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