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

[Jed Brown]

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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