Request for input: Laplace-based linear and bilinear integrators

[IdoAkkerman]

I am interested in implementing LinearFormIntegrator and BilinearFormIntegrator involving the Laplacian.

Any heads up on potential conflicting or duplicate effort is highly appreciated.
Any tips or warning for pitfalls are also welcome.
Last but not least, if I am skipping some useful functionality for other use cases please let me know.

My personal interest for these terms are stabilized methods. So first would be the scalar convection-diffusion case which would require the following:

• $(\Delta w, f)$ Linear form for a given Coefficient f
• $(w,q\Delta \phi)$ Bilinear form for a given Coefficient q
• $({\bf a}\cdot \nabla w,\Delta \phi)$ Bilinear form for a given VectorCoefficient a
• $(\Delta w,q \phi)$ Bilinear form for a given Coefficient q
• $(\Delta w,{\bf a}\cdot \nabla \phi)$ Bilinear form for a given VectorCoefficient a
• $(\Delta w,q\phi)$ Bilinear form for a given Coefficient q

Next would be vector based case, such as navier-stokes. Which would require the same as above but all the scalar shape functions become vector shape function. The coefficients will in principle be of Matrix form, probably more efficient routines are available for scalar or vector (= diagonal of matrix). I will consider implementing all three.

To enable the pressure and PSPG terms two mixed terms are required.

• $(\nabla q, Q \Delta {\bf u})$ Bilinear form for a given MatrixCoefficient Q
• $(\Delta {\bf w}, Q \nabla p)$ Bilinear form for a given MatrixCoefficient Q

Hessian based integrators will not be considered in this PR.

[IdoAkkerman]

Implementation effort can be reduced by using the TransposeIntegrator I assume?

[IdoAkkerman]

Is there a need for a Trans.LaplaceOrder(fe) ? As place holder I am going with fe.GetOrder() - 2 for now.

[v-dobrev]

This sounds great to me!

Is there a need for a Trans.LaplaceOrder(fe) ? As place holder I am going with fe.GetOrder() - 2 for now.

I'm not sure we really need to target exact integration of Laplacian terms, even if that is possible in some cases. I think using the lowest quadrature order (e.g., ignoring the transformation order) that works robustly is best. It is hard to test all possible use cases for "works robustly", so go with what you find in your tests or what is recommended in the literature, if there are such recommendations.

How do you plan to compute the Laplacian on a general element? Do you plan to use exact computation at quadrature points using the Hessian and the Jacobian of the transformation? An approximate alternative that only requires the Jacobian is to project the gradient (e.g. using nodal interpolation) back into the same FE space and then take the divergence. I think @bslazarov and/or @vladotomov wanted to try something like that but I'm not 100% certain.

[IdoAkkerman]

The Laplacian/Hessian is already implemented - in principle.
The base class has the (some) interfaces defined and for NURBS elements it should work.
I do not know about other elements.
There is likely still room for extensions, improvements and simplifications.

The Laplacian is compute as the trace of the Hessian which sounds expensive but is the easiest way to have the correct effect of the mapping . There is a check of to see if the Transformation is affine,
in that case a faster, more direct, way of computing the Laplacian is available.

[IdoAkkerman]

@michi002 My apologies, I pushed it to an other local repo. I will push it tomorrow, unfortunately I can not ssh into my machine due to overly zealous security...

[michi002]

@IdoAkkerman thank you. I had a look on your commit.
I think you only need non-mixed spaces for your purpose. However I would also need
$$ \langle \Delta u, v \rangle $$
for v piecewise constant.
Is it ok for you, if I just add the mixed version in your branch? Or should I branch out from your branch for that?
I will adpat to your workflow, as you started it :)

[IdoAkkerman]

@michi002 Convection-diffusion only requires the scalar implementation. However, the SUPG/PSPG terms for Stokes/Navier-Stokes involve terms where the velocity and pressure spaces are combined, or is there a way to bypass that?

Regarding, the term you are interested in, that can be obtained by by using the 4th term, viz.
$(\Delta w,q \phi)$ and choosing a zero order L2 finite element for $\phi$ (v in your comment) .

[michi002]

@IdoAkkerman
Yes, this is the bilinear form I need. But we also need to also implement the member-routine for mixed spaces
virtual void AssembleElementMatrix2(const FiniteElement &trial_fe, const FiniteElement &test_fe, ElementTransformation &Trans, DenseMatrix &elmat);
or not?
I have only seen the routine for the same ansatz and test-function
/// Support for use in BilinearForm. Can be used only when appropriate. virtual void AssembleElementMatrix(const FiniteElement &fe, ElementTransformation &Trans, DenseMatrix &elmat)

Concerning the coefficient:
I guess you are right, If the coefficients are differentiable one can just use the chain rule.

[IdoAkkerman]

Implementing AssembleElementMatrix2 should be minimal effort, so worth to include in the PR I guess.
However, the use case for AssembleElementMatrix2 is not clear to me. By itself not so much of a concern, but this would make it a bit harder to debug and test (unit_test).

Do you have a clear application in mind?

[michi002]

@IdoAkkerman
Yes, I want to apply it in distributed optimal control. I want to solve the equation
$-\Delta u = z$
for $z$, where $u$ is given. For implementing control constraints we want a simple connection between $z$ and $u$.
If we choose $u$ to be a $C^1$ spline and $z$ to be piecewise constant we can easily do that by
$\underline{z}_h = M_h^{-1} K_h \underline{u}_h$,
where $M_h$ is the (diagonal) mass for piecewise constants and $K_h$ is defined via
$\langle -\Delta u_h, p_h \rangle$
and here $p_h$ is also piecewise constant.

I do not know if it is an important case for the general userbase, but I think a mixed version (AssembleElementMatrix2) might come in handy for some users.

[IdoAkkerman]

So you are converting
$-\Delta u = z$
into a discrete weak form
$\langle-\Delta u^h, p_h \rangle =\langle z, p_h \rangle$
where $u^h$ are $C^1$ spline and $p_h$ are piecewise constants.
I was contemplating this as test case :)
But I do not see the logic of doing this... but perhaps it is something application specific.

Why not test with $C_1$ splines similar as used for $u^h$ -- and use integration by parts.
$\langle\nabla u^h, \nabla p_h \rangle - \langle\nabla u^h, n p_h \rangle_\Gamma=\langle z, p_h \rangle$
This will yield the standard optimal Galerkin formulation for the Poisson problem.

[michi002]

@IdoAkkerman
I am considering this for distributed optimal control, where $z$ (the control) is the quantity of interest.
We want to approximate the control $z_h$ via piecewise constants. The Laplace problem for the state, I solve as you suggest with $C^1$ test and ansatzfunctions.
So suppose the state $u_h$ is given (via a minimization problem), I want to reconstruct the right hand side, that gives this state.
Then I have to find $z_h \in Z_h$ such that
$\langle z_h, p_h \rangle = \langle -\Delta u_h, p_h\rangle \quad \forall p_h \in Y_h.$
If I now take z_h to be piecewise constant, I either have to take $u_h$ to be a $C^1$ function, to apply the strong Laplacian and $Y_h=Z_h$, or $Y_h$ to be piecewise linear continuous functions, to apply integration by parts, yielding a rectangular left hand side. In the second case I would have to solve some kind of saddle point problem, which I would like to avoid.

If you are really interested why we do that, you can look at our preprint: https://arxiv.org/abs/2404.10350
But I tried to summarize it. In the preprint we also did not consider control constraints yet, where this form is also advantageous.

I hope that answers your question.

[michi002]

@IdoAkkerman I have now implemented a mixed version of your LaplaceIntegrator called MixedLaplaceIntegrator.
I have pushed it to your branch (I hope that is ok).
I think in your implementation of it, there is a small error.
In line 1559 I think you want to calculate the laplacian of the trial functions, instead of its shape:

el.CalcPhysShape(Trans, shape);
el.CalcPhysShape(Trans, laplace);

I would like to test my routine, by projection the laplacian of a $C^1$ spline to piecewise constants defined on the same nurbs mesh. Then I would refine the mesh and look at the convergence order.
But I do not know how you do it at MFEM? I just saw big files for unit tests for bilinear form integrators in dimensions one to three, which were a little confusing for me.

[IdoAkkerman]

@michi002 Thanks for the input, I was to lazy to change the name. I think the functionality is okay, but you are right for a final commit this needs to be rectified.
I do not see the need for a a mixed class, doesn't the LaplaceIntegrator::AssembleElementMatrix2 member function not take care of that?