[petsc-dev] -pc_type gamg for vector PDE

[Alexander Grayver]

Mark, Jed,

Thanks a lot for your response.

> It looks like you have a largish \omega\mu\sigma because, as Jed 
> mentioned, one would not expect GAMG to work well for the curl-curl part.

Well in my case it is the other way around. I solve this equation for 
low \omega and possibly very low \sigma. The curl-curl term dominates 
always.

>
> A few things to try:
>
> 1) '-pc_gamg_type sa' will provide what should be a better solver for 
> SPD problems.

Unfortunately, the convergence with this option is worse for several 
operators I've tried.
Would it help to specify coordinates vertices?

>
> 2) Try:
>
> -pc_type hypre    ! instead of -pc_type gamg
> -pc_hypre_type boomeramg
> -pc_hypre_boomeramg_strong_threshold 0.6

When configuring with hypre I get "Cannot use hypre with complex numbers 
it is not coded for this capability".
I can reformulate my problem in terms of real matrix:

If C = A + iB
Cx=b  ==  [A -B; B A] [xr; xi] = [br; bi]

But I'm not sure that making spectra two times larger would not 
influence my condition number?

>
> 3) You say 'staggard' but I just see E here.  Do you have E on faces? 
>  I forget how staggering works here.  If E is cell centered then you 
> have a system of 3x3 blocks (with the right ordering) and GAMG might 
> benefit from setting the block size to tell it this:
>
> MatSetBlockSize(mat,3);

I have fields on the edges, but I can formulate another staggering 
scheme where fields are on the faces. Would it help?
My matrix is 3x3 block. For instance, if I have 100^3 grid, then matrix 
is 3*10^6 and each block is of 10^6 size.
Which block size should I pass to MatSetBlockSize? 10^6 or 3? Because 
from its description it is not obvious.

> And Jed's answer addresses your 2nd question about null-space.  These 
> solvers will degrade as \omega\mu\sigma gets smaller.

I do observe this for low \omega and low \sigma (e.g. in the air). I was 
thinking how could one project out this null-space.
Hitpmair, as Jed pointed, gives some clues. However it requires some 
efforts to implement and wanted first to try petsc built in stuff.

>
> Mark
>
> On Jan 19, 2012, at 5:37 PM, Jed Brown wrote:
>
>> On Wed, Jan 18, 2012 at 08:22, Alexander Grayver 
>> <agrayver at gfz-potsdam.de <mailto:agrayver at gfz-potsdam.de>> wrote:
>>
>>     Hello petsc team,
>>
>>     I solve 3D vector Helmholtz equation like following:
>>
>>     \nabla \times \nabla \times E + i\omega\mu\sigma E  = -J
>>
>>
>> Multigrid methods for curl-curl problems are pretty specialized. ML 
>> and Hypre have support for specific discretizations, I don't know if 
>> they support an imaginary shift. The PETSc interface to these 
>> packages does not currently support their special Maxwell interfaces.
>>
>> The methods are generally based on edge relaxation or auxiliary space 
>> preconditioning, see Hiptmair or Arnold, Falk, and Winther for the 
>> mathematical background.
>>
>>
>>     I use structured staggered grid and FD. The solution is a vector
>>     that consists of three parts E = {Ex Ey Ez}. The operator is
>>     symmetric matrix with complex numbers on diagonal.
>>     I'm interested in solving this system with iterative techniques.
>>     I applied newly presented gamg and it gives promising results,
>>     but all I did is just:
>>     -ksp_type tfqmr -pc_type gamg
>>
>>     I played with different ksp_type and gamg options which are
>>     listed on PCGAMG doc page, but nothing improved convergence.
>>     Could you please guide me a bit through usage of this technique?
>>     The precise questions are:
>>     1. Do I have to do something to say petsc that my equation is a
>>     vector equation? Is it important for gamg pc?
>>     2. Should I take into account null-space using KSPSetNullSpace?
>>     Since it is well known that as \omega or \sigma get small,
>>     null-space of geometric term (curl curl operator) starts to
>>     dominate and system gets more ill-conditioned.
>>     3. Which options for gamg may improve convergence in my case?
>>
>>
>>     Thanks a lot in advance.
>>
>>     -- 
>>     Regards,
>>     Alexander
>>
>>
>


-- 
Regards,
Alexander

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