[petsc-users] ML and -pc_factor_shift_nonzero

[Jed Brown]

On Mon, 19 Apr 2010 07:23:01 -0500, Matthew Knepley <knepley at gmail.com> wrote:
> So, to see if I understand correctly. You are saying that you can get
> away with more approximate solves if you do not do full reduction? I
> know the theory for the case of Stokes, but can you prove this in a
> general sense?

The theory is relatively general (as much as preconditioned GMRES is) if
you iterate in the full space with either block-diagonal or
block-triangular preconditioners.  Note that this formulation *never*
involves explicit application of a Schur complement.  Sometimes I get
better convergence with one subcycle on the Schur complement with a very
approximate inner solve (FGMRES outer).  I'm not sure if Dave sees this,
he seems to like doing a couple subcycles in multigrid smoothers.

The folks doing Q1-Q1 with ML are not doing *anything* with a Schur
complement (approxmate or otherwise).  They just coarsen on the full
indefinite system and use ASM (overlap 0 or 1) with ILU to precondition
the coupled system.  This makes a certain amount of sense because for
those stabilized formulations, this is similar in spirit to a Vanka
smoother (block SOR is a more precise analogue).

> This sounds like the black magic I expect :)

Yeah, this involves some sort of very local solve to produce the
aggregates and interpolations that are not transposes of each other (if
I understood Ray and Eric correctly).

> I still maintain that aggregation is a really crappy way to generate
> coarse systems, especially for mixed elements. We should be generating
> coarse systems geometrically, and then using a nice (maybe Black-Box)
> framework for calculating good projectors.

This whole framework doesn't work for mixed discretizations.

Jed