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

[Matthew Knepley]

On Fri, Oct 18, 2013 at 10:15 AM, Shiva Rudraraju <rudraa at umich.edu> wrote:

> >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?.
>

By "heavy" Jed means a powerful but expensive solver. By "correction based
on a discretization" Jed means
a smoother which is tailored to the specific higher order discretization
that probably will not work in the
general case (and would need to be coded up).

    Matt


> 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.
>>
>
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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