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

[Shiva Rudraraju]

Thanks Jed for the prompt reply.

> How do you distinguish "Lagrange" versus "Spectral" elements?  Are you referring
to nodal versus modal bases, choice of collocation quadrature, or something
else?
By Spectral Elements I mean spectral quadrilateral/hexahedral elements
based on tensor product lagrangian polynomials on Gauss Lobatto Legendre
points.

 >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.
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.

>Do you already have spectral element code or are you planning to write one
from scratch?  There are several unstructured FEM examples using DMPlex,
though they do not explicitly expose the tensor product that is
important for efficient high-order methods.  How would you plan to
precondition these things?

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.

And thanks for your paper reference.

> That is just -ksp_type richardson.  Or are you talking about nonlinear
multigrid (-snes_type fas)?
Cool..... -snes_type fas is what I was looking for.

Thanks,
Shiva


On Thu, Oct 17, 2013 at 10:50 AM, Jed Brown <jedbrown at mcs.anl.gov> wrote:

> Shiva Rudraraju <rudraa at umich.edu> writes:
>
> > I am planning to study the effects of geometric multigrid method both as
> a
> > Solver and Preconditioner on a system of transport problems involving
> > coupled diffusion + finite strain mechanics. Part of that study will be
> to
> > evaluate the performance of FD vs higher order FEM (Both Lagrange and
> > Spectral elements).
>
> How do you distinguish "Lagrange" versus "Spectral" elements?  Are you
> referring to nodal versus modal bases, choice of collocation quadrature,
> or something else?
>
> > So in this context I have a few questions:
> > 1)  To motivate the use of MG schemes in this coupled system of
> equations,
> > I plan to initially implement a Red-Black Gauss Siedel solver for FD to
> > serve as a benchmark. Is there a good example related to the Red-Black
> > Gauss Siedel implementation?.
>
> 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).
>
> > 2) I plan to implement a higher order Lagrange and Spectral element FEM
> > code. I found one related example in
> > petsc/src/ksp/ksp/examples/tutorials/ex59.c. Are there any better
> examples
> > of general FEM code implementations.
>
> Do you already have spectral element code or are you planning to write
> one from scratch?  There are several unstructured FEM examples using
> DMPlex, though they do not explicitly expose the tensor product that is
> important for efficient high-order methods.  How would you plan to
> precondition these things?
>
> I did some work on this using embedded low-order discretizations (see
> below) and the library used in that paper is available.  The paper used
> algebraic multigrid, but geometric and semi-geometric methods would be
> better in many settings.  (I'm hoping to merge the good attributes of
> that library into PETSc using the new DMPlex.)
>
> > 3) MG is often used as a preconditioner, but are there any examples
> showing
> > its use as a solver?.
>
> That is just -ksp_type richardson.  Or are you talking about nonlinear
> multigrid (-snes_type fas)?
>
> > I found a few examples here
> >
> http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/concepts/multigrid.html
> ,
> > but nothing in the current version of those codes refer to any MG
> > datastructures like DMMG which were part of the earlier versions.
>
> The DMMG object was a broken abstraction; its functionality was rolled
> into DM and SNES.
>
>
>
> @article {brown2010ens,
>   author = {Brown, Jed},
>   affiliation = {Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie
> (VAW), ETH Zürich, Zürich, Switzerland},
>   title = {Efficient Nonlinear Solvers for Nodal High-Order Finite
> Elements in {3D}},
>   journal = {Journal of Scientific Computing},
>   publisher = {Springer Netherlands},
>   issn = {0885-7474},
>   keyword = {Mathematics and Statistics},
>   pages = {48-63},
>   volume = {45},
>   issue = {1},
>   url = {http://dx.doi.org/10.1007/s10915-010-9396-8},
>   doi = {10.1007/s10915-010-9396-8},
>   year = {2010}
> }
>
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