[petsc-users] Ainsworth formula to solve saddle point problems / preconditioner for shell matrices

[Olivier Jamond]

On 03/10/2020 00:23, Barry Smith wrote:
>     I think what Jed is saying is that you should just actually build your preconditioner for your Ct*C + Qt*S*Q operator with S. Because Ct is tall and skinny the eigenstructure of Ct*C + Qt*S*Q is just the eigenstructure of S with a low rank "modification" and Krylov methods (GMRES) are good at solving problems where the eigenstructure of the preconditioner is only a small rank modification of the eigenstructure of the operator you are supply to GMRES. In the best situation each new iteration of GMRES corrects one more of the "rogue" eigen directions. I would first use a direct solver with S just to test how well it works as a preconditioner and then switch to GAMG or whatever should work efficiently for solving your particular S matrix.
>
>    I'd be interested in hearing how well the Ainsworth Formula works, it is something that might be worth adding to PCFIELDSPLIT.
>
>
>    Barry

Hi Barry,

Thanks for these clarifications.

To give some context, the test I am working on is a traction on an 
elastoplastic cube in large strain on which I apply 2% of strain at the 
first loading increment. The cube has 14739 dofs, and the number of rows 
of the C matrix is 867.

In this simple case, the C matrix just refers to simple dirichlet 
conditions. Then Q is diagonal with 1. on dofs without dirichlet on 0. 
for dofs with dirichlets. Q'*S*Q is like S with zeros on lines/columns 
referring to dofs with dirichlet, and then C'*C just re-add non null 
value on the diagonal for the dofs with dirichlet. In the end, I feel 
that in this case the ainsworth method just do exactly the same as 
row/column elimination that can be done with MatZeroRowsColumns and the 
x and b optional vectors provided.

On this test, with '-ksp_rtol 1.e-9' and '-ksp_type gmres', using S as a 
preconditionning matrix and a direct solver gives 65 iterations of the 
gmres for my first newton iteration (where S is SPD) and between 170 and 
290 for the next ones (S is still symmetric but has negative 
eigenvalues). If I use '-pc_type gamg', the number of iterations of the 
gmres for the first (SPD) newton iteration is (14 with Sp / 23 with S), 
and for the next ones (not SPD) it is (~45 with Sp / ~180 with S).

In this case with only simple dirichlets, I think I would like that the 
PCApply does something like: (I-Q)*jacobi(Ct*C)*(I-Q) + Q*precond(S)*Q. 
BUt I am not sure how to do that (I am quite newbie with petsc)... With 
a PCShell and PCShellSetApply?

In the end, if we found something that works well with the ainsworth 
formula, it would be nice to have it natively with PCFIELDSPLIT!

Many thanks,
Olivier