Dear Wolfgang,
Am 21.05.21 um 14:48 schrieb Wolfgang Bangerth:
On 5/21/21 2:23 AM, Sebastian Kinnewig wrote:
The problem you are considering, i.e.
curl ( curl ( E ) ) - k^2 E = 0
is respectively sometimes called the ill-posed Maxwell's equations or
the indefinite Maxwell's equations.
When you are considering only the 2D case a direct solver should be
enough.
But for the 3D case you will need more advanced methods. The most
common approach to solve these equations is via domain decomposition
method (ddm). The basic idea of the ddm is to divide the the domain
into smaller subdomains, where each subdomain becomes small enough so
it can be handled by a direct solver. The ddm for Maxwell is
explained in greater detail in this paper:A quasi-optimal domain
decomposition algorithm for the time-harmonic Maxwell's equations
<https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS0021999115001965&data=04%7C01%7CWolfgang.Bangerth%40colostate.edu%7Cae145f9e73094301870808d91c31afa9%7Cafb58802ff7a4bb1ab21367ff2ecfc8b%7C0%7C0%7C637571823692274784%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&sdata=ZE74xHiT8FLjtTNhQCA68VeVYp3BjnWxDl3o%2Brrn8nM%3D&reserved=0>.
But it is not easy to build a Maxwell solver based on ddm in deal.II
(I know this, since I am currently working on such a solver by myself).
I am sorry, there is no easy way to do this.
This is somewhat outside of my realm of experience, but I'll chime in
anyway:
Domain decomposition methods conceptually originated from the desire
to use serial codes that were developed in the 1990s and re-use them
in a parallel context. The question was how to connect them into a
parallel code. deal.II does not follow this paradigm: it just
distributes *all* data structures (mesh, matrices, vectors, etc). As a
consequence, it is difficult to press deal.II into service for DD
methods because they philosophically just don't quite fit into the
deal.II world view.
But, I believe you also don't have to because essentially everything
that has been developed in the DD world is also available in the
fully-distributed world deal.II lives in. In the case you describe
above, the key piece is the direct solver that can be run on each
subdomain sequentially. But there are good parallel direct solvers
that one could use -- specifically, I would suggest to explore MUMPS,
which you can access via its PETSc interfaces. (I think I recall that
there is also a Trilinos parallel direct solver, but I've forgotten
its name and we don't have an interface to it.) If you were to
consider something like step-40, where the domain is split into
subdomains, the matrix is split into corresponding sub-matrices, then
a parallel direct solver such as MUMPS can conceptually be thought of
as inverting the diagonal blocks of the matrix corresponding to each
subdomain -- i.e., not so different from what a direct solver applied
to one subdomain in the DD context does. The difference is that in the
DD context, you then have to think of how to stitch the individual
subdomains back together (the boundary or "transmission" conditions
between subdomains) whereas in the fully-distributed,
one-global-linear-system world this is unnecessary.
Thanks for your email concerning DD and deal.II. In general, we agree
that DD in the deal.II context
is not really necessary. Just using a parallel sparse direct solver
(e.g. MUMPS) is not an option because
of memory.
Maxwell is indeed very special and one cannot simply apply well-known
concepts. Of course, Hiptmair proposed in 1998 a multigrid method for
Maxwell. The key is a special
construction of the smoother (Gauß-Seidel-like / multiplicative Schwarz)
and a decomposition of the function spaces.
Of course this paper is highly cited and used, but it does not seem that
idea resolves all difficulties one may have
in Maxwell because of the indefinite structure.
Recently, over the last years, DD seems to be more competitive (e.g.,
Bouajaj et al. JCP 2015) and for exactly this reason, we started
developing a DD solution within deal.II. Our current results are promising.
We need to couple this later with Navier-Stokes and/or solids and for
this reason,
we decided to go with deal.II, despite that other packages (ngsolve for
instance) could
have been better for Maxwell alone.
Any further comments are specifically welcome of course.
Best regards,
Thomas
---
Thomas Wick
Leibniz University Hannover
https://thomaswick.org
Best
Wolfgang
--
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++++--------------------------------------------++++
Prof. Dr. Thomas Wick
Leibniz Universität Hannover (LUH)
Institut für Angewandte Mathematik (IfAM)
Arbeitsgruppe Wissenschaftliches Rechnen (GWR)
Welfengarten 1
30167 Hannover, Germany
Tel.: +49 511 762 3360
Email: [email protected]
www: https://ifam.uni-hannover.de/wick
www: https://thomaswick.org
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