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

[Jed Brown]

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.
>
> Regards,
> Fabian Gabel
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