[petsc-users] ML and -pc_factor_shift_nonzero
jed at 59a2.org
Mon Apr 19 07:36:42 CDT 2010
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.
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