On Nov 5, 2011, at 12:25 PM, Jed Brown wrote:
> On Sat, Nov 5, 2011 at 10:02, Mark F. Adams <mark.adams at columbia.edu>
> wrote:
> FYI: the yellow SIAM book on mixed FE methods by Brezi and Fortin has an
> excellent 2 page section on Uzawa that give, among other things, a precise
> recipe for Uzawa (page 99 I think) including preconditioning and a non-zero
> RHS for the constraint part.
>
> You can run Uzawa with -pc_fieldsplit_type schur and Richardson.
>
Uzawa is a great algorithm, very popular, dead simple (two lines),
embarrassingly robust and fast. I'm sure its easy to express in your
FieldSplit stuff. This paper does a good job and going through its variants,
including non-linear which I'd never seen before, and lots of convergence
theory. Eq 2.5 is probably the algorithm you want to look at.
In the past I have not written the first line in update form like this paper
and I use a Krylov solver with a rtol of ~1.e-3 for the first line of the
algorithm (more like their eq 2.2).
They have a zero K(2,2) block but you just need to modify the second line of
the algo to make it consistent.
Mark
> As a practical matter, I don't see any complication for Woodbury with/without
> nonzero RHS. I would do the Schur complement in the other direction and and
> as the preconditioner for the Schur complement that came from eliminating the
> (small number of) augmented variables, I would use the Woodbury formula with
> only a preconditioner for the A^{-1} that appear in that formula.
>
> If that inner preconditioner was a full solve, then this would provide the
> exact inverse, but that wouldn't gain anything because then CG on the Schur
> complement _in_ the augmented variables would converge without
> preconditioning in the same number of iterations.
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