On 02.05.2023 15:12, Matthew Knepley wrote:
On Tue, May 2, 2023 at 9:07 AM Blauth, Sebastian
<sebastian.bla...@itwm.fraunhofer.de
<mailto:sebastian.bla...@itwm.fraunhofer.de>> wrote:
Hello,____
__ __
I am having a problem using / configuring PETSc to obtain a scalable
solver for the incompressible Navier Stokes equations. I am
discretizing the equations using FEM (with the library fenics) and I
am using the stable P2-P1 Taylor-Hood elements. I have read and
tried a lot regarding preconditioners for incompressible Navier
Stokes and I am aware that this is very much an active research
field, but maybe I can get some hints / tips. ____
I am interested in solving large-scale 3D problems, but I cannot
even set up a scaleable 2D solver for the problems. All of my
approaches at the moment are trying to use a Schur Complement
approach, but I cannot get a “good” preconditioner for the Schur
complement matrix. For the velocity block, I am using the AMG
provided by hypre (which seems to work fine and is most likely not
the problem).____
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To test the solver, I am using a simple 2D channel flow problem with
do-nothing conditions at the outlet.____
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I am facing the following difficulties at the moment:____
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- First, I am having trouble with using
-pc_fieldsplit_schur_precondition selfp. With this setup, the cost
for solving the Schur complement part in the fieldsplit
preconditioner (approximately) increase when the mesh is refined. I
am using the following options for this setup (note that I am using
exact solves for the velocity part to debug, but using, e.g., gmres
with hypre boomeramg reaches a given tolerance with a number of
iterations that is independent of the mesh)
The diagonal of the momentum block is a bad preconditioner for the Schur
complement, because S is spectrally equivalent to the mass matrix. You
should build the mass matrix and use that as the preconditioning matrix
for the Schur part. The FEniCS people can show you how to do that. This
will provide mesh-independent convergence (you can see me doing this in
SNES ex69).
Thanks,
Matt
I agree with your comment for the Stokes equations - for these, I have
already tried and used the pressure mass matrix as part of a (additive)
block preconditioner and it gave mesh independent results.
However, for the Navier Stokes equations, is the Schur complement really
spectrally equivalent to the pressure mass matrix? And even if it is,
the convergence is only good for small Reynolds numbers, for moderately
high ones the convergence really deteriorates. This is why I am trying
to make fieldsplit_schur_precondition selfp work better (this is, if I
understand it correctly, a SIMPLE type preconditioner).
Best regards,
Sebastian
-ksp_type fgmres____
-ksp_rtol 1e-6____
-ksp_atol 1e-30____
-pc_type fieldsplit____
-pc_fieldsplit_type schur____
-pc_fieldsplit_schur_fact_type full____
-pc_fieldsplit_schur_precondition selfp____
-fieldsplit_0_ksp_type preonly____
-fieldsplit_0_pc_type lu____
-fieldsplit_1_ksp_type gmres____
-fieldsplit_1_ksp_pc_side right____
-fieldsplit_1_ksp_max_it 1000____
-fieldsplit_1_ksp_rtol 1e-1____
-fieldsplit_1_ksp_atol 1e-30____
-fieldsplit_1_pc_type lu____
-fieldsplit_1_ksp_converged_reason____
-ksp_converged_reason____
__ __
Note, that I use direct solvers for the subproblems to get an
“ideal” convergence. Even if I replace the direct solver with
boomeramg, the behavior is the same and the number of iterations
does not change much. ____
In particular, I get the following behavior:____
For a 8x8 mesh, I need, on average, 25 iterations to solve
fieldsplit_1____
For a 16x16 mesh, I need 40 iterations____
For a 32x32 mesh, I need 70 iterations____
For a 64x64 mesh, I need 100 iterations____
__ __
However, the outer fgmres requires, as expected, always the same
number of iterations to reach convergence (as expected).____
I do understand that the selfp preconditioner for the Schur
complement is expected to deteriorate as the Reynolds number
increases and the problem becomes more convective in nature, but I
had hoped that I can at least get a scaleable preconditioner with
respect to the mesh size out of it. Are there any tips on how to
achieve this?____
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My second problem is concerning the LSC preconditioner. When I am
using this, again both with exact solves of the linear problems or
when using boomeramg, I do not get a scalable solver with respect to
the mesh size. On the contrary, here the number of solves required
for solving fieldsplit_1 to a fixed relative tolerance seem to
behave linearly w.r.t. the problem size. For this problem, I suspect
that the issue lies in the scaling of the LSC preconditioner
matrices (in the book of Elman, Sylvester and Wathen, the matrices
are scaled with the inverse of the diagonal velocity mass matrix).
Is it possible to achieve this with PETSc? I started experimenting
with supplying the velocity mass matrix as preconditioner matrix and
using “use_amat”, but I am not sure where / how to do it this way.____
__ __
And finally, more of an observation and question: I noticed that the
AMG approximations for the velocity block became worse with increase
of the Reynolds number when using the default options. However, when
using -pc_hypre_boomeramg_relax_weight_all 0.0 I noticed that
boomeramg performed way more robustly w.r.t. the Reynolds number.
Are there any other ways to improve the AMG performance in this
regard?____
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Thanks a lot in advance and I am looking forward to your reply,____
Sebastian____
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--____
Dr. Sebastian Blauth____
Fraunhofer-Institut für____
Techno- und Wirtschaftsmathematik ITWM____
Abteilung Transportvorgänge____
Fraunhofer-Platz 1, 67663 Kaiserslautern____
Telefon: +49 631 31600-4968____
sebastian.bla...@itwm.fraunhofer.de
<mailto:sebastian.bla...@itwm.fraunhofer.de>____
https://www.itwm.fraunhofer.de <https://www.itwm.fraunhofer.de>____
__ __
--
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
https://www.cse.buffalo.edu/~knepley/ <http://www.cse.buffalo.edu/~knepley/>
--
Dr. Sebastian Blauth
Fraunhofer-Institut für
Techno- und Wirtschaftsmathematik ITWM
Abteilung Transportvorgänge
Fraunhofer-Platz 1, 67663 Kaiserslautern
Telefon: +49 631 31600-4968
sebastian.bla...@itwm.fraunhofer.de
www.itwm.fraunhofer.de
