[petsc-users] Convergence of transposed linear system.

[Gaetan Kenway]

That is a good idea to try Francois. Do you know if there is a easy way to
try that in PETSc? However, in my case, I'm not using an upwind scheme, but
rather a 2nd order JST scheme for the preconditioner. Also, we have
observed the same behavior even for Euler systems, although both the
direct/adjoint systems in this case are easier to solve and the difference
between the systems is less dramatic.

I also though about using left preconditioning for the adjoint system
instead of right preconditioning, but left preconditioning consistently
fails even for the untransposed system. I have no idea why left
preconditioning doesn't work.

Gaetan

On Mon, Nov 24, 2014 at 6:24 AM, francois Pacull <fpacull at hotmail.com>
wrote:

> Hello,
>
> This is just an idea but this might be due to the fact that the structure
> of the preconditioner is severely unsymmetrical when using a first-order
> upwind scheme without viscous terms: when building the overlap, the
> non-zero terms in the row-wise extra-diagonal blocks yield the list of
> vertices to add to each subdomain. If you use the transpose of the
> preconditioner, it still uses the row-wise and not the column-wise
> extra-diagonal blocks. So maybe you should build the ASM(1) preconditioner
> with the untransposed matrix first, and then transpose the preconditioning
> matrix? You may also change the side of the preconditioner, for the
> transposed system.
>
> Francois.
>
>
> ------------------------------
> Date: Sun, 23 Nov 2014 20:54:20 -0500
> From: gaetank at gmail.com
> To: petsc-users at mcs.anl.gov
> Subject: [petsc-users] Convergence of transposed linear system.
>
> Hi everyone
>
> I have a question relating to preconditioning effectiveness on large
> transposed systems. The linear system I'm trying to solve is jacobian
> matrix of 3D RANS CFD solver. The bock matrix consists of about 3 million
> block rows with a block size of 6: 5 for the inviscid part and 1 for the SA
> turbulence model.
>
> The preconditioning matrix is different from the linear system matrix in
> two ways: It uses a first order discretization (instead of second order)
> and the viscous fluxes are dropped.
>
> The untransposed system converges about 6 orders of magnitude with
> GRMES(100), ASM (overlap 1) and ILU(1) with RCM reordering. The test is run
> on 128 processors.  There are no convergence difficulties.
>
> However, when I try to solve the transpose of the same system, by either
> calling KSPSolveTranspose() or by assembling the transpose of the linear
> system and its preconditioner and calling KSPSolve(), GMRES stagnates after
> a negligible drop in the residual and no further progress is made.
>
> I have successfully solved this transpose system by using a different
> preconditioner that includes the complete linearization of the viscous
> terms (~4 times as many non-zeros in PC matrix) and a much much stronger
> preconditioner (ASM(2), ILU(2) with 200 GMRES reset.
>
> My question is why does the solution of the transpose system with the same
> method perform so terribly? Is it normal that vastly stronger
> preconditioning method is required to solve transpose systems?
>
> Any suggestions would be greatly appreciated
>
> Gaetan
>
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