Preconditioning for saddle point problems

[Jed Brown]

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 naïve, 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
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