Interface boundary attribute of partitioned meshes

[Tom Mathew]

Dear all,

I am a novice of dealii, but would like to work on it, if there is something which I am looking for. I would like to implement Optimized Restricted Additive Schwarz iterative solver or preconditioner for my problem, may be by inheriting the dealii linear algebra solver or iterative solver classes. Will it be possible to access the interface between the subdomains (interface boundary), so that I can impose optimized transmission (Robin) conditions for the iterative method. If not, any possible alternative directions or suggestions are welcomed. I am working on indefinite Maxwell's equations in complex algebra, so I guess I will be working with block matrices in dealii. Or if there is an example or project where the local subdomain matrices has been modified or any Schwarz methods has been tried can also be  wonderful place to start for me.

Thanks in advance, 

Best regards,

Tom

[Tom Mathew]

Is this a foolish question or this group don't entertain discussions ?

[Jean-Paul Pelteret]

Dear Tom,

You’ve come to the right place to ask your question but, occasionally, it takes some time for the right person to come along to be able to give a meaningful answer to it. I, for example, have no familiarity with Additive Schwarz methods, so I have no insights to share with you on that front. With regard to your intermediate question, when traversing a grid it is possible to ask a cell if a neighbour is not locally owned or is a ghosted cell, something like

for (auto cell : dof_handler.active_cell_iterators())

  for (unsigned int f=0; f<GeometryInfo<dim>::faces_per_cell; ++f)

    if (!cell->at_boundary(f) && cell->neighbour(f)->is_ghost())

      // Do whatever you want...

However, I don’t know if this is what you’re actually wanting to do in the context of the greater question. Maybe someone else with experience in this domain will chime in when they have the time (remember, this is a community driven by volunteers).

Best,

Jean-Paul

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[Tom Mathew]

Thank you for your reply. It's fine with me to wait, as I got quite skeptical when no one answered even something like this. Thanks for your kind reply.

The idea you said is wonderful for a starting point. It will take time also to implement my solver in dealii. I am very much looking forward to this. The answer I am expecting doesn't have to be specific to ORAS methods, so any suggestions or links to already attempted out of box linear solver or preconditioner are fully welcome.

It's completely fine, I can wait. Even sometime I thought this may not be the active group. 

Thanks again this is a wonderful starting point to think from.

So I guess I have to think from algebraic level than from a geometry level.

Thank you very much,

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[Wolfgang Bangerth]

On 6/5/19 10:24 AM, Tom Mathew wrote:
> Is this a foolish question or this group don't entertain discussions ?

Tom -- it's not a foolish question. As J-P already mentioned, it's often a
matter of everyone waiting for someone else more qualified to answer. The
questions that tend to linger are the ones where nobody seems to have the
requisite knowledge because it's a question a bit outside of what we all do.

J-P already gave you one answer. Let me try a different perspective that
augments his:

> I am a novice of dealii, but would like to work on it, if there is
> something which I am looking for. I would like to implement Optimized
> Restricted Additive Schwarz iterative solver or preconditioner for my
> problem, may be by inheriting the dealii linear algebra solver or
> iterative solver classes. Will it be possible to access the interface
> between the subdomains (interface boundary), so that I can impose
> optimized transmission (Robin) conditions for the iterative method. If
> not, any possible alternative directions or suggestions are welcomed. I am
> working on indefinite Maxwell's equations in complex algebra, so I guess I
> will be working with block matrices in dealii. Or if there is an example
> or project where the local subdomain matrices has been modified or any
> Schwarz methods has been tried can also be  wonderful place to start for me.

If you're interested in non-overlapping Schwarz methods (I suspect that's what
you are after, since you're interested in "transmission conditions"), then
that sort of implies that you are doing domain decomposition (DD) methods in
which you decompose the mesh into different partitions. One of the issues you
will encounter with deal.II if you do this is that non-overlapping DD methods
require you to duplicate degrees of freedom that lie on domain boundaries:
each side of the boundary has its own unique degree of freedom, even though
both sides' DoFs are located at the same physical point.

deal.II doesn't do this well. You can't do this easily on one Triangulation
object because that means that your finite element space is discontinuous
along the domain boundary. That is, unless you are willing to use
discontinuous Galerkin methods to begin with. The only way you can really
achieve that the space is discontinuous only along *some* faces, but not
*all*, is to use the hp framework and using FE_Nothing. step-46 shows up to
this; in that case, we actually solve for different physics on different
sub-domains (with the equivalent of "transmission boundary conditions", but
there is really nothing that prevents you from solving for the same physics on
all sub-domains.

Either way, I hope this is pointing you in the right direction. Feel free to
ask more questions if you run into trouble. As always, we'd be happy to take
any program that implements methods not yet covered in the tutorial, either
for the tutorial itself, or for the code gallery.

Best
W.

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

[Tom Mathew]

Dear all,

Thanks again for the reply. I was planing to follow actually this work https://arxiv.org/abs/1711.03789, where there is overlap. Though there exists non-overlapping versions too http://www.montefiore.ulg.ac.be/~geuzaine/preprints/ddm_double_sweep_preprint.pdf

I am a PhD student working in a thermal-fluids lab trying to model electromagnetic scattering for radiative transport via indefinite Maxwell's equations. My triangulations mostly comes from tomographied (micro-scale imaging techniques) STL files from industrial applications.

I am more interested in the finite element-linear algebra world, so this attempt is just for me to introduce myself as my helloworld to the linear algebra scientific computing community. I want to explore a little bit the underlying thing abstracting iterative Krylov methods, and I have at hand an indefinite Maxwell's problem, for which every state of the art linear algebra packages seems near obsolete.

Acknowledging that, everything is going fine for my thesis with FreeFEM++, and thanks to its ffddm framework, which seems quite efficient for my scattering problems. I have tried the hypre AMS quite briefly, regretting my poor mathematical background (I didn't knew it was only meant for definite Maxwell's problems. Only came to know after implementing it with MFEM. It was quite good on my workstation, but couldn't scale at the cluster).

So seeing for myself after the state of the art packages couldn't help me, I was flattered to give it a try, may be in my free time, as something which I aspire. 

Regarding your message, I guess I need sometime implementing my parallel solver in dealii. I need sometime also to undrestand how dealii functions. After I'll try to understand the linear algebra classes and try to inherit may be. 

I think the right direction for me is to try overlapping DDM, as https://arxiv.org/abs/1711.03789, discusses the influence of "generous overlap" (I couldn't read it completely though, it's highly mathematical). 

Thanks very much for  your reply, I will get in touch if I am stuck at some point.

Best regards,

Tom

[Tom Mathew]

for the non-overlapping version probably this one will be more apt: https://hal.archives-ouvertes.fr/hal-01244511/document

Regards,

Tom