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
