[petsc-users] Multigrid with defect correction

[Matt Landreman]

Thanks everyone for the help.

On item 1 (using Amat different from Pmat with geometric multigrid), I
tried Barry's suggestion but it did not seem to resolve the issue. For
example, in ksp ex25.c, I tried adding the following lines after line 112:
  if (J == jac) {

    ierr = PetscPrintf(PETSC_COMM_WORLD,"Creating a new Amat\n");

    ierr = DMCreateMatrix(da,&J);

    ierr = KSPSetOperators(ksp,J,jac);

  }
  ierr = MatShift(J,1.0);

This change should set Amat (J) to be different from Pmat (jac),
(specifically Amat=identity matrix), so the solution from KSP should be
completely different from the original ex25.c. But viewing the solution
vector, the solution is unchanged. It seems PETSc is ignoring the Amat
created in this approach.

Matt K's suggestion of switching from KSP to SNES does work, allowing Amat
to differ from Pmat on the finest multigrid level. (On coarser levels, it
seems PETSc still forces Amat=Pmat on entry to computeMatrix).

On Jed's comment, the application I have in mind is indeed a
convection-dominated equation (a steady linear 3D convection-diffusion
equation with smoothly varying anisotropic coefficients and recirculating
convection). Gamg and hypre-boomerAMG have been working on it ok if I
discretize with low-order upwind differences in Pmat and use Amat=Pmat, but
I'd like higher order accuracy. Using gmres with a higher-order
discretization in Amat and low-order Pmat works ok, but the number of KSP
iterations required gets large as the diffusion gets small compared to
convection, even with -pc_type lu. So I'm working to see if geometric mg
with defect correction at each level can do better.

Thanks,
Matt Landreman




On Thu, Feb 23, 2017 at 5:23 PM, Jed Brown <jed at jedbrown.org> wrote:

> Matt Landreman <matt.landreman at gmail.com> writes:
> > 3. Is it at all sensible to do this second kind of defect correction with
> > _algebraic_ multigrid? Perhaps Amat for each level could be formed from
> the
> > high-order matrix at the fine level by the Galerkin operator R A P, after
> > getting all the restriction matrices created by gamg for Pmat?
>
> Note that defect correction is most commonly used for
> transport-dominated processes for which the high order discretization is
> not h-elliptic.  AMG heuristics are typically really bad for such
> problems so stabilizing a smoother isn't really a relevant issue.  Also,
> it is usually used for strongly nonlinear problems where AMG's setup
> costs are likely overkill.  This is not to say that defect correction
> AMG won't work, but there is reason to believe that it doesn't matter
> for many important problems.
>
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