Preconditioning for saddle point problems

[Dave May]

Hey Jed,
    I took a look through the link you provided and wanted to ask a couple
of questions to clarity a couple of things.

1) How well does the the preconditioner you describe work with the power
law rheology implemented? What is the maximum viscosity contrast between
elements and across that box you've tried?

2) When you say you use AMG for the action of y = A^{-1} x, it appears from
your
README that this is defined using hypre. Is this correct? Have you found
boomeramg
to be superior to ML for the A block? In the past I tried both
preconditioners on the
A block for variable viscosity Stokes flow, but I never had much success.
They
either required massive amounts of setup time or the solution time has not
obtained
in O(N). Was there something fancy in the options configuration required to
make hypre
work for you?

Cheers,
    Dave


On Wed, Apr 30, 2008 at 1:14 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 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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