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 >
