[petsc-users] [DMMG] Stokes Solver

[Jed Brown]

On Fri, Apr 8, 2011 at 09:59, <domenico.borzacchiello at univ-st-etienne.fr>wrote:

> Hi,
> I'm trying to implement a 3d stokes solver on a simple cartesian staggered
> grid using the a MAC FV Discretisation. I'm following the example given in
> /snes/examples/tutorials/ex30.c. As the stokes solver is ready I'll add
> some more complex constitutive laws. For the moment I'm testing the solver
> with just 1-level MG and I'd like a few clarification to know if what I'm
> doing is correct.
>
> - The solver solves for u v w p. I'm using a single DA with 4 DOFs and due
> to the MAC arrangement I have some "extraboundary" nodes for u ( in the y
> and z directions) v (x & z dir) w ( x & y dir) and p (x,y & z dir). If
> what I'm getting from ex30.c is right I have to write a simple identity
> for each of these nodes (i.e. p_extra = anyvalue) as they are not coupled
> with the rest of the system. I'm doing the same for Dirichlet BCs nodes
> (i.e. u = Ubound). Is this correct?
>

yes


>
> - How does Petsc deal with the pressure-velocity coupling? Is it correct
> to try to solve the whole coupled system with DMMG as in ex30.c? At
> present time I'm getting no convergence by running dmmgsolve (snes) with
> all the default options on a very small system.
>
>  0 SNES Function norm 7.128085632250e+00
> Number of Newton iterations = 0
> Number of Linear iterations = 0
> Average Linear its / Newton = -nan
> Converged Reason = -3
>

You may as well check why the linear solve failed by running with
-ksp_converged_reason.

There are two challenges for solving Stokes in this way. First, the
interpolation operators that the DA gives you are probably not what you want
(pressure and velocity should be interpolated differently) and second, the
standard smoother of ILU is not expected to work with interlaced velocity
and pressure (you would want to either use a "Vanka smoother" that solves
small problems associated with each pressure cell and all adjacent
velocities, use field-split as a smoother, or (tricky and not guaranteed to
work) order the pressures last in each subdomain with ILU as a smoother).

Vanka smoothers are very problem-dependent so you would need to write that
part yourself. It would be nice to have an example for Stokes. An
alternative to coupled multigrid is to use PCFieldSplit at the top level and
multigrid for the viscous part separately (see e.g. Elman's many papers on
this approach). We don't currently have an example doing PCFieldSplit with
geometric multigrid inside the splits, but it should work with petsc-dev if
you follow the approach in src/ksp/ksp/examples/tutorials/ex45.c (which does
not use DMMG, we are working to fold DMMG's functionality into the existing
solver objects).
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