Preconditioning for saddle point problems

[Boyce Griffith]

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 
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.

(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