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

[Shiva Rudraraju]

>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?.


On Thu, Oct 17, 2013 at 10:36 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:

> Shiva Rudraraju <rudraa at umich.edu> writes:
>
> > By Spectral Elements I mean spectral quadrilateral/hexahedral elements
> > based on tensor product lagrangian polynomials on Gauss Lobatto Legendre
> > points.
>
> Okay "both Lagrange and Spectral elements" sounded like you wanted to
> distinguish between two classes of methods.
>
> >  >You could reorder your equations, but multicolor GS is not a very good
> or
> > representative algorithm on cache-based architectures, due to its poor
> > cache reuse.  I suggest just using standard GS smoothers (-pc_type sor
> with
> > default relaxation parameter of 1.0).
> > I plan to implement multicolor GS precisely to demonstrate its poor
> > performance as compared to other iterative and MG schemes, because in the
> > Phase Field community multicolor GS is still quite popular and lingers
> > around as a solver. The main point of this work is to clearly demonstrate
> > the ill-suitedness of GS for  these coupled transport problems.
>
> Block Jacobi/SOR is still popular and useful.
>
> >
> > So just wondering if there are any related examples showing multicolor
> > GS as a solver. Also, since you mentioned, are there any references
> > which demonstrate the poor cache reuse of multicolor GS or is it too
> > obvious?...  just curious.
>
> I though multicolor GS mostly died when cache-based architectures beat
> out vector machines.  One well-optimized application that uses
> multicolor GS is FUN3D, but it is doing nonlinear point-block
> Gauss-Seidel with a second order residual and first-order correction,
> and adds line smoothers for boundary layers.
>
> > Sorry I forgot to mention..... I am only interested in structured
> quad/hex
> > elements. I have my old implementations of higher order Lagrange elements
> > and also used deal.ii's Spectral elements.... but for this work I will
> more
> > or less write one from scratch. So any pointers to efficient tensor grid
> > FEM implementation will really help me.
>
> 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.
>
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