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

[Barry Smith]

On Nov 5, 2011, at 11:25 AM, 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.

   We (that is, you :-)  should document this clearly somewhere. Maybe in the index of the users manual? Definitely in the PCFIELDSPLIT manual page. Maybe in the FAQ?

   Barry

> 
> 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.