[petsc-users] Scalable Solver for Incompressible Flow

Tue May 2 08:12:25 CDT 2023

On Tue, May 2, 2023 at 9:07 AM Blauth, Sebastian <
sebastian.blauth at 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).
>
>
>
> To test the solver, I am using a simple 2D channel flow problem with
> do-nothing conditions at the outlet.
>
>
>
> I am facing the following difficulties at the moment:
>
>
>
> - 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

>     -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?
>
>
>
> 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?
>
>
>
> Thanks a lot in advance and I am looking forward to your reply,
>
> Sebastian
>
>
>
> --
>
> Dr. Sebastian Blauth
>
> Fraunhofer-Institut für
>
> Techno- und Wirtschaftsmathematik ITWM
>
> Abteilung Transportvorgänge
>
> Fraunhofer-Platz 1, 67663 Kaiserslautern
>
> Telefon: +49 631 31600-4968
>
> sebastian.blauth at 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/>
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