[petsc-users] Starting point for Stokes fieldsplit

Matthew Knepley knepley at gmail.com
Mon Feb 20 14:23:59 CST 2012

On Mon, Feb 20, 2012 at 2:05 PM, Max Rudolph <rudolph at berkeley.edu> wrote:

> Hi Dave,
> Thanks for your help.
> Max
> Hey Max,
> Without knowing anything about the specific application related to
> your Stokes problem, or information about the mesh you are using, I
> have a couple of questions and suggestions which might help.
> The test case that I am working with is isoviscous convection, benchmark
> case 1a from Blankenbach 1989.
> 1) If  A, is your stokes operator A = ( K,B ; B^T, 0 ), what is your
> precondition operator?
> Specifically, what is in the (2,2) slot in the precondioner? - i.e.
> what matrix are you you applying -stokes_fieldsplit_1_pc_type jacobi
> -stokes_fieldsplit_1_ksp_type preonly to?
> Is it the identity as in the SpeedUp notes?
> I think that this is the problem. The (2,2) slot in the LHS matrix is all
> zero (pressure does not appear in the continuity equation), so I think that
> the preconditioner is meaningless. I am still confused as to why this
> choice of preconditioner was suggested in the tutorial, and what is a
> better choice of preconditioner for this block? Should I be using one of
> the Schur complement methods instead of the additive or multiplicative
> field split?

Its not suggested, it is demonstrated. Its the first logical choice, since
Jacobi gives the identity for a 0 block (see
http://www.jstor.org/pss/2158202). Its
not meaningless. All the better preconditioners involve either a Schur
complement (also shown in the tutorial), or an auxiliary operator which is
difficult to setup and thus not shown.

> 2) This choice
> -stokes_fieldsplit_0_pc_type ml -stokes_fieldsplit_0_ksp_type preonly
> may simply not be a very effective and degrade the performance of the
> outer solver.
> I'd make the solver for the operator in the (1,1) slot much stronger,
> for example
>  -stokes_fieldsplit_0_ksp_type gmres
>  -stokes_fieldsplit_0_ksp_rtol 1.0e-4
>  -stokes_fieldsplit_0_mg_levels_ksp_type gmres
>  -stokes_fieldsplit_0_mg_levels_pc_type bjacobi
>  -stokes_fieldsplit_0_mg_levels_ksp_max_it 4
> Add a monitor on this solver (-stokes_fieldsplit_0_ksp_XXX) to see how
> ML is doing.
> 3) Using -stokes_pc_fieldsplit_type MULTIPLICATIVE should reduce the
> number of outer iterations by a factor of two, but it will use more
> memory.
> 4) You should use a flexible Krylov method on the outer most solve
> (-stokes_ksp_XXX) as the preconditioner is varying between each outer
> iteration. Use -stokes_ksp_type fgmres or -stokes_ksp_type gcr
> Thanks for pointing this out. I made that change.
> 5) Depending on how the physical problem is scaled
> (non-dimensionalised), the size of the residuals associated with the
> momentum and continuity equation make be quite different. You are
> currently use the entire residual from (u,p) to determine when to stop
> iterating. You might want to consider writing a monitor which examines
> the these residuals independently.
> I think that I have scaled the problem correctly. I (slowly) obtain a
> sufficiently accurate solution using as options only:
> -stokes_ksp_atol 1e-5 -stokes_ksp_rtol 1e-5
> -stokes_ksp_monitor_true_residual -stokes_ksp_norm_type UNPRECONDITIONED

How do you know the problem is scaled correctly? Have you looked at norms
of the residuals for the two systems?



> Cheers,
>  Dave

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
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