matrix-free FEM for electromagnetic problem

[yy.wayne]

Hello everyone!

I'm new to deal.II but find it really exciting.

The problem I plan to solve is the frequency-domain Maxwell wave equation with (curlcurl+k^2)E = f form. One of the advantages of deal.II is its matrix-free method for iterative solver, so I'd like to implement it into frequency-domain Maxwell problem. 

However step-37 says current matrix-free method only supports a portion of basis functions. My first question is are H1-curl conforming basis functions such as FE_NedelecSz and FE_Nedelec supported for matrix-free method in deal.II?

Besides I want to conform I understand matrix-free method. My second question is it correct that "matrix-free works for both structured (octree like) and unstructured hex mesh, but differs in Jacobian matrix for each cell has too be storaged for unstructured hex mesh" ?

In the end I'm looking for some vector indefinite Helmholtz like programs implement with deal.II but hardly find them. It's there any reference for similar deal.II projects on EM problems ?

Thank you!

[yy.wayne]

The discussion in 2019  interpolate FE_Nelelec is really similar to mine, both for physical background and concerns. I've try Domain Decomposition method as well, since it's one of the only good preconditioners for indefinite Maxwell problem with complex 3D geometries. However here I want to apply matrix-free method since they are appealing for large-scale computation, though they may not converge in this case, even combined with multigrid methods.

[yy.wayne]

Answer from any aspect is welcomed. Since it's just a yes-or-no question on deal.II's capability, I guess it won't take much time. 

[Martin Kronbichler]

At this moment, there is no support yet for Nedelec-type elements in the matrix-free framework. However, I would expect these to get ready within the next 2-3 months: We have recently added support for Raviart-Thomas elements, which have similar demands in terms of algorithms, therefore we have most of the necessary preparations. If you would be interested in this development, we could share what we have at this point and see how we would proceed.

Regarding your second question, it is correct that both structured and unstructured meshes are support by the matrix-free framework. As you say, what differs is the level of compression in the metric terms: On affine mesh elements (including Cartesian ones), we only need a single dim x dim tensor per mesh cell, whereas the general (curved) case has different terms in every point. As a result, the memory access and the achievable performance differs, but both are supported.

The big topic, as you say, is the question of multigrid methods. There is some reference on a problem I believe might be related to yours, https://doi.org/10.1002/nla.2348 - that solver builds on auxiliary space methods. Unfortunately, we do not have support for these kinds of methods in deal.II at this point, but I believe that the investment to get them running would not be huge.

Best regards,
Martin

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[yy.wayne]

Thank you Mr. Kronbichler. Looking forward to deal.II's development on matrix-free method.