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 // MFEM Example 27 // // Compile with: make ex27 // // Sample runs: ex27 // ex27 -dg // ex27 -dg -dbc 8 -nbc -2 // ex27 -rbc-a 1 -rbc-b 8 // // Description: This example code demonstrates the use of MFEM to define a // simple finite element discretization of the Laplace problem // -Delta u = 0 with a variety of boundary conditions. // // Specifically, we discretize using a FE space of the specified // order using a continuous or discontinuous space. We then apply // Dirichlet, Neumann (both homogeneous and inhomogeneous), Robin, // and Periodic boundary conditions on different portions of a // predefined mesh. // // The predefined mesh consists of a rectangle with two holes // removed (see below). The narrow ends of the mesh are connected // to form a Periodic boundary condition. The lower edge (tagged // with attribute 1) receives an inhomogeneous Neumann boundary // condition. A Robin boundary condition is applied to upper edge // (attribute 2). The circular hole on the left (attribute 3) // enforces a Dirichlet boundary condition. Finally, a natural // boundary condition, or homogeneous Neumann BC, is applied to // the circular hole on the right (attribute 4). // // Attribute 3 ^ y Attribute 2 // \ | / // +-----------+-----------+ // | \_ | _ | // | / \ | / \ | // <--+---+---+---+---+---+---+--> x // | \_/ | \_/ | // | | \ | // +-----------+-----------+ (hole radii are // / | \ adjustable) // Attribute 1 v Attribute 4 // // The boundary conditions are defined as (where u is the solution // field): // // Dirichlet: u = d // Neumann: n.Grad(u) = g // Robin: n.Grad(u) + a u = b // // The user can adjust the values of 'd', 'g', 'a', and 'b' with // command line options. // // This example highlights the differing implementations of // boundary conditions with continuous and discontinuous Galerkin // formulations of the Laplace problem. // // We recommend viewing Examples 1 and 14 before viewing this // example. #include "mfem.hpp" #include #include using namespace std; using namespace mfem; static double a_ = 0.2; // Normal to hole with boundary attribute 4 void n4Vec(const Vector &x, Vector &n) { n = x; n[0] -= 0.5; n /= -n.Norml2(); } Mesh * GenerateSerialMesh(int ref); // Compute the average value of alpha*n.Grad(sol) + beta*sol over the boundary // attributes marked in bdr_marker. Also computes the L2 norm of // alpha*n.Grad(sol) + beta*sol - gamma over the same boundary. double IntegrateBC(const GridFunction &sol, const Array &bdr_marker, double alpha, double beta, double gamma, double &error); int main(int argc, char *argv[]) { // 1. Parse command-line options. int ser_ref_levels = 2; int order = 1; double sigma = -1.0; double kappa = -1.0; bool h1 = true; bool visualization = true; double mat_val = 1.0; double dbc_val = 0.0; double nbc_val = 1.0; double rbc_a_val = 1.0; // du/dn + a * u = b double rbc_b_val = 1.0; OptionsParser args(argc, argv); args.AddOption(&h1, "-h1", "--continuous", "-dg", "--discontinuous", "Select continuous \"H1\" or discontinuous \"DG\" basis."); args.AddOption(&order, "-o", "--order", "Finite element order (polynomial degree) or -1 for" " isoparametric space."); args.AddOption(&sigma, "-s", "--sigma", "One of the two DG penalty parameters, typically +1/-1." " See the documentation of class DGDiffusionIntegrator."); args.AddOption(&kappa, "-k", "--kappa", "One of the two DG penalty parameters, should be positive." " Negative values are replaced with (order+1)^2."); args.AddOption(&ser_ref_levels, "-rs", "--refine-serial", "Number of times to refine the mesh uniformly in serial."); args.AddOption(&mat_val, "-mat", "--material-value", "Constant value for material coefficient " "in the Laplace operator."); args.AddOption(&dbc_val, "-dbc", "--dirichlet-value", "Constant value for Dirichlet Boundary Condition."); args.AddOption(&nbc_val, "-nbc", "--neumann-value", "Constant value for Neumann Boundary Condition."); args.AddOption(&rbc_a_val, "-rbc-a", "--robin-a-value", "Constant 'a' value for Robin Boundary Condition: " "du/dn + a * u = b."); args.AddOption(&rbc_b_val, "-rbc-b", "--robin-b-value", "Constant 'b' value for Robin Boundary Condition: " "du/dn + a * u = b."); args.AddOption(&a_, "-a", "--radius", "Radius of holes in the mesh."); args.AddOption(&visualization, "-vis", "--visualization", "-no-vis", "--no-visualization", "Enable or disable GLVis visualization."); args.Parse(); if (!args.Good()) { args.PrintUsage(mfem::out); return 1; } if (kappa < 0 && !h1) { kappa = (order+1)*(order+1); } args.PrintOptions(mfem::out); if (a_ < 0.01) { mfem::out << "Hole radius too small, resetting to 0.01.\n"; a_ = 0.01; } if (a_ > 0.49) { mfem::out << "Hole radius too large, resetting to 0.49.\n"; a_ = 0.49; } // 2. Construct the (serial) mesh and refine it if requested. Mesh *mesh = GenerateSerialMesh(ser_ref_levels); int dim = mesh->Dimension(); // 3. Define a finite element space on the serial mesh. Here we use either // continuous Lagrange finite elements or discontinuous Galerkin finite // elements of the specified order. FiniteElementCollection *fec = h1 ? (FiniteElementCollection*)new H1_FECollection(order, dim) : (FiniteElementCollection*)new DG_FECollection(order, dim); FiniteElementSpace fespace(mesh, fec); int size = fespace.GetTrueVSize(); mfem::out << "Number of finite element unknowns: " << size << endl; // 4. Create "marker arrays" to define the portions of boundary associated // with each type of boundary condition. These arrays have an entry // corresponding to each boundary attribute. Placing a '1' in entry i // marks attribute i+1 as being active, '0' is inactive. Array nbc_bdr(mesh->bdr_attributes.Max()); Array rbc_bdr(mesh->bdr_attributes.Max()); Array dbc_bdr(mesh->bdr_attributes.Max()); nbc_bdr = 0; nbc_bdr[0] = 1; rbc_bdr = 0; rbc_bdr[1] = 1; dbc_bdr = 0; dbc_bdr[2] = 1; Array ess_tdof_list(0); if (h1 && mesh->bdr_attributes.Size()) { // For a continuous basis the linear system must be modified to enforce an // essential (Dirichlet) boundary condition. In the DG case this is not // necessary as the boundary condition will only be enforced weakly. fespace.GetEssentialTrueDofs(dbc_bdr, ess_tdof_list); } // 5. Setup the various coefficients needed for the Laplace operator and the // various boundary conditions. In general these coefficients could be // functions of position but here we use only constants. ConstantCoefficient matCoef(mat_val); ConstantCoefficient dbcCoef(dbc_val); ConstantCoefficient nbcCoef(nbc_val); ConstantCoefficient rbcACoef(rbc_a_val); ConstantCoefficient rbcBCoef(rbc_b_val); // Since the n.Grad(u) terms arise by integrating -Div(m Grad(u)) by parts we // must introduce the coefficient 'm' into the boundary conditions. // Therefore, in the case of the Neumann BC, we actually enforce m n.Grad(u) // = m g rather than simply n.Grad(u) = g. ProductCoefficient m_nbcCoef(matCoef, nbcCoef); ProductCoefficient m_rbcACoef(matCoef, rbcACoef); ProductCoefficient m_rbcBCoef(matCoef, rbcBCoef); // 6. Define the solution vector u as a finite element grid function // corresponding to fespace. Initialize u with initial guess of zero. GridFunction u(&fespace); u = 0.0; // 7. Set up the bilinear form a(.,.) on the finite element space // corresponding to the Laplacian operator -Delta, by adding the Diffusion // domain integrator. BilinearForm a(&fespace); a.AddDomainIntegrator(new DiffusionIntegrator(matCoef)); if (h1) { // Add a Mass integrator on the Robin boundary a.AddBoundaryIntegrator(new MassIntegrator(m_rbcACoef), rbc_bdr); } else { // Add the interfacial portion of the Laplace operator a.AddInteriorFaceIntegrator(new DGDiffusionIntegrator(matCoef, sigma, kappa)); // Counteract the n.Grad(u) term on the Dirichlet portion of the boundary a.AddBdrFaceIntegrator(new DGDiffusionIntegrator(matCoef, sigma, kappa), dbc_bdr); // Augment the n.Grad(u) term with a*u on the Robin portion of boundary a.AddBdrFaceIntegrator(new BoundaryMassIntegrator(m_rbcACoef), rbc_bdr); } a.Assemble(); // 8. Assemble the linear form for the right hand side vector. LinearForm b(&fespace); if (h1) { // Set the Dirichlet values in the solution vector u.ProjectBdrCoefficient(dbcCoef, dbc_bdr); // Add the desired value for n.Grad(u) on the Neumann boundary b.AddBoundaryIntegrator(new BoundaryLFIntegrator(m_nbcCoef), nbc_bdr); // Add the desired value for n.Grad(u) + a*u on the Robin boundary b.AddBoundaryIntegrator(new BoundaryLFIntegrator(m_rbcBCoef), rbc_bdr); } else { // Add the desired value for the Dirichlet boundary b.AddBdrFaceIntegrator(new DGDirichletLFIntegrator(dbcCoef, matCoef, sigma, kappa), dbc_bdr); // Add the desired value for n.Grad(u) on the Neumann boundary b.AddBdrFaceIntegrator(new BoundaryLFIntegrator(m_nbcCoef), nbc_bdr); // Add the desired value for n.Grad(u) + a*u on the Robin boundary b.AddBdrFaceIntegrator(new BoundaryLFIntegrator(m_rbcBCoef), rbc_bdr); } b.Assemble(); // 9. Construct the linear system. OperatorPtr A; Vector B, X; a.FormLinearSystem(ess_tdof_list, u, b, A, X, B); #ifndef MFEM_USE_SUITESPARSE // 10. Define a simple symmetric Gauss-Seidel preconditioner and use it to // solve the system AX=B with PCG in the symmetric case, and GMRES in the // non-symmetric one. { GSSmoother M((SparseMatrix&)(*A)); if (sigma == -1.0) { PCG(*A, M, B, X, 1, 500, 1e-12, 0.0); } else { GMRES(*A, M, B, X, 1, 500, 10, 1e-12, 0.0); } } #else // 11. If MFEM was compiled with SuiteSparse, use UMFPACK to solve the // system. UMFPackSolver umf_solver; umf_solver.Control[UMFPACK_ORDERING] = UMFPACK_ORDERING_METIS; umf_solver.SetOperator(*A); umf_solver.Mult(B, X); #endif // 12. Recover the grid function corresponding to U. This is the local finite // element solution. a.RecoverFEMSolution(X, b, u); // 13. Compute the various boundary integrals. mfem::out << endl << "Verifying boundary conditions" << endl << "=============================" << endl; { // Integrate the solution on the Dirichlet boundary and compare to the // expected value. double error, avg = IntegrateBC(u, dbc_bdr, 0.0, 1.0, dbc_val, error); bool hom_dbc = (dbc_val == 0.0); error /= hom_dbc ? 1.0 : fabs(dbc_val); mfem::out << "Average of solution on Gamma_dbc:\t" << avg << ", \t" << (hom_dbc ? "absolute" : "relative") << " error " << error << endl; } { // Integrate n.Grad(u) on the inhomogeneous Neumann boundary and compare // to the expected value. double error, avg = IntegrateBC(u, nbc_bdr, 1.0, 0.0, nbc_val, error); bool hom_nbc = (nbc_val == 0.0); error /= hom_nbc ? 1.0 : fabs(nbc_val); mfem::out << "Average of n.Grad(u) on Gamma_nbc:\t" << avg << ", \t" << (hom_nbc ? "absolute" : "relative") << " error " << error << endl; } { // Integrate n.Grad(u) on the homogeneous Neumann boundary and compare to // the expected value of zero. Array nbc0_bdr(mesh->bdr_attributes.Max()); nbc0_bdr = 0; nbc0_bdr[3] = 1; double error, avg = IntegrateBC(u, nbc0_bdr, 1.0, 0.0, 0.0, error); bool hom_nbc = true; mfem::out << "Average of n.Grad(u) on Gamma_nbc0:\t" << avg << ", \t" << (hom_nbc ? "absolute" : "relative") << " error " << error << endl; } { // Integrate n.Grad(u) + a * u on the Robin boundary and compare to the // expected value. double error; double avg = IntegrateBC(u, rbc_bdr, 1.0, rbc_a_val, rbc_b_val, error); bool hom_rbc = (rbc_b_val == 0.0); error /= hom_rbc ? 1.0 : fabs(rbc_b_val); mfem::out << "Average of n.Grad(u)+a*u on Gamma_rbc:\t" << avg << ", \t" << (hom_rbc ? "absolute" : "relative") << " error " << error << endl; } // 14. Save the refined mesh and the solution. This output can be viewed // later using GLVis: "glvis -m refined.mesh -g sol.gf". { ofstream mesh_ofs("refined.mesh"); mesh_ofs.precision(8); mesh->Print(mesh_ofs); ofstream sol_ofs("sol.gf"); sol_ofs.precision(8); u.Save(sol_ofs); } // 15. Send the solution by socket to a GLVis server. if (visualization) { string title_str = h1 ? "H1" : "DG"; char vishost[] = "localhost"; int visport = 19916; socketstream sol_sock(vishost, visport); sol_sock.precision(8); sol_sock << "solution\n" << *mesh << u << "window_title '" << title_str << " Solution'" << " keys 'mmc'" << flush; } // 16. Free the used memory. delete fec; delete mesh; return 0; } void quad_trans(double u, double v, double &x, double &y, bool log = false) { double a = a_; // Radius of disc double d = 4.0 * a * (M_SQRT2 - 2.0 * a) * (1.0 - 2.0 * v); double v0 = (1.0 + M_SQRT2) * (M_SQRT2 * a - 2.0 * v) * ((4.0 - 3 * M_SQRT2) * a + (8.0 * (M_SQRT2 - 1.0) * a - 2.0) * v) / d; double r = 2.0 * ((M_SQRT2 - 1.0) * a * a * (1.0 - 4.0 *v) + 2.0 * (1.0 + M_SQRT2 * (1.0 + 2.0 * (2.0 * a - M_SQRT2 - 1.0) * a)) * v * v ) / d; double t = asin(v / r) * u / v; if (log) { mfem::out << "u, v, r, v0, t " << u << " " << v << " " << r << " " << v0 << " " << t << endl; } x = r * sin(t); y = r * cos(t) - v0; } void trans(const Vector &u, Vector &x) { double tol = 1e-4; if (u[1] > 0.5 - tol || u[1] < -0.5 + tol) { x = u; return; } if (u[0] > 1.0 - tol || u[0] < -1.0 + tol || fabs(u[0]) < tol) { x = u; return; } if (u[0] > 0.0) { if (u[1] > fabs(u[0] - 0.5)) { quad_trans(u[0] - 0.5, u[1], x[0], x[1]); x[0] += 0.5; return; } if (u[1] < -fabs(u[0] - 0.5)) { quad_trans(u[0] - 0.5, -u[1], x[0], x[1]); x[0] += 0.5; x[1] *= -1.0; return; } if (u[0] - 0.5 > fabs(u[1])) { quad_trans(u[1], u[0] - 0.5, x[1], x[0]); x[0] += 0.5; return; } if (u[0] - 0.5 < -fabs(u[1])) { quad_trans(u[1], 0.5 - u[0], x[1], x[0]); x[0] *= -1.0; x[0] += 0.5; return; } } else { if (u[1] > fabs(u[0] + 0.5)) { quad_trans(u[0] + 0.5, u[1], x[0], x[1]); x[0] -= 0.5; return; } if (u[1] < -fabs(u[0] + 0.5)) { quad_trans(u[0] + 0.5, -u[1], x[0], x[1]); x[0] -= 0.5; x[1] *= -1.0; return; } if (u[0] + 0.5 > fabs(u[1])) { quad_trans(u[1], u[0] + 0.5, x[1], x[0]); x[0] -= 0.5; return; } if (u[0] + 0.5 < -fabs(u[1])) { quad_trans(u[1], -0.5 - u[0], x[1], x[0]); x[0] *= -1.0; x[0] -= 0.5; return; } } x = u; } Mesh * GenerateSerialMesh(int ref) { Mesh * mesh = new Mesh(2, 29, 16, 24, 2); int vi[4]; for (int i=0; i<2; i++) { int o = 13 * i; vi[0] = o + 0; vi[1] = o + 3; vi[2] = o + 4; vi[3] = o + 1; mesh->AddQuad(vi); vi[0] = o + 1; vi[1] = o + 4; vi[2] = o + 5; vi[3] = o + 2; mesh->AddQuad(vi); vi[0] = o + 5; vi[1] = o + 8; vi[2] = o + 9; vi[3] = o + 2; mesh->AddQuad(vi); vi[0] = o + 8; vi[1] = o + 12; vi[2] = o + 15; vi[3] = o + 9; mesh->AddQuad(vi); vi[0] = o + 11; vi[1] = o + 14; vi[2] = o + 15; vi[3] = o + 12; mesh->AddQuad(vi); vi[0] = o + 10; vi[1] = o + 13; vi[2] = o + 14; vi[3] = o + 11; mesh->AddQuad(vi); vi[0] = o + 6; vi[1] = o + 13; vi[2] = o + 10; vi[3] = o + 7; mesh->AddQuad(vi); vi[0] = o + 0; vi[1] = o + 6; vi[2] = o + 7; vi[3] = o + 3; mesh->AddQuad(vi); } vi[0] = 0; vi[1] = 6; mesh->AddBdrSegment(vi, 1); vi[0] = 6; vi[1] = 13; mesh->AddBdrSegment(vi, 1); vi[0] = 13; vi[1] = 19; mesh->AddBdrSegment(vi, 1); vi[0] = 19; vi[1] = 26; mesh->AddBdrSegment(vi, 1); vi[0] = 28; vi[1] = 22; mesh->AddBdrSegment(vi, 2); vi[0] = 22; vi[1] = 15; mesh->AddBdrSegment(vi, 2); vi[0] = 15; vi[1] = 9; mesh->AddBdrSegment(vi, 2); vi[0] = 9; vi[1] = 2; mesh->AddBdrSegment(vi, 2); for (int i=0; i<2; i++) { int o = 13 * i; vi[0] = o + 7; vi[1] = o + 3; mesh->AddBdrSegment(vi, 3 + i); vi[0] = o + 10; vi[1] = o + 7; mesh->AddBdrSegment(vi, 3 + i); vi[0] = o + 11; vi[1] = o + 10; mesh->AddBdrSegment(vi, 3 + i); vi[0] = o + 12; vi[1] = o + 11; mesh->AddBdrSegment(vi, 3 + i); vi[0] = o + 8; vi[1] = o + 12; mesh->AddBdrSegment(vi, 3 + i); vi[0] = o + 5; vi[1] = o + 8; mesh->AddBdrSegment(vi, 3 + i); vi[0] = o + 4; vi[1] = o + 5; mesh->AddBdrSegment(vi, 3 + i); vi[0] = o + 3; vi[1] = o + 4; mesh->AddBdrSegment(vi, 3 + i); } double d[2]; double a = a_ / M_SQRT2; d[0] = -1.0; d[1] = -0.5; mesh->AddVertex(d); d[0] = -1.0; d[1] = 0.0; mesh->AddVertex(d); d[0] = -1.0; d[1] = 0.5; mesh->AddVertex(d); d[0] = -0.5 - a; d[1] = -a; mesh->AddVertex(d); d[0] = -0.5 - a; d[1] = 0.0; mesh->AddVertex(d); d[0] = -0.5 - a; d[1] = a; mesh->AddVertex(d); d[0] = -0.5; d[1] = -0.5; mesh->AddVertex(d); d[0] = -0.5; d[1] = -a; mesh->AddVertex(d); d[0] = -0.5; d[1] = a; mesh->AddVertex(d); d[0] = -0.5; d[1] = 0.5; mesh->AddVertex(d); d[0] = -0.5 + a; d[1] = -a; mesh->AddVertex(d); d[0] = -0.5 + a; d[1] = 0.0; mesh->AddVertex(d); d[0] = -0.5 + a; d[1] = a; mesh->AddVertex(d); d[0] = 0.0; d[1] = -0.5; mesh->AddVertex(d); d[0] = 0.0; d[1] = 0.0; mesh->AddVertex(d); d[0] = 0.0; d[1] = 0.5; mesh->AddVertex(d); d[0] = 0.5 - a; d[1] = -a; mesh->AddVertex(d); d[0] = 0.5 - a; d[1] = 0.0; mesh->AddVertex(d); d[0] = 0.5 - a; d[1] = a; mesh->AddVertex(d); d[0] = 0.5; d[1] = -0.5; mesh->AddVertex(d); d[0] = 0.5; d[1] = -a; mesh->AddVertex(d); d[0] = 0.5; d[1] = a; mesh->AddVertex(d); d[0] = 0.5; d[1] = 0.5; mesh->AddVertex(d); d[0] = 0.5 + a; d[1] = -a; mesh->AddVertex(d); d[0] = 0.5 + a; d[1] = 0.0; mesh->AddVertex(d); d[0] = 0.5 + a; d[1] = a; mesh->AddVertex(d); d[0] = 1.0; d[1] = -0.5; mesh->AddVertex(d); d[0] = 1.0; d[1] = 0.0; mesh->AddVertex(d); d[0] = 1.0; d[1] = 0.5; mesh->AddVertex(d); mesh->FinalizeTopology(); mesh->SetCurvature(1, true); // Stitch the ends of the stack together { Array v2v(mesh->GetNV()); for (int i = 0; i < v2v.Size() - 3; i++) { v2v[i] = i; } // identify vertices on the narrow ends of the rectangle v2v[v2v.Size() - 3] = 0; v2v[v2v.Size() - 2] = 1; v2v[v2v.Size() - 1] = 2; // renumber elements for (int i = 0; i < mesh->GetNE(); i++) { Element *el = mesh->GetElement(i); int *v = el->GetVertices(); int nv = el->GetNVertices(); for (int j = 0; j < nv; j++) { v[j] = v2v[v[j]]; } } // renumber boundary elements for (int i = 0; i < mesh->GetNBE(); i++) { Element *el = mesh->GetBdrElement(i); int *v = el->GetVertices(); int nv = el->GetNVertices(); for (int j = 0; j < nv; j++) { v[j] = v2v[v[j]]; } } mesh->RemoveUnusedVertices(); mesh->RemoveInternalBoundaries(); } mesh->SetCurvature(3, true); for (int l = 0; l < ref; l++) { mesh->UniformRefinement(); } mesh->Transform(trans); return mesh; } double IntegrateBC(const GridFunction &x, const Array &bdr, double alpha, double beta, double gamma, double &error) { double nrm = 0.0; double avg = 0.0; error = 0.0; const bool a_is_zero = alpha == 0.0; const bool b_is_zero = beta == 0.0; const FiniteElementSpace &fes = *x.FESpace(); MFEM_ASSERT(fes.GetVDim() == 1, ""); Mesh &mesh = *fes.GetMesh(); Vector shape, loc_dofs, w_nor; DenseMatrix dshape; Array dof_ids; for (int i = 0; i < mesh.GetNBE(); i++) { if (bdr[mesh.GetBdrAttribute(i)-1] == 0) { continue; } FaceElementTransformations *FTr = mesh.GetBdrFaceTransformations(i); if (FTr == nullptr) { continue; } const FiniteElement &fe = *fes.GetFE(FTr->Elem1No); MFEM_ASSERT(fe.GetMapType() == FiniteElement::VALUE, ""); const int int_order = 2*fe.GetOrder() + 3; const IntegrationRule &ir = IntRules.Get(FTr->FaceGeom, int_order); fes.GetElementDofs(FTr->Elem1No, dof_ids); x.GetSubVector(dof_ids, loc_dofs); if (!a_is_zero) { const int sdim = FTr->Face->GetSpaceDim(); w_nor.SetSize(sdim); dshape.SetSize(fe.GetDof(), sdim); } if (!b_is_zero) { shape.SetSize(fe.GetDof()); } for (int j = 0; j < ir.GetNPoints(); j++) { const IntegrationPoint &ip = ir.IntPoint(j); IntegrationPoint eip; FTr->Loc1.Transform(ip, eip); FTr->Face->SetIntPoint(&ip); double face_weight = FTr->Face->Weight(); double val = 0.0; if (!a_is_zero) { FTr->Elem1->SetIntPoint(&eip); fe.CalcPhysDShape(*FTr->Elem1, dshape); CalcOrtho(FTr->Face->Jacobian(), w_nor); val += alpha * dshape.InnerProduct(w_nor, loc_dofs) / face_weight; } if (!b_is_zero) { fe.CalcShape(eip, shape); val += beta * (shape * loc_dofs); } // Measure the length of the boundary nrm += ip.weight * face_weight; // Integrate alpha * n.Grad(x) + beta * x avg += val * ip.weight * face_weight; // Integrate |alpha * n.Grad(x) + beta * x - gamma|^2 val -= gamma; error += (val*val) * ip.weight * face_weight; } } // Normalize by the length of the boundary if (std::abs(nrm) > 0.0) { error /= nrm; avg /= nrm; } // Compute l2 norm of the error in the boundary condition (negative // quadrature weights may produce negative 'error') error = (error >= 0.0) ? sqrt(error) : -sqrt(-error); // Return the average value of alpha * n.Grad(x) + beta * x return avg; }