[petsc-users] Multigrid preconditioning of entire linear systems for discretized coupled multiphysics problems

[Matthew Knepley]

On Mon, Mar 2, 2015 at 5:29 PM, Jed Brown <jed at jedbrown.org> wrote:

> Fabian Gabel <gabel.fabian at gmail.com> writes:
>
> > Dear PETSc Team,
> >
> > I came across the following paragraph in your publication "Composable
> > Linear Solvers for Multiphysics" (2012):
> >
> > "Rather than splitting the matrix into large blocks and
> > forming a preconditioner from solvers (for example, multi-
> > grid) on each block, one can perform multigrid on the entire
> > system, basing the smoother on solves coming from the tiny
> > blocks coupling the degrees of freedom at a single point (or
> > small number of points). This approach is also handled in
> > PETSc, but we will not elaborate on it here."
> >
> > How would I use a multigrid preconditioner (GAMG)
>
> The heuristics in GAMG are not appropriate for indefinite/saddle-point
> systems such as arise from Navier-Stokes.  You can use geometric
> multigrid and use the fieldsplit techniques described in the paper as a
> smoother, for example.
>
> > from PETSc on linear systems of the form (after reordering the
> > variables):
> >
> > [A_uu   0     0   A_up  A_uT]
> > [0    A_vv    0   A_vp  A_vT]
> > [0      0   A_ww  A_up  A_wT]
> > [A_pu A_pv  A_pw  A_pp   0  ]
> > [A_Tu A_Tv  A_Tw  A_Tp  A_TT]
> >
> > where each of the block matrices A_ij, with i,j \in {u,v,w,p,T}, results
> > directly from a FVM discretization of the incompressible Navier-Stokes
> > equations and the temperature equation. The fifth row and column are
> > optional, depending on the method I choose to couple the temperature.
> > The Matrix is stored as one AIJ Matrix.
>

If you have an unstructured grid, we have examples using regular refinement
and GMG. We are working towards having irregular refinement using the
Pragmatic library from ICL.

  Thanks,

     Matt


> > Regards,
> > Fabian Gabel
>



-- 
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
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20150302/27cff010/attachment.html>