Preconditioning for saddle point problems
Barry Smith
bsmith at mcs.anl.gov
Tue Apr 29 12:28:13 CDT 2008
We are starting to provide this functionality in petsc-dev and it
will be in the
next release.
Barry
On Apr 29, 2008, at 6:54 AM, Jed Brown wrote:
> This issue arose while discussing improved preconditioners for a
> colleague's
> libmesh-based ice flow model. Since libmesh is one of several
> finite element
> libraries that use PETSc as the backend, perhaps the abstraction
> should be
> implemented in PETSc.
>
> As a model, consider the Stokes problem with mixed discretization in
> matrix form
>
> (1) [A, B1'; B2, 0] [u; p] = [f; 0]
>
> where A is the viscous velocity system, B1' is a gradient and B2 is
> divergence.
> The system (1) is indefinite so most preconditioners and many direct
> methods
> will fail, however A is a uniformly elliptic operator so we can use
> exotic
> preconditioners. An effective preconditioner is to solve the Schur
> complement
> problem
>
> (2) S p = -B2 A^{-1} f
>
> where S = -B2 A^{-1} B1' is the Schur complement. Once p is
> determined, we can
> find u by solving
>
> (3) A u = f - B1' p
>
> If we solve (2,3) exactly, this is the ultimate preconditioner and
> we converge
> in one iteration. It is important to note that applying S in (2)
> requires
> solving a system with A. This can be implemented in PETSc using a
> MatShell
> object. The classical Uzawa algorithm is a Richardson iteration on
> this
> problem. A better method (in my experience) is to precondition a
> Krylov
> iteration (FGMRES) on (1) with an approximate solve of (2,3). For
> instance I
> have had success with 3 iterations of GMRES on (2) where the A^{-1}
> in S is
> replaced by one V-cycle of algebraic multigrid and the A^{-1}
> appearing
> explicitly in (2,3) are replaced by 4 iterations of GMRES
> preconditioned with
> AMG.
>
> It is desirable to have all the details of this iteration
> controllable from the
> command line. For an example implementation (a spectral method) of
> this
> problem, check out StokesMatMultXXX() and StokesPCApply() at
>
> http://github.com/jedbrown/spectral-petsc/tree/master/stokes.C
>
> Note that typically, S is preconditioned with the mass matrix (or a
> ``pressure-laplacian'' at higher Reynolds number) but since this
> code uses a
> collocation method and is only intended for slow flow, there is no
> preconditioner for S.
>
>
> My question for the PETSc developers: how much of this can/should be
> abstracted
> out? While it's not difficult to code, perhaps there should at
> least be an
> example. If it would be useful, I can strip my experiment above
> down to make it
> suitable.
>
>
> PS: An excellent review paper:
> @article{benzi2005nss,
> title={{Numerical solution of saddle point problems}},
> author={Benzi, M. and Golub, G.H. and Liesen, J.},
> journal={Acta Numerica},
> volume={14},
> pages={1--137},
> year={2005},
> publisher={Cambridge Univ Press}
> }
>
> Jed
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