[petsc-users] ASM for Saddle-point problems

[Matthew Knepley]

On Wed, Feb 26, 2014 at 6:25 AM, Jed Brown <jed at jedbrown.org> wrote:

> Adriano Côrtes <adrimacortes at gmail.com> writes:
>
> > Dear all,
> >
> > I'm solving a saddle-point problem coming from a 2d stokes problem
> > discretized by an inf-sup stable mixed element.
> > By now I'm playing in serial, because I want to understand the
> > behavior of the solver. I have the whole matrix assembled in memory.
> > After playing with GMRES and ILU with different fill-ins,
>
> ILU is typically terrible for saddle point problems.
>
> > I started playing with ASM to see if I can get better results. By
> > using -pc_asm_blocks and -pc_asm_overlap I tried some variations none
> > giving better results.
> >
> > My questions are
> >
> > 1. how the blocks are built by PETSc, since my problem is a saddle-point
> one?
>
> It starts with the set of owned variables and adds overlap by taking all
> neighbors represented in the graph.  You generally need a minimum
> overlap of 1 for saddle point problems.
>
> > 2. From the theoretical point-of-view, block-factorizations, that is
> > using PCFieldsplit, are in general the best we can have in terms of
> > performance (number of iterations and scalability)?
>
> There is no consensus on this and I'm actually fond of "monolithic"
> multigrid methods, but it is harder to reuse components and harder to
> debug convergence.  PCFieldSplit is a good methodology.  Some of my
> talks have a comparison slide.  Here is a high-level one from last week.
>
> http://59a2.org/files/20140221-ExploitsInImplicitness.pdf
>

More specifically, you can use -pc_fieldsplit_detect_saddle_point on your
matrix, and then construct any of the block methods from options. I list
most
of the interesting ones in the Paris Tutorial on our website.

   Matt

-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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