comp < n_components; ++comp)
        for (unsigned int d = 0; d < dim; ++d)
          {
            grad_out[comp][d] =
              jac[d][0] * this->gradients_quad[(comp * dim) * nqp + q_point];
            for (unsigned int e = 1; e < dim; ++e)
              grad_out[comp][d] +=
                jac[d][e] *
                this->gradients_quad[(comp * dim + e) * nqp + q_point];
          }
    }
  return grad_out;
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE Tensor<1, n_components_, VectorizedArrayType>
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  get_normal_derivative(const unsigned int q_point) const
{
  AssertIndexRange(q_point, this->n_quadrature_points);
#  ifdef DEBUG
  Assert(this->gradients_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif

  Assert(this->normal_x_jacobian != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));

  const std::size_t                            nqp = this->n_quadrature_points;
  Tensor<1, n_components, VectorizedArrayType> grad_out;

  if (this->cell_type == internal::MatrixFreeFunctions::cartesian)
    for (unsigned int comp = 0; comp < n_components; ++comp)
      grad_out[comp] =
        this->gradients_quad[(comp * dim + dim - 1) * nqp + q_point] *
        (this->normal_x_jacobian[0][dim - 1]);
  else
    {
      const std::size_t index =
        this->cell_type <= internal::MatrixFreeFunctions::affine ? 0 : q_point;
      for (unsigned int comp = 0; comp < n_components; ++comp)
        {
          grad_out[comp] = this->gradients_quad[comp * dim * nqp + q_point] *
                           this->normal_x_jacobian[index][0];
          for (unsigned int d = 1; d < dim; ++d)
            grad_out[comp] +=
              this->gradients_quad[(comp * dim + d) * nqp + q_point] *
              this->normal_x_jacobian[index][d];
        }
    }
  return grad_out;
}



namespace internal
{
  // compute tmp = hess_unit(u) * J^T. do this manually because we do not
  // store the lower diagonal because of symmetry
  template <typename VectorizedArrayType>
  inline void
  hessian_unit_times_jac(const Tensor<2, 1, VectorizedArrayType> &jac,
                         const VectorizedArrayType *const         hessians,
                         const unsigned int,
                         VectorizedArrayType (&tmp)[1][1])
  {
    tmp[0][0] = jac[0][0] * hessians[0];
  }

  template <typename VectorizedArrayType>
  inline void
  hessian_unit_times_jac(const Tensor<2, 2, VectorizedArrayType> &jac,
                         const VectorizedArrayType *const         hessians,
                         const unsigned int                       nqp,
                         VectorizedArrayType (&tmp)[2][2])
  {
    for (unsigned int d = 0; d < 2; ++d)
      {
        tmp[0][d] = (jac[d][0] * hessians[0] + jac[d][1] * hessians[2 * nqp]);
        tmp[1][d] =
          (jac[d][0] * hessians[2 * nqp] + jac[d][1] * hessians[1 * nqp]);
      }
  }

  template <typename VectorizedArrayType>
  inline void
  hessian_unit_times_jac(const Tensor<2, 3, VectorizedArrayType> &jac,
                         const VectorizedArrayType *const         hessians,
                         const unsigned int                       nqp,
                         VectorizedArrayType (&tmp)[3][3])
  {
    for (unsigned int d = 0; d < 3; ++d)
      {
        tmp[0][d] =
          (jac[d][0] * hessians[0 * nqp] + jac[d][1] * hessians[3 * nqp] +
           jac[d][2] * hessians[4 * nqp]);
        tmp[1][d] =
          (jac[d][0] * hessians[3 * nqp] + jac[d][1] * hessians[1 * nqp] +
           jac[d][2] * hessians[5 * nqp]);
        tmp[2][d] =
          (jac[d][0] * hessians[4 * nqp] + jac[d][1] * hessians[5 * nqp] +
           jac[d][2] * hessians[2 * nqp]);
      }
  }
} // namespace internal



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline Tensor<1, n_components_, Tensor<2, dim, VectorizedArrayType>>
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  get_hessian(const unsigned int q_point) const
{
#  ifdef DEBUG
  Assert(this->hessians_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif
  AssertIndexRange(q_point, this->n_quadrature_points);

  Assert(this->jacobian != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_hessian"));
  const Tensor<2, dim, VectorizedArrayType> &jac =
    this->jacobian[this->cell_type <= internal::MatrixFreeFunctions::affine ?
                     0 :
                     q_point];

  Tensor<1, n_components, Tensor<2, dim, VectorizedArrayType>> hessian_out;

  const std::size_t      nqp  = this->n_quadrature_points;
  constexpr unsigned int hdim = (dim * (dim + 1)) / 2;

  // Cartesian cell
  if (!is_face && this->cell_type == internal::MatrixFreeFunctions::cartesian)
    {
      for (unsigned int comp = 0; comp < n_components; ++comp)
        {
          for (unsigned int d = 0; d < dim; ++d)
            hessian_out[comp][d][d] =
              this->hessians_quad[(comp * hdim + d) * nqp + q_point] *
              (jac[d][d] * jac[d][d]);
          switch (dim)
            {
              case 1:
                break;
              case 2:
                hessian_out[comp][0][1] =
                  this->hessians_quad[(comp * hdim + 2) * nqp + q_point] *
                  (jac[0][0] * jac[1][1]);
                break;
              case 3:
                hessian_out[comp][0][1] =
                  this->hessians_quad[(comp * hdim + 3) * nqp + q_point] *
                  (jac[0][0] * jac[1][1]);
                hessian_out[comp][0][2] =
                  this->hessians_quad[(comp * hdim + 4) * nqp + q_point] *
                  (jac[0][0] * jac[2][2]);
                hessian_out[comp][1][2] =
                  this->hessians_quad[(comp * hdim + 5) * nqp + q_point] *
                  (jac[1][1] * jac[2][2]);
                break;
              default:
                Assert(false, ExcNotImplemented());
            }
          for (unsigned int d = 0; d < dim; ++d)
            for (unsigned int e = d + 1; e < dim; ++e)
              hessian_out[comp][e][d] = hessian_out[comp][d][e];
        }
    }
  // cell with general Jacobian, but constant within the cell
  else if (this->cell_type <= internal::MatrixFreeFunctions::affine)
    {
      for (unsigned int comp = 0; comp < n_components; ++comp)
        {
          VectorizedArrayType tmp[dim][dim];
          internal::hessian_unit_times_jac(
            jac, this->hessians_quad + comp * hdim * nqp + q_point, nqp, tmp);

          // compute first part of hessian, J * tmp = J * hess_unit(u) * J^T
          for (unsigned int d = 0; d < dim; ++d)
            for (unsigned int e = d; e < dim; ++e)
              {
                hessian_out[comp][d][e] = jac[d][0] * tmp[0][e];
                for (unsigned int f = 1; f < dim; ++f)
                  hessian_out[comp][d][e] += jac[d][f] * tmp[f][e];
              }

          // no J' * grad(u) part here because the Jacobian is constant
          // throughout the cell and hence, its derivative is zero

          // take symmetric part
          for (unsigned int d = 0; d < dim; ++d)
            for (unsigned int e = d + 1; e < dim; ++e)
              hessian_out[comp][e][d] = hessian_out[comp][d][e];
        }
    }
  // cell with general Jacobian
  else
    {
      const auto &jac_grad =
        this->mapping_data->jacobian_gradients
          [1 - this->is_interior_face]
          [this->mapping_data->data_index_offsets[this->cell] + q_point];
      for (unsigned int comp = 0; comp < n_components; ++comp)
        {
          VectorizedArrayType tmp[dim][dim];
          internal::hessian_unit_times_jac(
            jac, this->hessians_quad + comp * hdim * nqp + q_point, nqp, tmp);

          // compute first part of hessian, J * tmp = J * hess_unit(u) * J^T
          for (unsigned int d = 0; d < dim; ++d)
            for (unsigned int e = d; e < dim; ++e)
              {
                hessian_out[comp][d][e] = jac[d][0] * tmp[0][e];
                for (unsigned int f = 1; f < dim; ++f)
                  hessian_out[comp][d][e] += jac[d][f] * tmp[f][e];
              }

          // add diagonal part of J' * grad(u)
          for (unsigned int d = 0; d < dim; ++d)
            for (unsigned int e = 0; e < dim; ++e)
              hessian_out[comp][d][d] +=
                jac_grad[d][e] *
                this->gradients_quad[(comp * dim + e) * nqp + q_point];

          // add off-diagonal part of J' * grad(u)
          for (unsigned int d = 0, count = dim; d < dim; ++d)
            for (unsigned int e = d + 1; e < dim; ++e, ++count)
              for (unsigned int f = 0; f < dim; ++f)
                hessian_out[comp][d][e] +=
                  jac_grad[count][f] *
                  this->gradients_quad[(comp * dim + f) * nqp + q_point];

          // take symmetric part
          for (unsigned int d = 0; d < dim; ++d)
            for (unsigned int e = d + 1; e < dim; ++e)
              hessian_out[comp][e][d] = hessian_out[comp][d][e];
        }
    }
  return hessian_out;
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline Tensor<1, n_components_, Tensor<1, dim, VectorizedArrayType>>
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  get_hessian_diagonal(const unsigned int q_point) const
{
  Assert(!is_face, ExcNotImplemented());
#  ifdef DEBUG
  Assert(this->hessians_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif
  AssertIndexRange(q_point, this->n_quadrature_points);

  Assert(this->jacobian != nullptr, ExcNotImplemented());
  const Tensor<2, dim, VectorizedArrayType> &jac =
    this->jacobian[this->cell_type <= internal::MatrixFreeFunctions::affine ?
                     0 :
                     q_point];

  const std::size_t      nqp  = this->n_quadrature_points;
  constexpr unsigned int hdim = (dim * (dim + 1)) / 2;
  Tensor<1, n_components_, Tensor<1, dim, VectorizedArrayType>> hessian_out;

  // Cartesian cell
  if (this->cell_type == internal::MatrixFreeFunctions::cartesian)
    {
      for (unsigned int comp = 0; comp < n_components; ++comp)
        for (unsigned int d = 0; d < dim; ++d)
          hessian_out[comp][d] =
            this->hessians_quad[(comp * hdim + d) * nqp + q_point] *
            (jac[d][d] * jac[d][d]);
    }
  // cell with general Jacobian, but constant within the cell
  else if (this->cell_type == internal::MatrixFreeFunctions::affine)
    {
      for (unsigned int comp = 0; comp < n_components; ++comp)
        {
          // compute laplacian before the gradient because it needs to access
          // unscaled gradient data
          VectorizedArrayType tmp[dim][dim];
          internal::hessian_unit_times_jac(
            jac, this->hessians_quad + comp * hdim * nqp + q_point, nqp, tmp);

          // compute only the trace part of hessian, J * tmp = J *
          // hess_unit(u) * J^T
          for (unsigned int d = 0; d < dim; ++d)
            {
              hessian_out[comp][d] = jac[d][0] * tmp[0][d];
              for (unsigned int f = 1; f < dim; ++f)
                hessian_out[comp][d] += jac[d][f] * tmp[f][d];
            }
        }
    }
  // cell with general Jacobian
  else
    {
      const Tensor<1, dim *(dim + 1) / 2, Tensor<1, dim, VectorizedArrayType>>
        &jac_grad =
          this->mapping_data->jacobian_gradients
            [0][this->mapping_data->data_index_offsets[this->cell] + q_point];
      for (unsigned int comp = 0; comp < n_components; ++comp)
        {
          // compute laplacian before the gradient because it needs to access
          // unscaled gradient data
          VectorizedArrayType tmp[dim][dim];
          internal::hessian_unit_times_jac(
            jac, this->hessians_quad + comp * hdim * nqp + q_point, nqp, tmp);

          // compute only the trace part of hessian, J * tmp = J *
          // hess_unit(u) * J^T
          for (unsigned int d = 0; d < dim; ++d)
            {
              hessian_out[comp][d] = jac[d][0] * tmp[0][d];
              for (unsigned int f = 1; f < dim; ++f)
                hessian_out[comp][d] += jac[d][f] * tmp[f][d];
            }

          for (unsigned int d = 0; d < dim; ++d)
            for (unsigned int e = 0; e < dim; ++e)
              hessian_out[comp][d] +=
                jac_grad[d][e] *
                this->gradients_quad[(comp * dim + e) * nqp + q_point];
        }
    }
  return hessian_out;
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline Tensor<1, n_components_, VectorizedArrayType>
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  get_laplacian(const unsigned int q_point) const
{
  Assert(is_face == false, ExcNotImplemented());
#  ifdef DEBUG
  Assert(this->hessians_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif
  AssertIndexRange(q_point, this->n_quadrature_points);

  Tensor<1, n_components_, VectorizedArrayType> laplacian_out;
  const auto hess_diag = get_hessian_diagonal(q_point);
  for (unsigned int comp = 0; comp < n_components; ++comp)
    {
      laplacian_out[comp] = hess_diag[comp][0];
      for (unsigned int d = 1; d < dim; ++d)
        laplacian_out[comp] += hess_diag[comp][d];
    }
  return laplacian_out;
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  submit_dof_value(const Tensor<1, n_components_, VectorizedArrayType> val_in,
                   const unsigned int                                  dof)
{
#  ifdef DEBUG
  this->dof_values_initialized = true;
#  endif
  const std::size_t dofs = this->data->dofs_per_component_on_cell;
  AssertIndexRange(dof, this->data->dofs_per_component_on_cell);
  for (unsigned int comp = 0; comp < n_components; ++comp)
    this->values_dofs[comp * dofs + dof] = val_in[comp];
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  submit_value(const Tensor<1, n_components_, VectorizedArrayType> val_in,
               const unsigned int                                  q_point)
{
  Assert(this->cell != numbers::invalid_unsigned_int, ExcNotInitialized());
  AssertIndexRange(q_point, this->n_quadrature_points);
  Assert(this->J_value != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_values"));
#  ifdef DEBUG
  this->values_quad_submitted = true;
#  endif

  const std::size_t nqp = this->n_quadrature_points;
  if (this->cell_type <= internal::MatrixFreeFunctions::affine)
    {
      const VectorizedArrayType JxW =
        this->J_value[0] * this->quadrature_weights[q_point];
      for (unsigned int comp = 0; comp < n_components; ++comp)
        this->values_quad[comp * nqp + q_point] = val_in[comp] * JxW;
    }
  else
    {
      const VectorizedArrayType JxW = this->J_value[q_point];
      for (unsigned int comp = 0; comp < n_components; ++comp)
        this->values_quad[comp * nqp + q_point] = val_in[comp] * JxW;
    }
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  submit_gradient(
    const Tensor<1, n_components_, Tensor<1, dim, VectorizedArrayType>> grad_in,
    const unsigned int                                                  q_point)
{
  Assert(this->cell != numbers::invalid_unsigned_int, ExcNotInitialized());
  AssertIndexRange(q_point, this->n_quadrature_points);
  Assert(this->J_value != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));
  Assert(this->jacobian != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));
#  ifdef DEBUG
  this->gradients_quad_submitted = true;
#  endif

  const std::size_t nqp = this->n_quadrature_points;
  if (!is_face && this->cell_type == internal::MatrixFreeFunctions::cartesian)
    {
      const VectorizedArrayType JxW =
        this->J_value[0] * this->quadrature_weights[q_point];
      for (unsigned int d = 0; d < dim; ++d)
        {
          const VectorizedArrayType factor = this->jacobian[0][d][d] * JxW;
          for (unsigned int comp = 0; comp < n_components; ++comp)
            this->gradients_quad[(comp * dim + d) * nqp + q_point] =
              grad_in[comp][d] * factor;
        }
    }
  else
    {
      const Tensor<2, dim, VectorizedArrayType> jac =
        this->cell_type > internal::MatrixFreeFunctions::affine ?
          this->jacobian[q_point] :
          this->jacobian[0];
      const VectorizedArrayType JxW =
        this->cell_type > internal::MatrixFreeFunctions::affine ?
          this->J_value[q_point] :
          this->J_value[0] * this->quadrature_weights[q_point];
      for (unsigned int comp = 0; comp < n_components; ++comp)
        for (unsigned int d = 0; d < dim; ++d)
          {
            VectorizedArrayType new_val = jac[0][d] * grad_in[comp][0];
            for (unsigned int e = 1; e < dim; ++e)
              new_val += (jac[e][d] * grad_in[comp][e]);
            this->gradients_quad[(comp * dim + d) * nqp + q_point] =
              new_val * JxW;
          }
    }
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  submit_normal_derivative(
    const Tensor<1, n_components_, VectorizedArrayType> grad_in,
    const unsigned int                                  q_point)
{
  AssertIndexRange(q_point, this->n_quadrature_points);
  Assert(this->normal_x_jacobian != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));
#  ifdef DEBUG
  this->gradients_quad_submitted = true;
#  endif

  const std::size_t nqp = this->n_quadrature_points;
  if (this->cell_type == internal::MatrixFreeFunctions::cartesian)
    for (unsigned int comp = 0; comp < n_components; ++comp)
      {
        for (unsigned int d = 0; d < dim - 1; ++d)
          this->gradients_quad[(comp * dim + d) * nqp + q_point] =
            VectorizedArrayType();
        this->gradients_quad[(comp * dim + dim - 1) * nqp + q_point] =
          grad_in[comp] *
          (this->normal_x_jacobian[0][dim - 1] * this->J_value[0] *
           this->quadrature_weights[q_point]);
      }
  else
    {
      const unsigned int index =
        this->cell_type <= internal::MatrixFreeFunctions::affine ? 0 : q_point;
      const Tensor<1, dim, VectorizedArrayType> jac =
        this->normal_x_jacobian[index];
      for (unsigned int comp = 0; comp < n_components; ++comp)
        {
          VectorizedArrayType factor = grad_in[comp] * this->J_value[index];
          if (this->cell_type <= internal::MatrixFreeFunctions::affine)
            factor = factor * this->quadrature_weights[q_point];
          for (unsigned int d = 0; d < dim; ++d)
            this->gradients_quad[(comp * dim + d) * nqp + q_point] =
              factor * jac[d];
        }
    }
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  submit_hessian(
    const Tensor<1, n_components_, Tensor<2, dim, VectorizedArrayType>>
                       hessian_in,
    const unsigned int q_point)
{
  Assert(this->cell != numbers::invalid_unsigned_int, ExcNotInitialized());
  AssertIndexRange(q_point, this->n_quadrature_points);
  Assert(this->J_value != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_hessians"));
  Assert(this->jacobian != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_hessians"));
#  ifdef DEBUG
  this->hessians_quad_submitted = true;
#  endif

  // compute hessian_unit = J^T * hessian_in(u) * J
  const std::size_t      nqp  = this->n_quadrature_points;
  constexpr unsigned int hdim = (dim * (dim + 1)) / 2;
  if (!is_face && this->cell_type == internal::MatrixFreeFunctions::cartesian)
    {
      const VectorizedArrayType JxW =
        this->J_value[0] * this->quadrature_weights[q_point];

      // diagonal part
      for (unsigned int d = 0; d < dim; ++d)
        {
          const auto                jac_d  = this->jacobian[0][d][d];
          const VectorizedArrayType factor = jac_d * jac_d * JxW;
          for (unsigned int comp = 0; comp < n_components; ++comp)
            this->hessians_quad[(comp * hdim + d) * nqp + q_point] =
              hessian_in[comp][d][d] * factor;
        }

      // off diagonal part
      for (unsigned int d = 1, off_dia = dim; d < dim; ++d)
        for (unsigned int e = 0; e < d; ++e, ++off_dia)
          {
            const auto                jac_d  = this->jacobian[0][d][d];
            const auto                jac_e  = this->jacobian[0][e][e];
            const VectorizedArrayType factor = jac_d * jac_e * JxW;
            for (unsigned int comp = 0; comp < n_components; ++comp)
              this->hessians_quad[(comp * hdim + off_dia) * nqp + q_point] =
                (hessian_in[comp][d][e] + hessian_in[comp][e][d]) * factor;
          }
    }
  // cell with general Jacobian, but constant within the cell
  else if (this->cell_type <= internal::MatrixFreeFunctions::affine)
    {
      const Tensor<2, dim, VectorizedArrayType> jac = this->jacobian[0];
      const VectorizedArrayType                 JxW =
        this->J_value[0] * this->quadrature_weights[q_point];
      for (unsigned int comp = 0; comp < n_components; ++comp)
        {
          // 1. tmp = hessian_in(u) * J
          VectorizedArrayType tmp[dim][dim];
          for (unsigned int i = 0; i < dim; ++i)
            for (unsigned int j = 0; j < dim; ++j)
              {
                tmp[i][j] = hessian_in[comp][i][0] * jac[0][j];
                for (unsigned int k = 1; k < dim; ++k)
                  tmp[i][j] += hessian_in[comp][i][k] * jac[k][j];
              }

          // 2. hessian_unit = J^T * tmp
          VectorizedArrayType tmp2[dim][dim];
          for (unsigned int i = 0; i < dim; ++i)
            for (unsigned int j = 0; j < dim; ++j)
              {
                tmp2[i][j] = jac[0][i] * tmp[0][j];
                for (unsigned int k = 1; k < dim; ++k)
                  tmp2[i][j] += jac[k][i] * tmp[k][j];
              }

          // diagonal part
          for (unsigned int d = 0; d < dim; ++d)
            this->hessians_quad[(comp * hdim + d) * nqp + q_point] =
              tmp2[d][d] * JxW;

          // off diagonal part
          for (unsigned int d = 0, off_diag = dim; d < dim; ++d)
            for (unsigned int e = d + 1; e < dim; ++e, ++off_diag)
              this->hessians_quad[(comp * hdim + off_diag) * nqp + q_point] =
                (tmp2[d][e] + tmp2[e][d]) * JxW;
        }
    }
  else
    {
      const Tensor<2, dim, VectorizedArrayType> jac = this->jacobian[q_point];
      const VectorizedArrayType                 JxW = this->J_value[q_point];
      const auto &                              jac_grad =
        this->mapping_data->jacobian_gradients
          [1 - this->is_interior_face]
          [this->mapping_data->data_index_offsets[this->cell] + q_point];
      for (unsigned int comp = 0; comp < n_components; ++comp)
        {
          // 1. tmp = hessian_in(u) * J
          VectorizedArrayType tmp[dim][dim];
          for (unsigned int i = 0; i < dim; ++i)
            for (unsigned int j = 0; j < dim; ++j)
              {
                tmp[i][j] = hessian_in[comp][i][0] * jac[0][j];
                for (unsigned int k = 1; k < dim; ++k)
                  tmp[i][j] += hessian_in[comp][i][k] * jac[k][j];
              }

          // 2. hessian_unit = J^T * tmp
          VectorizedArrayType tmp2[dim][dim];
          for (unsigned int i = 0; i < dim; ++i)
            for (unsigned int j = 0; j < dim; ++j)
              {
                tmp2[i][j] = jac[0][i] * tmp[0][j];
                for (unsigned int k = 1; k < dim; ++k)
                  tmp2[i][j] += jac[k][i] * tmp[k][j];
              }

          // diagonal part
          for (unsigned int d = 0; d < dim; ++d)
            this->hessians_quad[(comp * hdim + d) * nqp + q_point] =
              tmp2[d][d] * JxW;

          // off diagonal part
          for (unsigned int d = 0, off_diag = dim; d < dim; ++d)
            for (unsigned int e = d + 1; e < dim; ++e, ++off_diag)
              this->hessians_quad[(comp * hdim + off_diag) * nqp + q_point] =
                (tmp2[d][e] + tmp2[e][d]) * JxW;

          // 3. gradient_unit = J' ** hessian_in
          for (unsigned int d = 0; d < dim; ++d)
            {
              VectorizedArrayType sum = 0;
              for (unsigned int e = 0; e < dim; ++e)
                sum += hessian_in[comp][e][e] * jac_grad[e][d];
              for (unsigned int e = 0, count = dim; e < dim; ++e)
                for (unsigned int f = e + 1; f < dim; ++f, ++count)
                  sum += (hessian_in[comp][e][f] + hessian_in[comp][f][e]) *
                         jac_grad[count][d];
              this->gradients_from_hessians_quad[(comp * dim + d) * nqp +
                                                 q_point] = sum * JxW;
            }
        }
    }
}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline Tensor<1, n_components_, VectorizedArrayType>
FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>::
  integrate_value() const
{
  Assert(this->cell != numbers::invalid_unsigned_int, ExcNotInitialized());
#  ifdef DEBUG
  Assert(this->values_quad_submitted == true,
         internal::ExcAccessToUninitializedField());
#  endif

  Tensor<1, n_components_, VectorizedArrayType> return_value;
  const std::size_t                             nqp = this->n_quadrature_points;
  for (unsigned int q = 0; q < nqp; ++q)
    for (unsigned int comp = 0; comp < n_components; ++comp)
      return_value[comp] += this->values_quad[comp * nqp + q];
  return (return_value);
}



/*----------------------- FEEvaluationAccess --------------------------------*/


template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline FEEvaluationAccess<dim,
                          n_components_,
                          Number,
                          is_face,
                          VectorizedArrayType>::
  FEEvaluationAccess(
    const MatrixFree<dim, Number, VectorizedArrayType> &data_in,
    const unsigned int                                  dof_no,
    const unsigned int first_selected_component,
    const unsigned int quad_no,
    const unsigned int fe_degree,
    const unsigned int n_q_points,
    const bool         is_interior_face,
    const unsigned int active_fe_index,
    const unsigned int active_quad_index,
    const unsigned int face_type)
  : FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>(
      data_in,
      dof_no,
      first_selected_component,
      quad_no,
      fe_degree,
      n_q_points,
      is_interior_face,
      active_fe_index,
      active_quad_index,
      face_type)
{}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline FEEvaluationAccess<dim,
                          n_components_,
                          Number,
                          is_face,
                          VectorizedArrayType>::
  FEEvaluationAccess(
    const Mapping<dim> &      mapping,
    const FiniteElement<dim> &fe,
    const Quadrature<1> &     quadrature,
    const UpdateFlags         update_flags,
    const unsigned int        first_selected_component,
    const FEEvaluationData<dim, VectorizedArrayType, is_face> *other)
  : FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>(
      mapping,
      fe,
      quadrature,
      update_flags,
      first_selected_component,
      other)
{}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline FEEvaluationAccess<dim,
                          n_components_,
                          Number,
                          is_face,
                          VectorizedArrayType>::
  FEEvaluationAccess(const FEEvaluationAccess<dim,
                                              n_components_,
                                              Number,
                                              is_face,
                                              VectorizedArrayType> &other)
  : FEEvaluationBase<dim, n_components_, Number, is_face, VectorizedArrayType>(
      other)
{}



template <int dim,
          int n_components_,
          typename Number,
          bool is_face,
          typename VectorizedArrayType>
inline FEEvaluationAccess<dim,
                          n_components_,
                          Number,
                          is_face,
                          VectorizedArrayType> &
FEEvaluationAccess<dim, n_components_, Number, is_face, VectorizedArrayType>::
operator=(const FEEvaluationAccess<dim,
                                   n_components_,
                                   Number,
                                   is_face,
                                   VectorizedArrayType> &other)
{
  this->FEEvaluationBase<dim,
                         n_components_,
                         Number,
                         is_face,
                         VectorizedArrayType>::operator=(other);
  return *this;
}



/*-------------------- FEEvaluationAccess scalar ----------------------------*/


template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::
  FEEvaluationAccess(
    const MatrixFree<dim, Number, VectorizedArrayType> &data_in,
    const unsigned int                                  dof_no,
    const unsigned int first_selected_component,
    const unsigned int quad_no,
    const unsigned int fe_degree,
    const unsigned int n_q_points,
    const bool         is_interior_face,
    const unsigned int active_fe_index,
    const unsigned int active_quad_index,
    const unsigned int face_type)
  : FEEvaluationBase<dim, 1, Number, is_face, VectorizedArrayType>(
      data_in,
      dof_no,
      first_selected_component,
      quad_no,
      fe_degree,
      n_q_points,
      is_interior_face,
      active_fe_index,
      active_quad_index,
      face_type)
{}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::
  FEEvaluationAccess(
    const Mapping<dim> &      mapping,
    const FiniteElement<dim> &fe,
    const Quadrature<1> &     quadrature,
    const UpdateFlags         update_flags,
    const unsigned int        first_selected_component,
    const FEEvaluationData<dim, VectorizedArrayType, is_face> *other)
  : FEEvaluationBase<dim, 1, Number, is_face, VectorizedArrayType>(
      mapping,
      fe,
      quadrature,
      update_flags,
      first_selected_component,
      other)
{}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::
  FEEvaluationAccess(
    const FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>
      &other)
  : FEEvaluationBase<dim, 1, Number, is_face, VectorizedArrayType>(other)
{}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType> &
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::operator=(
  const FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType> &other)
{
  this
    ->FEEvaluationBase<dim, 1, Number, is_face, VectorizedArrayType>::operator=(
      other);
  return *this;
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE VectorizedArrayType
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::get_dof_value(
  const unsigned int dof) const
{
  AssertIndexRange(dof, this->data->dofs_per_component_on_cell);
  return this->values_dofs[dof];
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE VectorizedArrayType
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::get_value(
  const unsigned int q_point) const
{
#  ifdef DEBUG
  Assert(this->values_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif
  AssertIndexRange(q_point, this->n_quadrature_points);
  return this->values_quad[q_point];
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE VectorizedArrayType
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::
  get_normal_derivative(const unsigned int q_point) const
{
  return BaseClass::get_normal_derivative(q_point)[0];
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE Tensor<1, dim, VectorizedArrayType>
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::get_gradient(
  const unsigned int q_point) const
{
  // could use the base class gradient, but that involves too many expensive
  // initialization operations on tensors

#  ifdef DEBUG
  Assert(this->gradients_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif
  AssertIndexRange(q_point, this->n_quadrature_points);

  Assert(this->jacobian != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));

  Tensor<1, dim, VectorizedArrayType> grad_out;

  const std::size_t nqp = this->n_quadrature_points;
  if (!is_face && this->cell_type == internal::MatrixFreeFunctions::cartesian)
    {
      for (unsigned int d = 0; d < dim; ++d)
        grad_out[d] =
          this->gradients_quad[d * nqp + q_point] * this->jacobian[0][d][d];
    }
  // cell with general/affine Jacobian
  else
    {
      const Tensor<2, dim, VectorizedArrayType> &jac =
        this->jacobian[this->cell_type > internal::MatrixFreeFunctions::affine ?
                         q_point :
                         0];
      for (unsigned int d = 0; d < dim; ++d)
        {
          grad_out[d] = jac[d][0] * this->gradients_quad[q_point];
          for (unsigned int e = 1; e < dim; ++e)
            grad_out[d] += jac[d][e] * this->gradients_quad[e * nqp + q_point];
        }
    }
  return grad_out;
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline Tensor<2, dim, VectorizedArrayType>
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::get_hessian(
  const unsigned int q_point) const
{
  return BaseClass::get_hessian(q_point)[0];
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline Tensor<1, dim, VectorizedArrayType>
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::
  get_hessian_diagonal(const unsigned int q_point) const
{
  return BaseClass::get_hessian_diagonal(q_point)[0];
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline VectorizedArrayType
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::get_laplacian(
  const unsigned int q_point) const
{
  return BaseClass::get_laplacian(q_point)[0];
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline void DEAL_II_ALWAYS_INLINE
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::
  submit_dof_value(const VectorizedArrayType val_in, const unsigned int dof)
{
#  ifdef DEBUG
  this->dof_values_initialized = true;
  AssertIndexRange(dof, this->data->dofs_per_component_on_cell);
#  endif
  this->values_dofs[dof] = val_in;
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline void DEAL_II_ALWAYS_INLINE
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::submit_value(
  const VectorizedArrayType val_in,
  const unsigned int        q_point)
{
  Assert(this->cell != numbers::invalid_unsigned_int, ExcNotInitialized());
  AssertIndexRange(q_point, this->n_quadrature_points);
  Assert(this->J_value != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_value"));
#  ifdef DEBUG
  this->values_quad_submitted = true;
#  endif

  if (this->cell_type <= internal::MatrixFreeFunctions::affine)
    {
      const VectorizedArrayType JxW =
        this->J_value[0] * this->quadrature_weights[q_point];
      this->values_quad[q_point] = val_in * JxW;
    }
  else // if (this->cell_type < internal::MatrixFreeFunctions::general)
    {
      this->values_quad[q_point] = val_in * this->J_value[q_point];
    }
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::submit_value(
  const Tensor<1, 1, VectorizedArrayType> val_in,
  const unsigned int                      q_point)
{
  submit_value(val_in[0], q_point);
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::
  submit_normal_derivative(const VectorizedArrayType grad_in,
                           const unsigned int        q_point)
{
  Tensor<1, 1, VectorizedArrayType> grad;
  grad[0] = grad_in;
  BaseClass::submit_normal_derivative(grad, q_point);
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::
  submit_gradient(const Tensor<1, dim, VectorizedArrayType> grad_in,
                  const unsigned int                        q_point)
{
  Assert(this->cell != numbers::invalid_unsigned_int, ExcNotInitialized());
  AssertIndexRange(q_point, this->n_quadrature_points);
  Assert(this->J_value != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));
  Assert(this->jacobian != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));
#  ifdef DEBUG
  this->gradients_quad_submitted = true;
#  endif

  const std::size_t nqp = this->n_quadrature_points;
  if (!is_face && this->cell_type == internal::MatrixFreeFunctions::cartesian)
    {
      const VectorizedArrayType JxW =
        this->J_value[0] * this->quadrature_weights[q_point];
      for (unsigned int d = 0; d < dim; ++d)
        this->gradients_quad[d * nqp + q_point] =
          (grad_in[d] * this->jacobian[0][d][d] * JxW);
    }
  // general/affine cell type
  else
    {
      const Tensor<2, dim, VectorizedArrayType> &jac =
        this->cell_type > internal::MatrixFreeFunctions::affine ?
          this->jacobian[q_point] :
          this->jacobian[0];
      const VectorizedArrayType JxW =
        this->cell_type > internal::MatrixFreeFunctions::affine ?
          this->J_value[q_point] :
          this->J_value[0] * this->quadrature_weights[q_point];
      for (unsigned int d = 0; d < dim; ++d)
        {
          VectorizedArrayType new_val = jac[0][d] * grad_in[0];
          for (unsigned int e = 1; e < dim; ++e)
            new_val += jac[e][d] * grad_in[e];
          this->gradients_quad[d * nqp + q_point] = new_val * JxW;
        }
    }
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::
  submit_hessian(const Tensor<2, dim, VectorizedArrayType> hessian_in,
                 const unsigned int                        q_point)
{
  Tensor<1, 1, Tensor<2, dim, VectorizedArrayType>> hessian;
  hessian[0] = hessian_in;
  BaseClass::submit_hessian(hessian, q_point);
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline VectorizedArrayType
FEEvaluationAccess<dim, 1, Number, is_face, VectorizedArrayType>::
  integrate_value() const
{
  return BaseClass::integrate_value()[0];
}



/*----------------- FEEvaluationAccess vector-valued ------------------------*/


template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::
  FEEvaluationAccess(
    const MatrixFree<dim, Number, VectorizedArrayType> &data_in,
    const unsigned int                                  dof_no,
    const unsigned int first_selected_component,
    const unsigned int quad_no,
    const unsigned int fe_degree,
    const unsigned int n_q_points,
    const bool         is_interior_face,
    const unsigned int active_fe_index,
    const unsigned int active_quad_index,
    const unsigned int face_type)
  : FEEvaluationBase<dim, dim, Number, is_face, VectorizedArrayType>(
      data_in,
      dof_no,
      first_selected_component,
      quad_no,
      fe_degree,
      n_q_points,
      is_interior_face,
      active_fe_index,
      active_quad_index,
      face_type)
{}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::
  FEEvaluationAccess(
    const Mapping<dim> &      mapping,
    const FiniteElement<dim> &fe,
    const Quadrature<1> &     quadrature,
    const UpdateFlags         update_flags,
    const unsigned int        first_selected_component,
    const FEEvaluationData<dim, VectorizedArrayType, is_face> *other)
  : FEEvaluationBase<dim, dim, Number, is_face, VectorizedArrayType>(
      mapping,
      fe,
      quadrature,
      update_flags,
      first_selected_component,
      other)
{}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::
  FEEvaluationAccess(
    const FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>
      &other)
  : FEEvaluationBase<dim, dim, Number, is_face, VectorizedArrayType>(other)
{}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType> &
FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::operator=(
  const FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>
    &other)
{
  this->FEEvaluationBase<dim, dim, Number, is_face, VectorizedArrayType>::
  operator=(other);
  return *this;
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE Tensor<2, dim, VectorizedArrayType>
FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::
  get_gradient(const unsigned int q_point) const
{
  return BaseClass::get_gradient(q_point);
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE VectorizedArrayType
FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::
  get_divergence(const unsigned int q_point) const
{
#  ifdef DEBUG
  Assert(this->gradients_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif
  AssertIndexRange(q_point, this->n_quadrature_points);
  Assert(this->jacobian != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));

  VectorizedArrayType divergence;
  const std::size_t   nqp = this->n_quadrature_points;

  // Cartesian cell
  if (!is_face && this->cell_type == internal::MatrixFreeFunctions::cartesian)
    {
      divergence = this->gradients_quad[q_point] * this->jacobian[0][0][0];
      for (unsigned int d = 1; d < dim; ++d)
        divergence += this->gradients_quad[(dim * d + d) * nqp + q_point] *
                      this->jacobian[0][d][d];
    }
  // cell with general/constant Jacobian
  else
    {
      const Tensor<2, dim, VectorizedArrayType> &jac =
        this->cell_type == internal::MatrixFreeFunctions::general ?
          this->jacobian[q_point] :
          this->jacobian[0];
      divergence = jac[0][0] * this->gradients_quad[q_point];
      for (unsigned int e = 1; e < dim; ++e)
        divergence += jac[0][e] * this->gradients_quad[e * nqp + q_point];
      for (unsigned int d = 1; d < dim; ++d)
        for (unsigned int e = 0; e < dim; ++e)
          divergence +=
            jac[d][e] * this->gradients_quad[(d * dim + e) * nqp + q_point];
    }
  return divergence;
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE SymmetricTensor<2, dim, VectorizedArrayType>
FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::
  get_symmetric_gradient(const unsigned int q_point) const
{
  // copy from generic function into dim-specialization function
  const auto          grad = get_gradient(q_point);
  VectorizedArrayType symmetrized[(dim * dim + dim) / 2];
  VectorizedArrayType half = Number(0.5);
  for (unsigned int d = 0; d < dim; ++d)
    symmetrized[d] = grad[d][d];
  switch (dim)
    {
      case 1:
        break;
      case 2:
        symmetrized[2] = grad[0][1] + grad[1][0];
        symmetrized[2] *= half;
        break;
      case 3:
        symmetrized[3] = grad[0][1] + grad[1][0];
        symmetrized[3] *= half;
        symmetrized[4] = grad[0][2] + grad[2][0];
        symmetrized[4] *= half;
        symmetrized[5] = grad[1][2] + grad[2][1];
        symmetrized[5] *= half;
        break;
      default:
        Assert(false, ExcNotImplemented());
    }
  return SymmetricTensor<2, dim, VectorizedArrayType>(symmetrized);
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE
  Tensor<1, (dim == 2 ? 1 : dim), VectorizedArrayType>
  FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::get_curl(
    const unsigned int q_point) const
{
  // copy from generic function into dim-specialization function
  const Tensor<2, dim, VectorizedArrayType> grad = get_gradient(q_point);
  Tensor<1, (dim == 2 ? 1 : dim), VectorizedArrayType> curl;
  switch (dim)
    {
      case 1:
        Assert(false,
               ExcMessage(
                 "Computing the curl in 1d is not a useful operation"));
        break;
      case 2:
        curl[0] = grad[1][0] - grad[0][1];
        break;
      case 3:
        curl[0] = grad[2][1] - grad[1][2];
        curl[1] = grad[0][2] - grad[2][0];
        curl[2] = grad[1][0] - grad[0][1];
        break;
      default:
        Assert(false, ExcNotImplemented());
    }
  return curl;
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE Tensor<2, dim, VectorizedArrayType>
FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::
  get_hessian_diagonal(const unsigned int q_point) const
{
  return BaseClass::get_hessian_diagonal(q_point);
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE Tensor<3, dim, VectorizedArrayType>
FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::get_hessian(
  const unsigned int q_point) const
{
#  ifdef DEBUG
  Assert(this->hessians_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif
  AssertIndexRange(q_point, this->n_quadrature_points);
  return BaseClass::get_hessian(q_point);
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::
  submit_gradient(const Tensor<2, dim, VectorizedArrayType> grad_in,
                  const unsigned int                        q_point)
{
  BaseClass::submit_gradient(grad_in, q_point);
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::
  submit_gradient(
    const Tensor<1, dim, Tensor<1, dim, VectorizedArrayType>> grad_in,
    const unsigned int                                        q_point)
{
  BaseClass::submit_gradient(grad_in, q_point);
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::
  submit_divergence(const VectorizedArrayType div_in,
                    const unsigned int        q_point)
{
  Assert(this->cell != numbers::invalid_unsigned_int, ExcNotInitialized());
  AssertIndexRange(q_point, this->n_quadrature_points);
  Assert(this->J_value != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));
  Assert(this->jacobian != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));
#  ifdef DEBUG
  this->gradients_quad_submitted = true;
#  endif

  const std::size_t nqp = this->n_quadrature_points;
  if (!is_face && this->cell_type == internal::MatrixFreeFunctions::cartesian)
    {
      const VectorizedArrayType fac =
        this->J_value[0] * this->quadrature_weights[q_point] * div_in;
      for (unsigned int d = 0; d < dim; ++d)
        {
          this->gradients_quad[(d * dim + d) * nqp + q_point] =
            (fac * this->jacobian[0][d][d]);
          for (unsigned int e = d + 1; e < dim; ++e)
            {
              this->gradients_quad[(d * dim + e) * nqp + q_point] =
                VectorizedArrayType();
              this->gradients_quad[(e * dim + d) * nqp + q_point] =
                VectorizedArrayType();
            }
        }
    }
  else
    {
      const Tensor<2, dim, VectorizedArrayType> &jac =
        this->cell_type == internal::MatrixFreeFunctions::general ?
          this->jacobian[q_point] :
          this->jacobian[0];
      const VectorizedArrayType fac =
        (this->cell_type == internal::MatrixFreeFunctions::general ?
           this->J_value[q_point] :
           this->J_value[0] * this->quadrature_weights[q_point]) *
        div_in;
      for (unsigned int d = 0; d < dim; ++d)
        {
          for (unsigned int e = 0; e < dim; ++e)
            this->gradients_quad[(d * dim + e) * nqp + q_point] =
              jac[d][e] * fac;
        }
    }
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::
  submit_symmetric_gradient(
    const SymmetricTensor<2, dim, VectorizedArrayType> sym_grad,
    const unsigned int                                 q_point)
{
  // could have used base class operator, but that involves some overhead
  // which is inefficient. it is nice to have the symmetric tensor because
  // that saves some operations
  Assert(this->cell != numbers::invalid_unsigned_int, ExcNotInitialized());
  AssertIndexRange(q_point, this->n_quadrature_points);
  Assert(this->J_value != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));
  Assert(this->jacobian != nullptr,
         internal::ExcMatrixFreeAccessToUninitializedMappingField(
           "update_gradients"));
#  ifdef DEBUG
  this->gradients_quad_submitted = true;
#  endif

  const std::size_t nqp = this->n_quadrature_points;
  if (!is_face && this->cell_type == internal::MatrixFreeFunctions::cartesian)
    {
      const VectorizedArrayType JxW =
        this->J_value[0] * this->quadrature_weights[q_point];
      for (unsigned int d = 0; d < dim; ++d)
        this->gradients_quad[(d * dim + d) * nqp + q_point] =
          (sym_grad.access_raw_entry(d) * JxW * this->jacobian[0][d][d]);
      for (unsigned int e = 0, counter = dim; e < dim; ++e)
        for (unsigned int d = e + 1; d < dim; ++d, ++counter)
          {
            const VectorizedArrayType value =
              sym_grad.access_raw_entry(counter) * JxW;
            this->gradients_quad[(e * dim + d) * nqp + q_point] =
              value * this->jacobian[0][d][d];
            this->gradients_quad[(d * dim + e) * nqp + q_point] =
              value * this->jacobian[0][e][e];
          }
    }
  // general/affine cell type
  else
    {
      const VectorizedArrayType JxW =
        this->cell_type == internal::MatrixFreeFunctions::general ?
          this->J_value[q_point] :
          this->J_value[0] * this->quadrature_weights[q_point];
      const Tensor<2, dim, VectorizedArrayType> &jac =
        this->cell_type == internal::MatrixFreeFunctions::general ?
          this->jacobian[q_point] :
          this->jacobian[0];
      VectorizedArrayType weighted[dim][dim];
      for (unsigned int i = 0; i < dim; ++i)
        weighted[i][i] = sym_grad.access_raw_entry(i) * JxW;
      for (unsigned int i = 0, counter = dim; i < dim; ++i)
        for (unsigned int j = i + 1; j < dim; ++j, ++counter)
          {
            const VectorizedArrayType value =
              sym_grad.access_raw_entry(counter) * JxW;
            weighted[i][j] = value;
            weighted[j][i] = value;
          }
      for (unsigned int comp = 0; comp < dim; ++comp)
        for (unsigned int d = 0; d < dim; ++d)
          {
            VectorizedArrayType new_val = jac[0][d] * weighted[comp][0];
            for (unsigned int e = 1; e < dim; ++e)
              new_val += jac[e][d] * weighted[comp][e];
            this->gradients_quad[(comp * dim + d) * nqp + q_point] = new_val;
          }
    }
}



template <int dim, typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<dim, dim, Number, is_face, VectorizedArrayType>::submit_curl(
  const Tensor<1, dim == 2 ? 1 : dim, VectorizedArrayType> curl,
  const unsigned int                                       q_point)
{
  Tensor<2, dim, VectorizedArrayType> grad;
  switch (dim)
    {
      case 1:
        Assert(false,
               ExcMessage(
                 "Testing by the curl in 1d is not a useful operation"));
        break;
      case 2:
        grad[1][0] = curl[0];
        grad[0][1] = -curl[0];
        break;
      case 3:
        grad[2][1] = curl[0];
        grad[1][2] = -curl[0];
        grad[0][2] = curl[1];
        grad[2][0] = -curl[1];
        grad[1][0] = curl[2];
        grad[0][1] = -curl[2];
        break;
      default:
        Assert(false, ExcNotImplemented());
    }
  submit_gradient(grad, q_point);
}


/*-------------------- FEEvaluationAccess scalar for 1d ---------------------*/


template <typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::
  FEEvaluationAccess(const MatrixFree<1, Number, VectorizedArrayType> &data_in,
                     const unsigned int                                dof_no,
                     const unsigned int first_selected_component,
                     const unsigned int quad_no,
                     const unsigned int fe_degree,
                     const unsigned int n_q_points,
                     const bool         is_interior_face,
                     const unsigned int active_fe_index,
                     const unsigned int active_quad_index,
                     const unsigned int face_type)
  : FEEvaluationBase<1, 1, Number, is_face, VectorizedArrayType>(
      data_in,
      dof_no,
      first_selected_component,
      quad_no,
      fe_degree,
      n_q_points,
      is_interior_face,
      active_fe_index,
      active_quad_index,
      face_type)
{}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::
  FEEvaluationAccess(
    const Mapping<1> &      mapping,
    const FiniteElement<1> &fe,
    const Quadrature<1> &   quadrature,
    const UpdateFlags       update_flags,
    const unsigned int      first_selected_component,
    const FEEvaluationData<1, VectorizedArrayType, is_face> *other)
  : FEEvaluationBase<1, 1, Number, is_face, VectorizedArrayType>(
      mapping,
      fe,
      quadrature,
      update_flags,
      first_selected_component,
      other)
{}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::
  FEEvaluationAccess(
    const FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType> &other)
  : FEEvaluationBase<1, 1, Number, is_face, VectorizedArrayType>(other)
{}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType> &
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::operator=(
  const FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType> &other)
{
  this->FEEvaluationBase<1, 1, Number, is_face, VectorizedArrayType>::operator=(
    other);
  return *this;
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE VectorizedArrayType
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::get_dof_value(
  const unsigned int dof) const
{
  AssertIndexRange(dof, this->data->dofs_per_component_on_cell);
  return this->values_dofs[dof];
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE VectorizedArrayType
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::get_value(
  const unsigned int q_point) const
{
#  ifdef DEBUG
  Assert(this->values_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif
  AssertIndexRange(q_point, this->n_quadrature_points);
  return this->values_quad[q_point];
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE Tensor<1, 1, VectorizedArrayType>
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::get_gradient(
  const unsigned int q_point) const
{
  // could use the base class gradient, but that involves too many inefficient
  // initialization operations on tensors

#  ifdef DEBUG
  Assert(this->gradients_quad_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif
  AssertIndexRange(q_point, this->n_quadrature_points);

  const Tensor<2, 1, VectorizedArrayType> &jac =
    this->cell_type == internal::MatrixFreeFunctions::general ?
      this->jacobian[q_point] :
      this->jacobian[0];

  Tensor<1, 1, VectorizedArrayType> grad_out;
  grad_out[0] = jac[0][0] * this->gradients_quad[q_point];

  return grad_out;
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE VectorizedArrayType
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::get_divergence(
  const unsigned int q_point) const
{
  return get_gradient(q_point)[0];
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE VectorizedArrayType
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::
  get_normal_derivative(const unsigned int q_point) const
{
  return BaseClass::get_normal_derivative(q_point)[0];
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE Tensor<2, 1, VectorizedArrayType>
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::get_hessian(
  const unsigned int q_point) const
{
  return BaseClass::get_hessian(q_point)[0];
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE Tensor<1, 1, VectorizedArrayType>
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::
  get_hessian_diagonal(const unsigned int q_point) const
{
  return BaseClass::get_hessian_diagonal(q_point)[0];
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE VectorizedArrayType
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::get_laplacian(
  const unsigned int q_point) const
{
  return BaseClass::get_laplacian(q_point)[0];
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void DEAL_II_ALWAYS_INLINE
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::
  submit_dof_value(const VectorizedArrayType val_in, const unsigned int dof)
{
#  ifdef DEBUG
  this->dof_values_initialized = true;
  AssertIndexRange(dof, this->data->dofs_per_component_on_cell);
#  endif
  this->values_dofs[dof] = val_in;
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::submit_value(
  const VectorizedArrayType val_in,
  const unsigned int        q_point)
{
  Assert(this->cell != numbers::invalid_unsigned_int, ExcNotInitialized());
  AssertIndexRange(q_point, this->n_quadrature_points);
#  ifdef DEBUG
  this->values_quad_submitted = true;
#  endif

  if (this->cell_type == internal::MatrixFreeFunctions::general)
    {
      const VectorizedArrayType JxW = this->J_value[q_point];
      this->values_quad[q_point]    = val_in * JxW;
    }
  else // if (this->cell_type == internal::MatrixFreeFunctions::general)
    {
      const VectorizedArrayType JxW =
        this->J_value[0] * this->quadrature_weights[q_point];
      this->values_quad[q_point] = val_in * JxW;
    }
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::submit_value(
  const Tensor<1, 1, VectorizedArrayType> val_in,
  const unsigned int                      q_point)
{
  submit_value(val_in[0], q_point);
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::submit_gradient(
  const Tensor<1, 1, VectorizedArrayType> grad_in,
  const unsigned int                      q_point)
{
  submit_gradient(grad_in[0], q_point);
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::submit_gradient(
  const VectorizedArrayType grad_in,
  const unsigned int        q_point)
{
  Assert(this->cell != numbers::invalid_unsigned_int, ExcNotInitialized());
  AssertIndexRange(q_point, this->n_quadrature_points);
#  ifdef DEBUG
  this->gradients_quad_submitted = true;
#  endif

  const Tensor<2, 1, VectorizedArrayType> &jac =
    this->cell_type == internal::MatrixFreeFunctions::general ?
      this->jacobian[q_point] :
      this->jacobian[0];
  const VectorizedArrayType JxW =
    this->cell_type == internal::MatrixFreeFunctions::general ?
      this->J_value[q_point] :
      this->J_value[0] * this->quadrature_weights[q_point];

  this->gradients_quad[q_point] = jac[0][0] * grad_in * JxW;
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::submit_gradient(
  const Tensor<2, 1, VectorizedArrayType> grad_in,
  const unsigned int                      q_point)
{
  submit_gradient(grad_in[0][0], q_point);
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::
  submit_normal_derivative(const VectorizedArrayType grad_in,
                           const unsigned int        q_point)
{
  Tensor<1, 1, VectorizedArrayType> grad;
  grad[0] = grad_in;
  BaseClass::submit_normal_derivative(grad, q_point);
}



template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::
  submit_normal_derivative(const Tensor<1, 1, VectorizedArrayType> grad_in,
                           const unsigned int                      q_point)
{
  BaseClass::submit_normal_derivative(grad_in, q_point);
}


template <typename Number, bool is_face, typename VectorizedArrayType>
inline DEAL_II_ALWAYS_INLINE void
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::submit_hessian(
  const Tensor<2, 1, VectorizedArrayType> hessian_in,
  const unsigned int                      q_point)
{
  Tensor<1, 1, Tensor<2, 1, VectorizedArrayType>> hessian;
  hessian[0] = hessian_in;
  BaseClass::submit_hessian(hessian, q_point);
}


template <typename Number, bool is_face, typename VectorizedArrayType>
inline VectorizedArrayType
FEEvaluationAccess<1, 1, Number, is_face, VectorizedArrayType>::
  integrate_value() const
{
  return BaseClass::integrate_value()[0];
}



/*-------------------------- FEEvaluation -----------------------------------*/


template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline FEEvaluation<dim,
                    fe_degree,
                    n_q_points_1d,
                    n_components_,
                    Number,
                    VectorizedArrayType>::
  FEEvaluation(const MatrixFree<dim, Number, VectorizedArrayType> &data_in,
               const unsigned int                                  fe_no,
               const unsigned int                                  quad_no,
               const unsigned int first_selected_component,
               const unsigned int active_fe_index,
               const unsigned int active_quad_index)
  : BaseClass(data_in,
              fe_no,
              first_selected_component,
              quad_no,
              fe_degree,
              static_n_q_points,
              true /*note: this is not a face*/,
              active_fe_index,
              active_quad_index)
  , dofs_per_component(this->data->dofs_per_component_on_cell)
  , dofs_per_cell(this->data->dofs_per_component_on_cell * n_components_)
  , n_q_points(this->data->n_q_points)
{
  check_template_arguments(fe_no, 0);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline FEEvaluation<dim,
                    fe_degree,
                    n_q_points_1d,
                    n_components_,
                    Number,
                    VectorizedArrayType>::
  FEEvaluation(const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
               const std::pair<unsigned int, unsigned int> &       range,
               const unsigned int                                  dof_no,
               const unsigned int                                  quad_no,
               const unsigned int first_selected_component)
  : FEEvaluation(matrix_free,
                 dof_no,
                 quad_no,
                 first_selected_component,
                 matrix_free.get_cell_active_fe_index(range))
{}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline FEEvaluation<dim,
                    fe_degree,
                    n_q_points_1d,
                    n_components_,
                    Number,
                    VectorizedArrayType>::
  FEEvaluation(const Mapping<dim> &      mapping,
               const FiniteElement<dim> &fe,
               const Quadrature<1> &     quadrature,
               const UpdateFlags         update_flags,
               const unsigned int        first_selected_component)
  : BaseClass(mapping,
              fe,
              quadrature,
              update_flags,
              first_selected_component,
              nullptr)
  , dofs_per_component(this->data->dofs_per_component_on_cell)
  , dofs_per_cell(this->data->dofs_per_component_on_cell * n_components_)
  , n_q_points(this->data->n_q_points)
{
  check_template_arguments(numbers::invalid_unsigned_int, 0);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline FEEvaluation<dim,
                    fe_degree,
                    n_q_points_1d,
                    n_components_,
                    Number,
                    VectorizedArrayType>::
  FEEvaluation(const FiniteElement<dim> &fe,
               const Quadrature<1> &     quadrature,
               const UpdateFlags         update_flags,
               const unsigned int        first_selected_component)
  : BaseClass(StaticMappingQ1<dim>::mapping,
              fe,
              quadrature,
              update_flags,
              first_selected_component,
              nullptr)
  , dofs_per_component(this->data->dofs_per_component_on_cell)
  , dofs_per_cell(this->data->dofs_per_component_on_cell * n_components_)
  , n_q_points(this->data->n_q_points)
{
  check_template_arguments(numbers::invalid_unsigned_int, 0);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline FEEvaluation<dim,
                    fe_degree,
                    n_q_points_1d,
                    n_components_,
                    Number,
                    VectorizedArrayType>::
  FEEvaluation(const FiniteElement<dim> &                               fe,
               const FEEvaluationData<dim, VectorizedArrayType, false> &other,
               const unsigned int first_selected_component)
  : BaseClass(other.mapped_geometry->get_fe_values().get_mapping(),
              fe,
              other.mapped_geometry->get_quadrature(),
              other.mapped_geometry->get_fe_values().get_update_flags(),
              first_selected_component,
              &other)
  , dofs_per_component(this->data->dofs_per_component_on_cell)
  , dofs_per_cell(this->data->dofs_per_component_on_cell * n_components_)
  , n_q_points(this->data->n_q_points)
{
  check_template_arguments(numbers::invalid_unsigned_int, 0);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline FEEvaluation<dim,
                    fe_degree,
                    n_q_points_1d,
                    n_components_,
                    Number,
                    VectorizedArrayType>::FEEvaluation(const FEEvaluation
                                                         &other)
  : BaseClass(other)
  , dofs_per_component(this->data->dofs_per_component_on_cell)
  , dofs_per_cell(this->data->dofs_per_component_on_cell * n_components_)
  , n_q_points(this->data->n_q_points)
{
  check_template_arguments(numbers::invalid_unsigned_int, 0);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline FEEvaluation<dim,
                    fe_degree,
                    n_q_points_1d,
                    n_components_,
                    Number,
                    VectorizedArrayType> &
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::operator=(const FEEvaluation &other)
{
  BaseClass::operator=(other);
  check_template_arguments(numbers::invalid_unsigned_int, 0);
  return *this;
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::
  check_template_arguments(const unsigned int dof_no,
                           const unsigned int first_selected_component)
{
  (void)dof_no;
  (void)first_selected_component;

#  ifdef DEBUG
  // print error message when the dimensions do not match. Propose a possible
  // fix
  if ((static_cast<unsigned int>(fe_degree) != numbers::invalid_unsigned_int &&
       static_cast<unsigned int>(fe_degree) !=
         this->data->data.front().fe_degree) ||
      n_q_points != this->n_quadrature_points)
    {
      std::string message =
        "-------------------------------------------------------\n";
      message += "Illegal arguments in constructor/wrong template arguments!\n";
      message += "    Called -->   FEEvaluation<dim,";
      message += Utilities::int_to_string(fe_degree) + ",";
      message += Utilities::int_to_string(n_q_points_1d);
      message += "," + Utilities::int_to_string(n_components);
      message += ",Number>(data";
      if (first_selected_component != numbers::invalid_unsigned_int)
        {
          message += ", " + Utilities::int_to_string(dof_no) + ", ";
          message += Utilities::int_to_string(this->quad_no) + ", ";
          message += Utilities::int_to_string(first_selected_component);
        }
      message += ")\n";

      // check whether some other vector component has the correct number of
      // points
      unsigned int proposed_dof_comp  = numbers::invalid_unsigned_int,
                   proposed_fe_comp   = numbers::invalid_unsigned_int,
                   proposed_quad_comp = numbers::invalid_unsigned_int;
      if (dof_no != numbers::invalid_unsigned_int)
        {
          if (static_cast<unsigned int>(fe_degree) ==
              this->data->data.front().fe_degree)
            {
              proposed_dof_comp = dof_no;
              proposed_fe_comp  = first_selected_component;
            }
          else
            for (unsigned int no = 0; no < this->matrix_free->n_components();
                 ++no)
              for (unsigned int nf = 0;
                   nf < this->matrix_free->n_base_elements(no);
                   ++nf)
                if (this->matrix_free
                      ->get_shape_info(no, 0, nf, this->active_fe_index, 0)
                      .data.front()
                      .fe_degree == static_cast<unsigned int>(fe_degree))
                  {
                    proposed_dof_comp = no;
                    proposed_fe_comp  = nf;
                    break;
                  }
          if (n_q_points ==
              this->mapping_data->descriptor[this->active_quad_index]
                .n_q_points)
            proposed_quad_comp = this->quad_no;
          else
            for (unsigned int no = 0;
                 no < this->matrix_free->get_mapping_info().cell_data.size();
                 ++no)
              if (this->matrix_free->get_mapping_info()
                    .cell_data[no]
                    .descriptor[this->active_quad_index]
                    .n_q_points == n_q_points)
                {
                  proposed_quad_comp = no;
                  break;
                }
        }
      if (proposed_dof_comp != numbers::invalid_unsigned_int &&
          proposed_quad_comp != numbers::invalid_unsigned_int)
        {
          if (proposed_dof_comp != first_selected_component)
            message += "Wrong vector component selection:\n";
          else
            message += "Wrong quadrature formula selection:\n";
          message += "    Did you mean FEEvaluation<dim,";
          message += Utilities::int_to_string(fe_degree) + ",";
          message += Utilities::int_to_string(n_q_points_1d);
          message += "," + Utilities::int_to_string(n_components);
          message += ",Number>(data";
          if (dof_no != numbers::invalid_unsigned_int)
            {
              message +=
                ", " + Utilities::int_to_string(proposed_dof_comp) + ", ";
              message += Utilities::int_to_string(proposed_quad_comp) + ", ";
              message += Utilities::int_to_string(proposed_fe_comp);
            }
          message += ")?\n";
          std::string correct_pos;
          if (proposed_dof_comp != dof_no)
            correct_pos = " ^ ";
          else
            correct_pos = "   ";
          if (proposed_quad_comp != this->quad_no)
            correct_pos += " ^ ";
          else
            correct_pos += "   ";
          if (proposed_fe_comp != first_selected_component)
            correct_pos += " ^\n";
          else
            correct_pos += "  \n";
          message += "                                                     " +
                     correct_pos;
        }
      // ok, did not find the numbers specified by the template arguments in
      // the given list. Suggest correct template arguments
      const unsigned int proposed_n_q_points_1d = static_cast<unsigned int>(
        std::pow(1.001 * this->n_quadrature_points, 1. / dim));
      message += "Wrong template arguments:\n";
      message += "    Did you mean FEEvaluation<dim,";
      message +=
        Utilities::int_to_string(this->data->data.front().fe_degree) + ",";
      message += Utilities::int_to_string(proposed_n_q_points_1d);
      message += "," + Utilities::int_to_string(n_components);
      message += ",Number>(data";
      if (dof_no != numbers::invalid_unsigned_int)
        {
          message += ", " + Utilities::int_to_string(dof_no) + ", ";
          message += Utilities::int_to_string(this->quad_no);
          message += ", " + Utilities::int_to_string(first_selected_component);
        }
      message += ")?\n";
      std::string correct_pos;
      if (this->data->data.front().fe_degree !=
          static_cast<unsigned int>(fe_degree))
        correct_pos = " ^";
      else
        correct_pos = "  ";
      if (proposed_n_q_points_1d != n_q_points_1d)
        correct_pos += " ^\n";
      else
        correct_pos += "  \n";
      message += "                                 " + correct_pos;

      Assert(static_cast<unsigned int>(fe_degree) ==
                 this->data->data.front().fe_degree &&
               n_q_points == this->n_quadrature_points,
             ExcMessage(message));
    }
  if (dof_no != numbers::invalid_unsigned_int)
    AssertDimension(
      n_q_points,
      this->mapping_data->descriptor[this->active_quad_index].n_q_points);
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::reinit(const unsigned int cell_index)
{
  Assert(this->mapped_geometry == nullptr,
         ExcMessage("FEEvaluation was initialized without a matrix-free object."
                    " Integer indexing is not possible"));
  if (this->mapped_geometry != nullptr)
    return;

  Assert(this->dof_info != nullptr, ExcNotInitialized());
  Assert(this->mapping_data != nullptr, ExcNotInitialized());
  this->cell = cell_index;
  this->cell_type =
    this->matrix_free->get_mapping_info().get_cell_type(cell_index);

  const unsigned int offsets =
    this->mapping_data->data_index_offsets[cell_index];
  this->jacobian = &this->mapping_data->jacobians[0][offsets];
  this->J_value  = &this->mapping_data->JxW_values[offsets];

  for (unsigned int i = 0; i < VectorizedArrayType::size(); ++i)
    this->cell_ids[i] = cell_index * VectorizedArrayType::size() + i;

#  ifdef DEBUG
  this->dof_values_initialized     = false;
  this->values_quad_initialized    = false;
  this->gradients_quad_initialized = false;
  this->hessians_quad_initialized  = false;
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
template <bool level_dof_access>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::
  reinit(const TriaIterator<DoFCellAccessor<dim, dim, level_dof_access>> &cell)
{
  Assert(this->matrix_free == nullptr,
         ExcMessage("Cannot use initialization from cell iterator if "
                    "initialized from MatrixFree object. Use variant for "
                    "on the fly computation with arguments as for FEValues "
                    "instead"));
  Assert(this->mapped_geometry.get() != nullptr, ExcNotInitialized());
  this->mapped_geometry->reinit(
    static_cast<typename Triangulation<dim>::cell_iterator>(cell));
  this->local_dof_indices.resize(cell->get_fe().n_dofs_per_cell());
  if (level_dof_access)
    cell->get_mg_dof_indices(this->local_dof_indices);
  else
    cell->get_dof_indices(this->local_dof_indices);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::
  reinit(const typename Triangulation<dim>::cell_iterator &cell)
{
  Assert(this->matrix_free == 0,
         ExcMessage("Cannot use initialization from cell iterator if "
                    "initialized from MatrixFree object. Use variant for "
                    "on the fly computation with arguments as for FEValues "
                    "instead"));
  Assert(this->mapped_geometry.get() != 0, ExcNotInitialized());
  this->mapped_geometry->reinit(cell);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline Point<dim, VectorizedArrayType>
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::quadrature_point(const unsigned int q) const
{
  if (this->matrix_free == nullptr)
    {
      Assert((this->mapped_geometry->get_fe_values().get_update_flags() |
              update_quadrature_points),
             internal::ExcMatrixFreeAccessToUninitializedMappingField(
               "update_quadrature_points"));
    }
  else
    {
      Assert(this->mapping_data->quadrature_point_offsets.empty() == false,
             internal::ExcMatrixFreeAccessToUninitializedMappingField(
               "update_quadrature_points"));
    }

  AssertIndexRange(q, n_q_points);

  const Point<dim, VectorizedArrayType> *quadrature_points =
    &this->mapping_data->quadrature_points
       [this->mapping_data->quadrature_point_offsets[this->cell]];

  // Cartesian/affine mesh: only first vertex of cell is stored, we must
  // compute it through the Jacobian (which is stored in non-inverted and
  // non-transposed form as index '1' in the jacobian field)
  if (this->cell_type <= internal::MatrixFreeFunctions::affine)
    {
      Assert(this->jacobian != nullptr, ExcNotInitialized());
      Point<dim, VectorizedArrayType> point = quadrature_points[0];

      const Tensor<2, dim, VectorizedArrayType> &jac = this->jacobian[1];
      if (this->cell_type == internal::MatrixFreeFunctions::cartesian)
        for (unsigned int d = 0; d < dim; ++d)
          point[d] += jac[d][d] * static_cast<Number>(
                                    this->descriptor->quadrature.point(q)[d]);
      else
        for (unsigned int d = 0; d < dim; ++d)
          for (unsigned int e = 0; e < dim; ++e)
            point[d] += jac[d][e] * static_cast<Number>(
                                      this->descriptor->quadrature.point(q)[e]);
      return point;
    }
  else
    return quadrature_points[q];
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::evaluate(const bool evaluate_values,
                                            const bool evaluate_gradients,
                                            const bool evaluate_hessians)
{
#  ifdef DEBUG
  Assert(this->dof_values_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif
  evaluate(this->values_dofs,
           evaluate_values,
           evaluate_gradients,
           evaluate_hessians);
}


template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::
  evaluate(const EvaluationFlags::EvaluationFlags evaluation_flags)
{
#  ifdef DEBUG
  Assert(this->dof_values_initialized == true,
         internal::ExcAccessToUninitializedField());
#  endif
  evaluate(this->values_dofs, evaluation_flags);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::evaluate(const VectorizedArrayType
                                              *        values_array,
                                            const bool evaluate_values,
                                            const bool evaluate_gradients,
                                            const bool evaluate_hessians)
{
  const EvaluationFlags::EvaluationFlags flag =
    ((evaluate_values) ? EvaluationFlags::values : EvaluationFlags::nothing) |
    ((evaluate_gradients) ? EvaluationFlags::gradients :
                            EvaluationFlags::nothing) |
    ((evaluate_hessians) ? EvaluationFlags::hessians :
                           EvaluationFlags::nothing);

  evaluate(values_array, flag);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::
  evaluate(const VectorizedArrayType *            values_array,
           const EvaluationFlags::EvaluationFlags evaluation_flag)
{
  const bool hessians_on_general_cells =
    evaluation_flag & EvaluationFlags::hessians &&
    (this->cell_type > internal::MatrixFreeFunctions::affine);
  EvaluationFlags::EvaluationFlags evaluation_flag_actual = evaluation_flag;
  if (hessians_on_general_cells)
    evaluation_flag_actual |= EvaluationFlags::gradients;

  if (fe_degree > -1)
    {
      SelectEvaluator<dim, fe_degree, n_q_points_1d, VectorizedArrayType>::
        evaluate(n_components, evaluation_flag_actual, values_array, *this);
    }
  else
    {
      internal::FEEvaluationFactory<dim, VectorizedArrayType>::evaluate(
        n_components,
        evaluation_flag_actual,
        const_cast<VectorizedArrayType *>(values_array),
        *this);
    }

#  ifdef DEBUG
  if (evaluation_flag_actual & EvaluationFlags::values)
    this->values_quad_initialized = true;
  if (evaluation_flag_actual & EvaluationFlags::gradients)
    this->gradients_quad_initialized = true;
  if (evaluation_flag_actual & EvaluationFlags::hessians)
    this->hessians_quad_initialized = true;
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEEvaluation<
  dim,
  fe_degree,
  n_q_points_1d,
  n_components_,
  Number,
  VectorizedArrayType>::gather_evaluate(const VectorType &input_vector,
                                        const bool        evaluate_values,
                                        const bool        evaluate_gradients,
                                        const bool        evaluate_hessians)
{
  const EvaluationFlags::EvaluationFlags flag =
    ((evaluate_values) ? EvaluationFlags::values : EvaluationFlags::nothing) |
    ((evaluate_gradients) ? EvaluationFlags::gradients :
                            EvaluationFlags::nothing) |
    ((evaluate_hessians) ? EvaluationFlags::hessians :
                           EvaluationFlags::nothing);

  gather_evaluate(input_vector, flag);
}


namespace internal
{
  /**
   * Implementation for standard vectors (that have the begin() methods).
   */
  template <typename Number,
            typename VectorizedArrayType,
            typename VectorType,
            typename EvaluatorType,
            typename std::enable_if<
              internal::has_begin<VectorType>::value &&
                std::is_same<decltype(std::declval<VectorType>().begin()),
                             Number *>::value,
              VectorType>::type * = nullptr>
  VectorizedArrayType *
  check_vector_access_inplace(const EvaluatorType &fe_eval, VectorType &vector)
  {
    const unsigned int cell     = fe_eval.get_cell_or_face_batch_id();
    const auto &       dof_info = fe_eval.get_dof_info();

    // If the index storage is interleaved and contiguous and the vector
    // storage has the correct alignment, we can directly pass the pointer
    // into the vector to the evaluate() and integrate() calls, without
    // reading the vector entries into a separate data field. This saves some
    // operations.
    if (std::is_same<typename VectorType::value_type, Number>::value &&
        dof_info.index_storage_variants
            [internal::MatrixFreeFunctions::DoFInfo::dof_access_cell][cell] ==
          internal::MatrixFreeFunctions::DoFInfo::IndexStorageVariants::
            interleaved_contiguous &&
        reinterpret_cast<std::size_t>(
          vector.begin() +
          dof_info.dof_indices_contiguous
            [internal::MatrixFreeFunctions::DoFInfo::dof_access_cell]
            [cell * VectorizedArrayType::size()]) %
            sizeof(VectorizedArrayType) ==
          0)
      {
        return reinterpret_cast<VectorizedArrayType *>(
          vector.begin() +
          dof_info.dof_indices_contiguous
            [internal::MatrixFreeFunctions::DoFInfo::dof_access_cell]
            [cell * VectorizedArrayType::size()] +
          dof_info.component_dof_indices_offset
              [fe_eval.get_active_fe_index()]
              [fe_eval.get_first_selected_component()] *
            VectorizedArrayType::size());
      }
    else
      return nullptr;
  }

  /**
   * Implementation for block vectors.
   */
  template <typename Number,
            typename VectorizedArrayType,
            typename VectorType,
            typename EvaluatorType,
            typename std::enable_if<
              !internal::has_begin<VectorType>::value ||
                !std::is_same<decltype(std::declval<VectorType>().begin()),
                              Number *>::value,
              VectorType>::type * = nullptr>
  VectorizedArrayType *
  check_vector_access_inplace(const EvaluatorType &, VectorType &)
  {
    return nullptr;
  }
} // namespace internal



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::
  gather_evaluate(const VectorType &                     input_vector,
                  const EvaluationFlags::EvaluationFlags evaluation_flag)
{
  const VectorizedArrayType *src_ptr =
    internal::check_vector_access_inplace<Number, const VectorizedArrayType>(
      *this, input_vector);
  if (src_ptr != nullptr)
    evaluate(src_ptr, evaluation_flag);
  else
    {
      this->read_dof_values(input_vector);
      evaluate(this->begin_dof_values(), evaluation_flag);
    }
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::integrate(const bool integrate_values,
                                             const bool integrate_gradients)
{
  integrate(integrate_values, integrate_gradients, this->values_dofs);

#  ifdef DEBUG
  this->dof_values_initialized = true;
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::
  integrate(const EvaluationFlags::EvaluationFlags integration_flag)
{
  integrate(integration_flag, this->values_dofs);

#  ifdef DEBUG
  this->dof_values_initialized = true;
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::integrate(const bool integrate_values,
                                             const bool integrate_gradients,
                                             VectorizedArrayType *values_array)
{
  EvaluationFlags::EvaluationFlags flag =
    (integrate_values ? EvaluationFlags::values : EvaluationFlags::nothing) |
    (integrate_gradients ? EvaluationFlags::gradients :
                           EvaluationFlags::nothing);
  integrate(flag, values_array);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::
  integrate(const EvaluationFlags::EvaluationFlags integration_flag,
            VectorizedArrayType *                  values_array,
            const bool                             sum_into_values_array)
{
#  ifdef DEBUG
  if (integration_flag & EvaluationFlags::values)
    Assert(this->values_quad_submitted == true,
           internal::ExcAccessToUninitializedField());
  if (integration_flag & EvaluationFlags::gradients)
    Assert(this->gradients_quad_submitted == true,
           internal::ExcAccessToUninitializedField());
  if (integration_flag & EvaluationFlags::hessians)
    Assert(this->hessians_quad_submitted == true,
           internal::ExcAccessToUninitializedField());
#  endif
  Assert(this->matrix_free != nullptr ||
           this->mapped_geometry->is_initialized(),
         ExcNotInitialized());

  Assert(
    (integration_flag & ~(EvaluationFlags::values | EvaluationFlags::gradients |
                          EvaluationFlags::hessians)) == 0,
    ExcMessage("Only EvaluationFlags::values, EvaluationFlags::gradients, and "
               "EvaluationFlags::hessians are supported."));

  EvaluationFlags::EvaluationFlags integration_flag_actual = integration_flag;
  if (integration_flag & EvaluationFlags::hessians &&
      (this->cell_type > internal::MatrixFreeFunctions::affine))
    {
      unsigned int size = n_components * dim * n_q_points;
      if (integration_flag & EvaluationFlags::gradients)
        {
          for (unsigned int i = 0; i < size; ++i)
            this->gradients_quad[i] += this->gradients_from_hessians_quad[i];
        }
      else
        {
          for (unsigned int i = 0; i < size; ++i)
            this->gradients_quad[i] = this->gradients_from_hessians_quad[i];
          integration_flag_actual |= EvaluationFlags::gradients;
        }
    }

  if (fe_degree > -1)
    {
      SelectEvaluator<dim, fe_degree, n_q_points_1d, VectorizedArrayType>::
        integrate(n_components,
                  integration_flag_actual,
                  values_array,
                  *this,
                  sum_into_values_array);
    }
  else
    {
      internal::FEEvaluationFactory<dim, VectorizedArrayType>::integrate(
        n_components,
        integration_flag_actual,
        values_array,
        *this,
        sum_into_values_array);
    }

#  ifdef DEBUG
  this->dof_values_initialized = true;
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEEvaluation<
  dim,
  fe_degree,
  n_q_points_1d,
  n_components_,
  Number,
  VectorizedArrayType>::integrate_scatter(const bool  integrate_values,
                                          const bool  integrate_gradients,
                                          VectorType &destination)
{
  const EvaluationFlags::EvaluationFlags flag =
    ((integrate_values) ? EvaluationFlags::values : EvaluationFlags::nothing) |
    ((integrate_gradients) ? EvaluationFlags::gradients :
                             EvaluationFlags::nothing);

  integrate_scatter(flag, destination);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::
  integrate_scatter(const EvaluationFlags::EvaluationFlags integration_flag,
                    VectorType &                           destination)
{
  VectorizedArrayType *dst_ptr =
    internal::check_vector_access_inplace<Number, VectorizedArrayType>(
      *this, destination);
  if (dst_ptr != nullptr)
    integrate(integration_flag, dst_ptr, true);
  else
    {
      integrate(integration_flag, this->begin_dof_values());
      this->distribute_local_to_global(destination);
    }
}



/*-------------------------- FEFaceEvaluation ---------------------------*/



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline FEFaceEvaluation<dim,
                        fe_degree,
                        n_q_points_1d,
                        n_components_,
                        Number,
                        VectorizedArrayType>::
  FEFaceEvaluation(
    const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
    const bool                                          is_interior_face,
    const unsigned int                                  dof_no,
    const unsigned int                                  quad_no,
    const unsigned int first_selected_component,
    const unsigned int active_fe_index,
    const unsigned int active_quad_index,
    const unsigned int face_type)
  : BaseClass(matrix_free,
              dof_no,
              first_selected_component,
              quad_no,
              fe_degree,
              static_n_q_points,
              is_interior_face,
              active_fe_index,
              active_quad_index,
              face_type)
  , dofs_per_component(this->data->dofs_per_component_on_cell)
  , dofs_per_cell(this->data->dofs_per_component_on_cell * n_components_)
  , n_q_points(this->n_quadrature_points)
{}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline FEFaceEvaluation<dim,
                        fe_degree,
                        n_q_points_1d,
                        n_components_,
                        Number,
                        VectorizedArrayType>::
  FEFaceEvaluation(
    const MatrixFree<dim, Number, VectorizedArrayType> &matrix_free,
    const std::pair<unsigned int, unsigned int> &       range,
    const bool                                          is_interior_face,
    const unsigned int                                  dof_no,
    const unsigned int                                  quad_no,
    const unsigned int first_selected_component)
  : FEFaceEvaluation(matrix_free,
                     is_interior_face,
                     dof_no,
                     quad_no,
                     first_selected_component,
                     matrix_free.get_face_active_fe_index(range,
                                                          is_interior_face),
                     numbers::invalid_unsigned_int,
                     matrix_free.get_face_info(range.first).face_type)
{}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components_,
                 Number,
                 VectorizedArrayType>::reinit(const unsigned int face_index)
{
  Assert(this->mapped_geometry == nullptr,
         ExcMessage("FEEvaluation was initialized without a matrix-free object."
                    " Integer indexing is not possible"));
  if (this->mapped_geometry != nullptr)
    return;

  this->cell = face_index;
  this->dof_access_index =
    this->is_interior_face ?
      internal::MatrixFreeFunctions::DoFInfo::dof_access_face_interior :
      internal::MatrixFreeFunctions::DoFInfo::dof_access_face_exterior;
  Assert(this->mapping_data != nullptr, ExcNotInitialized());

  if (face_index >=
        this->matrix_free->get_task_info().face_partition_data.back() &&
      face_index <
        this->matrix_free->get_task_info().boundary_partition_data.back())
    Assert(this->is_interior_face,
           ExcMessage(
             "Boundary faces do not have a neighbor. When looping over "
             "boundary faces use FEFaceEvaluation with the parameter "
             "is_interior_face set to true. "));

  this->reinit_face(this->matrix_free->get_face_info(face_index));
  for (unsigned int i = 0; i < VectorizedArrayType::size(); ++i)
    this->cell_or_face_ids[i] = face_index * VectorizedArrayType::size() + i;

  this->cell_type = this->matrix_free->get_mapping_info().face_type[face_index];
  const unsigned int offsets =
    this->mapping_data->data_index_offsets[face_index];
  this->J_value        = &this->mapping_data->JxW_values[offsets];
  this->normal_vectors = &this->mapping_data->normal_vectors[offsets];
  this->jacobian =
    &this->mapping_data->jacobians[!this->is_interior_face][offsets];
  this->normal_x_jacobian =
    &this->mapping_data
       ->normals_times_jacobians[!this->is_interior_face][offsets];

#  ifdef DEBUG
  this->dof_values_initialized     = false;
  this->values_quad_initialized    = false;
  this->gradients_quad_initialized = false;
  this->hessians_quad_initialized  = false;
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components_,
                 Number,
                 VectorizedArrayType>::reinit(const unsigned int cell_index,
                                              const unsigned int face_number)
{
  Assert(
    this->quad_no <
      this->matrix_free->get_mapping_info().face_data_by_cells.size(),
    ExcMessage(
      "You must set MatrixFree::AdditionalData::mapping_update_flags_faces_by_cells to use the present reinit method."));
  AssertIndexRange(face_number, GeometryInfo<dim>::faces_per_cell);
  AssertIndexRange(cell_index,
                   this->matrix_free->get_mapping_info().cell_type.size());
  Assert(this->mapped_geometry == nullptr,
         ExcMessage("FEEvaluation was initialized without a matrix-free object."
                    " Integer indexing is not possible"));
  if (this->mapped_geometry != nullptr)
    return;
  Assert(this->matrix_free != nullptr, ExcNotInitialized());

  this->cell_type = this->matrix_free->get_mapping_info().cell_type[cell_index];
  this->cell      = cell_index;
  this->subface_index = GeometryInfo<dim>::max_children_per_cell;
  this->dof_access_index =
    internal::MatrixFreeFunctions::DoFInfo::dof_access_cell;

  constexpr unsigned int n_lanes = VectorizedArrayType::size();

  if (this->is_interior_face == false)
    {
      // for this case, we need to look into the FaceInfo field that collects
      // information from both sides of a face once for the global mesh, and
      // pick the face id that is not the local one (cell_this).
      for (unsigned int i = 0; i < n_lanes; ++i)
        {
          // compute actual (non vectorized) cell ID
          const unsigned int cell_this = cell_index * n_lanes + i;
          // compute face ID
          unsigned int face_index =
            this->matrix_free->get_cell_and_face_to_plain_faces()(cell_index,
                                                                  face_number,
                                                                  i);

          if (face_index == numbers::invalid_unsigned_int)
            {
              this->cell_ids[i]              = numbers::invalid_unsigned_int;
              this->all_face_numbers[i]      = static_cast<std::uint8_t>(-1);
              this->all_face_orientations[i] = static_cast<std::uint8_t>(-1);
              continue; // invalid face ID: no neighbor on boundary
            }

          const auto &faces =
            this->matrix_free->get_face_info(face_index / n_lanes);
          // get cell ID on both sides of face
          auto cell_m = faces.cells_interior[face_index % n_lanes];
          auto cell_p = faces.cells_exterior[face_index % n_lanes];

          const bool face_identifies_as_interior = cell_m != cell_this;

          Assert(cell_m == cell_this || cell_p == cell_this,
                 ExcInternalError());

          // compare the IDs with the given cell ID
          if (face_identifies_as_interior)
            {
              this->cell_ids[i]         = cell_m; // neighbor has the other ID
              this->all_face_numbers[i] = faces.interior_face_no;
            }
          else
            {
              this->cell_ids[i]         = cell_p;
              this->all_face_numbers[i] = faces.exterior_face_no;
            }

          const bool   orientation_interior_face = faces.face_orientation >= 8;
          unsigned int face_orientation          = faces.face_orientation % 8;
          if (face_identifies_as_interior != orientation_interior_face)
            {
              constexpr std::array<std::uint8_t, 8> table{
                {0, 1, 2, 3, 6, 5, 4, 7}};
              face_orientation = table[face_orientation];
            }
          this->all_face_orientations[i] = face_orientation;
        }

      this->face_no          = this->all_face_numbers[0];
      this->face_orientation = this->all_face_orientations[0];
    }
  else
    {
      this->face_orientation = 0;
      this->face_no          = face_number;
      for (unsigned int i = 0; i < n_lanes; ++i)
        this->cell_ids[i] = cell_index * n_lanes + i;
    }

  const unsigned int offsets =
    this->matrix_free->get_mapping_info()
      .face_data_by_cells[this->quad_no]
      .data_index_offsets[cell_index * GeometryInfo<dim>::faces_per_cell +
                          face_number];
  AssertIndexRange(offsets,
                   this->matrix_free->get_mapping_info()
                     .face_data_by_cells[this->quad_no]
                     .JxW_values.size());
  this->J_value = &this->matrix_free->get_mapping_info()
                     .face_data_by_cells[this->quad_no]
                     .JxW_values[offsets];
  this->normal_vectors = &this->matrix_free->get_mapping_info()
                            .face_data_by_cells[this->quad_no]
                            .normal_vectors[offsets];
  this->jacobian = &this->matrix_free->get_mapping_info()
                      .face_data_by_cells[this->quad_no]
                      .jacobians[!this->is_interior_face][offsets];
  this->normal_x_jacobian =
    &this->matrix_free->get_mapping_info()
       .face_data_by_cells[this->quad_no]
       .normals_times_jacobians[!this->is_interior_face][offsets];

#  ifdef DEBUG
  this->dof_values_initialized     = false;
  this->values_quad_initialized    = false;
  this->gradients_quad_initialized = false;
  this->hessians_quad_initialized  = false;
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components,
          typename Number,
          typename VectorizedArrayType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components,
                 Number,
                 VectorizedArrayType>::evaluate(const bool evaluate_values,
                                                const bool evaluate_gradients)
{
#  ifdef DEBUG
  Assert(this->dof_values_initialized, ExcNotInitialized());
#  endif

  evaluate(this->values_dofs, evaluate_values, evaluate_gradients);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components,
          typename Number,
          typename VectorizedArrayType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components,
                 Number,
                 VectorizedArrayType>::
  evaluate(const EvaluationFlags::EvaluationFlags evaluation_flag)
{
#  ifdef DEBUG
  Assert(this->dof_values_initialized, ExcNotInitialized());
#  endif

  evaluate(this->values_dofs, evaluation_flag);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components,
          typename Number,
          typename VectorizedArrayType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components,
                 Number,
                 VectorizedArrayType>::evaluate(const VectorizedArrayType
                                                  *        values_array,
                                                const bool evaluate_values,
                                                const bool evaluate_gradients)
{
  const EvaluationFlags::EvaluationFlags flag =
    ((evaluate_values) ? EvaluationFlags::values : EvaluationFlags::nothing) |
    ((evaluate_gradients) ? EvaluationFlags::gradients :
                            EvaluationFlags::nothing);

  evaluate(values_array, flag);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components,
          typename Number,
          typename VectorizedArrayType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components,
                 Number,
                 VectorizedArrayType>::
  evaluate(const VectorizedArrayType *            values_array,
           const EvaluationFlags::EvaluationFlags evaluation_flag)
{
  Assert((evaluation_flag &
          ~(EvaluationFlags::values | EvaluationFlags::gradients |
            EvaluationFlags::hessians)) == 0,
         ExcMessage("Only EvaluationFlags::values, EvaluationFlags::gradients, "
                    "and EvaluationFlags::hessians are supported."));

  const bool hessians_on_general_cells =
    evaluation_flag & EvaluationFlags::hessians &&
    (this->cell_type > internal::MatrixFreeFunctions::affine);
  EvaluationFlags::EvaluationFlags evaluation_flag_actual = evaluation_flag;
  if (hessians_on_general_cells)
    evaluation_flag_actual |= EvaluationFlags::gradients;

  if (fe_degree > -1)
    internal::FEFaceEvaluationImplEvaluateSelector<dim, VectorizedArrayType>::
      template run<fe_degree, n_q_points_1d>(n_components,
                                             evaluation_flag_actual,
                                             values_array,
                                             *this);
  else
    internal::FEFaceEvaluationFactory<dim, VectorizedArrayType>::evaluate(
      n_components, evaluation_flag_actual, values_array, *this);

#  ifdef DEBUG
  if (evaluation_flag_actual & EvaluationFlags::values)
    this->values_quad_initialized = true;
  if (evaluation_flag_actual & EvaluationFlags::gradients)
    this->gradients_quad_initialized = true;
  if (evaluation_flag_actual & EvaluationFlags::hessians)
    this->hessians_quad_initialized = true;
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components,
          typename Number,
          typename VectorizedArrayType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components,
                 Number,
                 VectorizedArrayType>::
  integrate(const EvaluationFlags::EvaluationFlags integration_flag)
{
  integrate(integration_flag, this->values_dofs);

#  ifdef DEBUG
  this->dof_values_initialized = true;
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components,
          typename Number,
          typename VectorizedArrayType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components,
                 Number,
                 VectorizedArrayType>::integrate(const bool integrate_values,
                                                 const bool integrate_gradients)
{
  integrate(integrate_values, integrate_gradients, this->values_dofs);

#  ifdef DEBUG
  this->dof_values_initialized = true;
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components,
          typename Number,
          typename VectorizedArrayType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components,
                 Number,
                 VectorizedArrayType>::integrate(const bool integrate_values,
                                                 const bool integrate_gradients,
                                                 VectorizedArrayType
                                                   *values_array)
{
  const EvaluationFlags::EvaluationFlags flag =
    ((integrate_values) ? EvaluationFlags::values : EvaluationFlags::nothing) |
    ((integrate_gradients) ? EvaluationFlags::gradients :
                             EvaluationFlags::nothing);

  integrate(flag, values_array);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components,
          typename Number,
          typename VectorizedArrayType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components,
                 Number,
                 VectorizedArrayType>::
  integrate(const EvaluationFlags::EvaluationFlags integration_flag,
            VectorizedArrayType *                  values_array)
{
  Assert((integration_flag &
          ~(EvaluationFlags::values | EvaluationFlags::gradients |
            EvaluationFlags::hessians)) == 0,
         ExcMessage("Only EvaluationFlags::values, EvaluationFlags::gradients, "
                    "and EvaluationFlags::hessians are supported."));

  EvaluationFlags::EvaluationFlags integration_flag_actual = integration_flag;
  if (integration_flag & EvaluationFlags::hessians &&
      (this->cell_type > internal::MatrixFreeFunctions::affine))
    {
      unsigned int size = n_components * dim * n_q_points;
      if (integration_flag & EvaluationFlags::gradients)
        {
          for (unsigned int i = 0; i < size; ++i)
            this->gradients_quad[i] += this->gradients_from_hessians_quad[i];
        }
      else
        {
          for (unsigned int i = 0; i < size; ++i)
            this->gradients_quad[i] = this->gradients_from_hessians_quad[i];
          integration_flag_actual |= EvaluationFlags::gradients;
        }
    }

  if (fe_degree > -1)
    internal::FEFaceEvaluationImplIntegrateSelector<dim, VectorizedArrayType>::
      template run<fe_degree, n_q_points_1d>(n_components,
                                             integration_flag_actual,
                                             values_array,
                                             *this);
  else
    internal::FEFaceEvaluationFactory<dim, VectorizedArrayType>::integrate(
      n_components, integration_flag_actual, values_array, *this);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEFaceEvaluation<
  dim,
  fe_degree,
  n_q_points_1d,
  n_components_,
  Number,
  VectorizedArrayType>::gather_evaluate(const VectorType &input_vector,
                                        const bool        evaluate_values,
                                        const bool        evaluate_gradients)
{
  const EvaluationFlags::EvaluationFlags flag =
    ((evaluate_values) ? EvaluationFlags::values : EvaluationFlags::nothing) |
    ((evaluate_gradients) ? EvaluationFlags::gradients :
                            EvaluationFlags::nothing);

  gather_evaluate(input_vector, flag);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components_,
                 Number,
                 VectorizedArrayType>::
  gather_evaluate(const VectorType &                     input_vector,
                  const EvaluationFlags::EvaluationFlags evaluation_flag)
{
  Assert((evaluation_flag &
          ~(EvaluationFlags::values | EvaluationFlags::gradients |
            EvaluationFlags::hessians)) == 0,
         ExcMessage("Only EvaluationFlags::values, EvaluationFlags::gradients, "
                    "and EvaluationFlags::hessians are supported."));

  const auto shared_vector_data = internal::get_shared_vector_data(
    input_vector,
    this->dof_access_index ==
      internal::MatrixFreeFunctions::DoFInfo::dof_access_cell,
    this->active_fe_index,
    this->dof_info);

  if (this->data->data.front().fe_degree > 0 &&
      fast_evaluation_supported(this->data->data.front().fe_degree,
                                this->data->data.front().n_q_points_1d) &&
      internal::FEFaceEvaluationImplGatherEvaluateSelector<
        dim,
        Number,
        VectorizedArrayType>::supports(evaluation_flag,
                                       *this->data,
                                       internal::get_beginning<Number>(
                                         input_vector),
                                       this->dof_info->index_storage_variants
                                         [this->dof_access_index][this->cell]))
    {
      if (fe_degree > -1)
        {
          internal::FEFaceEvaluationImplGatherEvaluateSelector<
            dim,
            Number,
            VectorizedArrayType>::template run<fe_degree,
                                               n_q_points_1d>(
            n_components,
            evaluation_flag,
            internal::get_beginning<Number>(input_vector),
            shared_vector_data,
            *this);
        }
      else
        {
          internal::FEFaceEvaluationGatherFactory<
            dim,
            Number,
            VectorizedArrayType>::evaluate(n_components,
                                           evaluation_flag,
                                           internal::get_beginning<Number>(
                                             input_vector),
                                           shared_vector_data,
                                           *this);
        }
    }
  else
    {
      this->read_dof_values(input_vector);
      this->evaluate(evaluation_flag);
    }

#  ifdef DEBUG
  if (evaluation_flag & EvaluationFlags::values)
    this->values_quad_initialized = true;
  if (evaluation_flag & EvaluationFlags::gradients)
    this->gradients_quad_initialized = true;
  if (evaluation_flag & EvaluationFlags::hessians)
    this->hessians_quad_initialized = true;
#  endif
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEFaceEvaluation<
  dim,
  fe_degree,
  n_q_points_1d,
  n_components_,
  Number,
  VectorizedArrayType>::integrate_scatter(const bool  integrate_values,
                                          const bool  integrate_gradients,
                                          VectorType &destination)
{
  const EvaluationFlags::EvaluationFlags flag =
    ((integrate_values) ? EvaluationFlags::values : EvaluationFlags::nothing) |
    ((integrate_gradients) ? EvaluationFlags::gradients :
                             EvaluationFlags::nothing);

  integrate_scatter(flag, destination);
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
template <typename VectorType>
inline void
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components_,
                 Number,
                 VectorizedArrayType>::
  integrate_scatter(const EvaluationFlags::EvaluationFlags integration_flag,
                    VectorType &                           destination)
{
  Assert((this->dof_access_index ==
            internal::MatrixFreeFunctions::DoFInfo::dof_access_cell &&
          this->is_interior_face == false) == false,
         ExcNotImplemented());

  const auto shared_vector_data = internal::get_shared_vector_data(
    destination,
    this->dof_access_index ==
      internal::MatrixFreeFunctions::DoFInfo::dof_access_cell,
    this->active_fe_index,
    this->dof_info);

  if (this->data->data.front().fe_degree > 0 &&
      fast_evaluation_supported(this->data->data.front().fe_degree,
                                this->data->data.front().n_q_points_1d) &&
      internal::FEFaceEvaluationImplGatherEvaluateSelector<
        dim,
        Number,
        VectorizedArrayType>::supports(integration_flag,
                                       *this->data,
                                       internal::get_beginning<Number>(
                                         destination),
                                       this->dof_info->index_storage_variants
                                         [this->dof_access_index][this->cell]))
    {
      if (fe_degree > -1)
        {
          internal::FEFaceEvaluationImplIntegrateScatterSelector<
            dim,
            Number,
            VectorizedArrayType>::template run<fe_degree,
                                               n_q_points_1d>(
            n_components,
            integration_flag,
            internal::get_beginning<Number>(destination),
            shared_vector_data,
            *this);
        }
      else
        {
          internal::FEFaceEvaluationGatherFactory<
            dim,
            Number,
            VectorizedArrayType>::integrate(n_components,
                                            integration_flag,
                                            internal::get_beginning<Number>(
                                              destination),
                                            shared_vector_data,
                                            *this);
        }
    }
  else
    {
      integrate(integration_flag);
      this->distribute_local_to_global(destination);
    }
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
inline Point<dim, VectorizedArrayType>
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components_,
                 Number,
                 VectorizedArrayType>::quadrature_point(const unsigned int q)
  const
{
  AssertIndexRange(q, n_q_points);
  if (this->dof_access_index < 2)
    {
      Assert(this->mapping_data->quadrature_point_offsets.empty() == false,
             internal::ExcMatrixFreeAccessToUninitializedMappingField(
               "update_quadrature_points"));
      AssertIndexRange(this->cell,
                       this->mapping_data->quadrature_point_offsets.size());
      return this->mapping_data->quadrature_points
        [this->mapping_data->quadrature_point_offsets[this->cell] + q];
    }
  else
    {
      Assert(this->matrix_free->get_mapping_info()
                 .face_data_by_cells[this->quad_no]
                 .quadrature_point_offsets.empty() == false,
             internal::ExcMatrixFreeAccessToUninitializedMappingField(
               "update_quadrature_points"));
      const unsigned int index =
        this->cell * GeometryInfo<dim>::faces_per_cell + this->face_no;
      AssertIndexRange(index,
                       this->matrix_free->get_mapping_info()
                         .face_data_by_cells[this->quad_no]
                         .quadrature_point_offsets.size());
      return this->matrix_free->get_mapping_info()
        .face_data_by_cells[this->quad_no]
        .quadrature_points[this->matrix_free->get_mapping_info()
                             .face_data_by_cells[this->quad_no]
                             .quadrature_point_offsets[index] +
                           q];
    }
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
bool
FEEvaluation<dim,
             fe_degree,
             n_q_points_1d,
             n_components_,
             Number,
             VectorizedArrayType>::
  fast_evaluation_supported(const unsigned int given_degree,
                            const unsigned int give_n_q_points_1d)
{
  return fe_degree == -1 ?
           internal::FEEvaluationFactory<dim, VectorizedArrayType>::
             fast_evaluation_supported(given_degree, give_n_q_points_1d) :
           true;
}



template <int dim,
          int fe_degree,
          int n_q_points_1d,
          int n_components_,
          typename Number,
          typename VectorizedArrayType>
bool
FEFaceEvaluation<dim,
                 fe_degree,
                 n_q_points_1d,
                 n_components_,
                 Number,
                 VectorizedArrayType>::
  fast_evaluation_supported(const unsigned int given_degree,
                            const unsigned int give_n_q_points_1d)
{
  return fe_degree == -1 ?
           internal::FEEvaluationFactory<dim, VectorizedArrayType>::
             fast_evaluation_supported(given_degree, give_n_q_points_1d) :
           true;
}



/*------------------------- end FEFaceEvaluation ------------------------- */


#endif // ifndef DOXYGEN


DEAL_II_NAMESPACE_CLOSE

#endif