[petsc-users] multigrid questions

[Jed Brown]

Matt Landreman <matt.landreman at gmail.com> writes:

> Hi, Petsc developers,
>
> A few basic questions about geometric and algebraic multigrid in petsc:
>
> Based on the output of -ksp_view, I see that for both -pc_type mg and
> -pc_type gamg, the smoothing on each multigrid level is implemented using a
> ksp with default type chebyshev, associated with a pc of default type sor.
> I also see in section 4.4.5 of the petsc manual that switching to
> -mg_levels_ksp_type richardson is sometimes recommended, and I see in the
> ksp examples makefile that ksp ex28.c uses -mg_levels_ksp_type gmres.
>
> 1. Does “textbook” Gauss-Seidel smoothing correspond in petsc to
>  -mg_levels_ksp_type richardson?

Yeah, default Richardson/SOR is plain GS.

> 2. It says in
> http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/KSP/KSPCHEBYSHEV.html
> that PETSc’s chebyshev iteration only works for symmetric matrices. So to
> confirm, for any problem with non-symmetric matrices such as
> convection-diffusion, setting -mg_levels_ksp_type to some non-default value
> like richardson or gmres is *mandatory* then?  (If so, it would be helpful
> if this were stated in the manual.)

A lot of nonsymmetric problems are "not too non-symmetric".
Chebyshev(1) is actually just Richardson.  High-degree Chebyshev has a
tighter smoothing profile on the real axis so problems with eigenvalues
far from the real axis can be unstable.  The defaults were chosen
because they tend to be robust on a variety of problems.

> 3. For smoothing with geometric multigrid, is there a way in petsc to do
> red-black Gauss Seidel? Line relaxation?

Multi-color GS would need a coloring which is difficult in parallel and
the performance tends to be poor when there are many colors.  Block
Jacobi/ILU smoothers often behave similarly to line smoothers (depends
on ordering).  Usually people write custom code for
discretization-dependent smoothers.
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