[petsc-users] Geometric MG as Solver & Preconditioner for FEM/Spectral/FD

[Jed Brown]

Shiva Rudraraju <rudraa at umich.edu> writes:

>>I did unstructured hexes.  You still haven't said what you'll use for relaxation.
>  High-order discretizations tend to have poor h-ellipticity, so they either
> need heavy smoothers or a correction based on a discretization with better
> h-ellipticity.
> Quite frankly, I was not aware of the poor h-ellipticity of higher order
> elements and I was assuming I would use the regular GS/GMRES/etc for
> relaxation. I looked up h-ellipticity of higher order elements and now this
> adds to my worries :(. I may be asking for too much here.... but what do
> you mean by heavy smoothers? or correction based on a discretization?.

You can use one discretization for evaluating residuals, but then use an
embedded low-order discretization to apply the correction.  See "defect
correction" in Achi's multigrid guide or in Trottenberg.

The paper I mentioned earlier was doing something simpler and less
intrusive: assemble the embedded low-order operator and feed it to
algebraic multigrid, but evaluate residuals matrix-free using the
high-order discretization.  But if you have a reasonable geometric
hierarchy, the defect correction schemes are better.
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