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. Matt > > Thanks, > > -- Boyce > > > > > Matthew Knepley wrote: > > > 1) I believe the Wathen-Elman-Silvester stuff is the best out there of the > > shelf. I love the review > > > > A.C. de Niet and F.W. Wubs "Two preconditioners for saddle point > > problems in fluid flows" > > Int. J. Num. Meth. Fluids 2007: 54: 355-377 > > > > 2) Note in there that Augmented Lagrangian preconditioning (ala Axelsson) > > works > > even better, but the system is harder to invert. I like these > > because you only need > > one field in the formulation, instead of having a mixed system. > > This is detailed in > > Brenner & Scott (Iterated Penalty method). > > > > 3) If you want a mixed system, there is new code in PETSc > > (PCFieldSplit) to do exactly > > what you want, and it works automatically with DAs. If you do it > > by hand, you provide > > explicitly the IS for each field. > > > > Matt > > > > On Tue, Apr 29, 2008 at 10:14 AM, Jed Brown <jed at 59a2.org> wrote: > > > > > On Tue 2008-04-29 10:44, Lisandro Dalcin wrote: > > > > Well, I've worked hard on similar methods, but for incompressible NS > > > > equations (pressure-convection preconditioners, Elman et al.). I > > > > abandoned temporarily this research, but I was not able to get decent > > > > results. However, for Stokes flow it seens to work endeed, but never > > > > studied this seriously. > > > > > > My experiments with the Stokes problem shows that it takes about four > > > times as > > > long to solve the indefinite Stokes system as it takes to solve a poisson > > > problem with the same number of degrees of freedom. For instance, in 3D > > > with > > > half a million degrees of freedom, the Stokes problem takes 2 minutes on > > > my > > > laptop while the poisson problem takes 30 seconds (both are using > > > algebraic > > > multigrid as the preconditioner). Note that these tests are for a > > > Chebyshev > > > spectral method where the (unformed because it is dense) system matrix is > > > applied via DCT, but a low-order finite difference or finite element > > > approximation on the collocation nodes is used to obtain a sparse matrix > > > with > > > equivalent spectral properties, to which AMG is applied. With a finite > > > difference discretization (src/ksp/ksp/examples/tutorials/ex22.c) the > > > same sized > > > 3D poisson problem takes 13 seconds with AMG and 8 with geometric > > > multigrid. > > > This is not a surprise since the conditioning of the spectral system is > > > much > > > worse, O(p^4) versus O(n^2), since the collocation nodes are > > > quadratically > > > clustered. > > > > > > I've read Elman et al. 2002 ``Performance and analysis of saddle point > > > preconditioners for the discrete steady-state Navier-Stokes equations'' > > > but I > > > haven't implemented anything there since I'm mostly interested in slow > > > flow. > > > Did your method work well for the Stokes problem, but poorly for NS? I > > > found > > > that performance was quite dependent on the number of iterations at each > > > level > > > and the strength of the viscous preconditioner. I thought my approach > > > was > > > completely na?ve, but it seems to work reasonably well. Certainly it is > > > much > > > faster than SPAI/ParaSails which is the alternative. > > > > > > > > > > I'll comment you the degree of abstraction I could achieve. In my base > > > > FEM code, I have a global [F, G; D C] matrix (I use stabilized > > > > methods) built from standard linear elements and partitioned across > > > > processors in a way inherited by the mesh partitioner (metis). So the > > > > F, G, D, C entries are all 'interleaved' at each proc. > > > > > > > > In order to extract the blocks as parallel matrices from the goblal > > > > saddle-point parallel matrix, I used MatGetSubmatrix, for this I > > > > needed to build two index set for momentum eqs and continuity eqs > > > > local at each proc but in global numbering. Those index set are the > > > > only input required (apart from the global matrix) to build the > > > > preconditioner. > > > > > > This seems like the right approach. I am extending my collocation > > > approach to a > > > hp-element version, so the code you wrote might be very helpful. How > > > difficult > > > would it be to extend to the case where the matrices could be MatShell? > > > That > > > is, to form the preconditioners, we only need entries for approximations > > > S' and > > > F' to S and F respectively; the rest can be MatShell. In my case, F' is > > > a > > > finite difference or Q1 finite element discretization on the collocation > > > nodes > > > and S' is the mass matrix (which is the identity for collocation). > > > > > > Would it be useful for me to strip my code down to make an example? > > > It's not > > > parallel since it does DCTs of the entire domain, but it is a spectrally > > > accurate, fully iterative solver for the 3D Stokes problem with nonlinear > > > rheology. I certainly learned a lot about PETSc while writing it and > > > there > > > aren't any examples which do something similar. > > > > > > Jed > > > > > > > > > > > > > > > > -- 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
