[petsc-users] Poor multigrid convergence in parallel

[Lawrence Mitchell]

On 29 Jul 2014, at 16:58, Jed Brown <jed at jedbrown.org> wrote:

> Lawrence Mitchell <lawrence.mitchell at imperial.ac.uk> writes:
>> No, I'm L2-projecting (with mass-lumping) for the restriction.  So if I
>> weren't lumping, I think this is the dual of the prolongation.  
> 
> A true L2 projection is a dense operation (involves the inverse mass
> matrix).

Sure, hence the lumping.

>  But here, we're trying to map residuals, not states, so
> _integration_ rather than "projection" is the desired operation.

So my approach was to transfer using projection and then use riesz representation to get the residual from the dual space back into the primal space so I can apply the operator at the next level.  Is there an obvious reason why this is worse than the transpose of prolongation?  The analysis I've read typically approaches from an FD angle, which doesn't always obviously (at least to me) map into FE speak.  In particular, moving forward, I'm interested in GMG for H(div) problems, for which the analysis I'm aware of uses projections to move residuals between levels (for example, Arnold, Falk and Winther Num Math 85:197 (2000)).

> Integration is the transpose of your prolongation.  We can do a precise
> analysis, but I expect that you'll see slightly better convergence if
> you modify this operator.

So I coded this up, I only have the matrix-free application of the prolongation, so it was not just taking the transpose.  I get marginally better convergence for this problem, although it's almost a wash.

> I would also recommend trying FMG soon.  You should be able to converge
> in one F-cycle, and that will be faster than V-cycles with Krylov.

This is just -pc_mg_type full, right?  Given that I only have matrix-free transfer operators do I need to explicitly set do PCMGSetRscale such that the state is restricted correctly?
If I just select -pc_mg_type full, I take more than one Krylov iteration to converge.

Cheers,

Lawrence
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