[petsc-users] How to use multigrid?

Jed Brown jedbrown at mcs.anl.gov
Mon Nov 5 21:44:07 CST 2012

On Sun, Nov 4, 2012 at 7:52 PM, w_ang_temp <w_ang_temp at 163.com> wrote:

> >At 2012-11-04 21:52:23,"Mark F. Adams" <mark.adams at columbia.edu> wrote:
> >Just to add to Jed and Matt's comments:
> >1) What are your equations (I really don't care what physics you are
> modeling, its the equations that we see).  Is is a scalar div (alpha(x)
> grad ) u?
> The linear system Ax=b is resulted from the finite element discretization.
> It is a algebraic equation, based from a elliptic equation, my opinion.
If you use the formulation in the first cited reference, then you have a
saddle point problem. This is extremely important and explains why naive
application of algebraic multigrid did not work. AMG can handle the
variable coefficients (and "ill-conditioning"), but it can't
"automatically" handle the saddle. You can use PCFIELDSPLIT for these
problems, but it's not a black box and it requires that you _understand_
what you are trying to do.

You need to choose a respectable reference on preconditioners for saddle
point problems. Best would be a review paper written since 2000 and with
more than 100 citations. (There are a handful, almost any will be fine.)
Read it first, then read a couple papers specific to poroelasticity and
frame those methods in the context from the saddle point paper. Also read
section 4.5 of the PETSc user's manual on PCFIELDSPLIT.

Yes, this takes time, but if you want to solve these problems robustly, you
have to understand enough about the methods to troubleshoot.

>2) Try using hyper (-pc_type hyper -pc_hypre_type boomeramg).  Look at "PCApply
> 448", this should be like 10-20.  Configure your systems with
> -->download-hyper=1 if you do not have it.
> >3) Why do you say it is ill conditioned?  Stretched grids? large jumps in
> material coefficients?  Why do you think it is illoconditioned?
> Pragmatic geotechnical problems often involve materials with  highly
> varied material zones, such as in soil-structure interaction problems or
> soil-water
> coupled problems.([1]summary;[2-4])
> Reference:
> [1]Krishna Bahadur Chaudhary. Preconditioners for soil-structure
> interaction problems with significant material stiffness contrast[D].
> [2]S.H.Chan. A modified Jacobi preconditioner for solving ill-conditioned
> Biot's consolidation equations using symmetric quasi-minimal residual
> method.International journal for numerical and analytical methods in
> geomechanics[J].
> [3]Kok-Kwang Phoon. Iterative solution of large-scale consolidation and
> constraint finite element equations for 3D problems.International
> e-Conference on
> Modern Trends in Foundation Engineering[J].
> [4]Massimiliano, etc. Ill-conditioning of finite element poroelasticity
> equations. International Journal of Solids and Structures[J].
> Thanks.
> Jim
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