[petsc-users] Solving/creating SPD systems

[Justin Chang]

So I am wanting to compare the performance of various FEM discretization
with their respective "best" possible solver/pre conditioner. There
are saddle-point systems which HDiv formulations like RT0 work, but then
there are others like LSFEM that are naturally SPD and so the CG solver can
be used (though finding a good preconditioner is still an open problem).

I have read and learned that the advantage of LSFEM is that it will always
give you an SPD system, even for non-linear problems (because what you do
is linearize the problem first and then minimize/take the Gateaux
derivative to get the weak form). But after talking to some people and
reading some stuff online, it seems one could also make non SPD systems SPD
(hence eliminating what may be the only advantage of LSFEM).

Two of said people happen to be PETSc developers but I forgot to ask them
how one would achieve that. Or if one really only can achieve S or PD and
not both :)

On Saturday, November 28, 2015, Patrick Sanan <patrick.sanan at gmail.com>
wrote:

> On Sat, Nov 28, 2015 at 06:31:31AM -0600, Matthew Knepley wrote:
> > On Sat, Nov 28, 2015 at 12:10 AM, Justin Chang <jychang48 at gmail.com
> <javascript:;>> wrote:
> >
> > > Hi all,
> > >
> > > Say I have a saddle-point system for the mixed-poisson equation:
> > >
> > > [I  -grad] [u]  = [0]
> > > [-div  0  ] [p]     [-f]
> > >
> > > The above is symmetric but indefinite. I have heard that one could make
> > > the above symmetric and positive definite (SPD). How would I do that?
> And
> > > if that's the case, would this allow me to use CG instead of GMRES?
> > >
> >
> > I believe you just multiply the bottom row by -1. You can use CG for an
> SPD
> > system, but you can
> > use MINRES for symmetric indefinite.
> If I'm remembering correctly, flipping that sign lets you make your system
> alternately P.D. or
> symmetric, but not both. Maybe you were hearing about the Bramble-Pasciak
> preconditioner or a related approach?
> >
> >    Matt
> >
> >
> > > Thanks,
> > > Justin
> > >
> >
> >
> >
> > --
> > 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
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20151128/7772a067/attachment.html>