[petsc-users] ML and -pc_factor_shift_nonzero

[tribur at vision.ee.ethz.ch]

Hi Jed and Matt,

thanks a lot for your help and the interesting discussion.

Kathrin


Quoting "Jed Brown" <jed at 59a2.org>:

> 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
>