[petsc-dev] Bordered systems and low-rank corrections

[Mark F. Adams]

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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