MtrixFree for biharmonic problem: submit_laplacian

[Michał Wichrowski]

Dear all,

I was trying to implement MatrixFree biharmonic solver. We already have most of functionality needed to do it (DG,  computing laplacian on both faces and cells), but the one thing missing is extending FEEvaluation and FEFaceEvaluation to support integration of 2nd order formulae: submit_laplacian() and extending integrate() would resolve the problem and allow matrix-free version  of step-47. 

The biharmonic solver may be also useful in other parts of library, especially handling mesh deformation and transforms.

Best,

Michał
 

[Martin Kronbichler]

Dear Michal,

That is great news. There is interest in this functionality also by Julius Witte (I added him here) and there were some steps done.

The main difficulty in terms of user interface is the fact that the submit_laplacian() is a bit nastier than the other functions: on non-affine geometries, the Laplacian (or Hessian) of the test function in real space gives contributions to both the Laplacian and the gradient in reference space. So we either need a `submit_laplacian()` function that both inserts the desired contribution to test by the gradient and the Laplacian - which breaks symmetry with the get_gradient()/get_laplacian() methods - or we need to have a second field that stores the accumulated part for the reference-cell gradient of the test function and merges this inside 'integrate'.

When you say "we already have most functionality needed to to it (DG, computing laplacian on both faces and cells)", do you imply that you have already started with the work needed for get_laplacian() on faces? At least on dealii master, we lack both the ability to evaluate the second derivatives in tangential directions in reference space and the Jacobian gradient needed for non-affine cell shapes.

In any case, I would be happy to help and join the efforts, and I think so is Julius. We have an imminent release in a few weeks for deal.II (scheduled feature freeze May 8), so the goal would be to try to achieve this soon after the release.

Best,
Martin

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[Michał Wichrowski]

I've just looked through the possible solution strategies and what could be done inside existing deal.II functionality. While implementing I noticed that there is missing option of subimiting laplacian and I thought that will be something easy to do. Now I see where the problem is.

I need a quick fix for my FSI solver (moving mesh) so I'm rewriting step-47 in parallel with multigrid solver. It required some fixes in deal.ii but think it is going to work. I'll post the patch when it will be working. 

Thanks,
Michał

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[Wolfgang Bangerth]

On 4/26/20 4:53 PM, Michał Wichrowski wrote:
> I've just looked through the possible solution strategies and what could be
> done inside existing deal.II functionality. While implementing I noticed that
> there is missing option of subimiting laplacian and I thought that will be
> something easy to do. Now I see where the problem is.
>
> I need a quick fix for my FSI solver (moving mesh) so I'm rewriting step-47 in
> parallel with multigrid solver. It required some fixes in deal.ii but think it
> is going to work. I'll post the patch when it will be working.

It would definitely be very interesting to see the matrix-free implementation.
That would make for an excellent "Possibilities for extensions" sub-section
for step-47!

Best
W.

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Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/