Preconditioning for saddle point problems

[Matthew Knepley]

On Tue, Apr 29, 2008 at 12:54 PM, Boyce Griffith <griffith at cims.nyu.edu> wrote:
>
>
>  Matthew Knepley wrote:
>
> > On Tue, Apr 29, 2008 at 12:28 PM, Boyce Griffith <griffith at cims.nyu.edu>
> wrote:
> >
> >
> > > Hi, Matt et al. --
> > >
> > > Do people ever use standard projection methods as preconditioners for
> these kinds of problems?
> > >
> > > I have been playing around with doing this in the context of a staggered
> grid (MAC) finite difference scheme.  It is probably not much of a surprise,
> but for problems where an exact projection method is actually an exact
> Stokes solver (e.g., in the case of periodic boundary conditions), one can
> obtain convergence with a single application of the projection
> preconditioner when it is paired up with FGMRES.  I'm still working on
> implementing physical boundaries and local mesh refinement for this
> formulation, so it isn't clear how well this approach works for less trivial
> situations.
> > >
> >
> > If I understand you correctly, Wathen and Golub have a paper on this.
> > Basically, it says using
> >
> >  / \hat A    B \
> >  \     B^T  0 /
> >
> > as a preconditioner is great since all the eigenvalues for the
> > constraint are preserved.
> >
>
>  Hi, Matt --
>
>  Are you referring to Golub & Wathen, SIAM J. Sci. Comput. 1998?  I think

Could be. It sound sright.

> they are doing something different.  I am solving the time-dependent Stokes
> equations, and am preconditioning via a fully second-order accurate version
> of the Kim-Moin projection method, i.e., following the approach of Brown,
> Cortez, and Minion, J. Comput. Phys. 2001.

These all look different, but I think they are really the same thing.
Its also the same
as what Vivek Sarin does. All of them project exactly onto the
constraint manifold.
They only differ in how A is preconditioned. I mention Wathen&Gloub because in
their analysis, you can use any preconditioner for A, which is the most general.
However, they do not give a prescription for inverting the preconditioner, which
Vivek does (in O(N) time and space).

  Matt

>  (Note that at this point, I am not trying to treat the advection terms
> implicitly; this is really just a warm-up to doing implicit timestepping for
> fluid-structure interaction.)
>
>  -- Boyce
-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which
their experiments lead.
-- Norbert Wiener