[petsc-users] Solving/creating SPD systems

[Justin Chang]

Yes you are correct matt

On Saturday, November 28, 2015, Justin Chang <jychang48 at gmail.com> wrote:

> 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
> <javascript:_e(%7B%7D,'cvml','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>
>> 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
>>
>
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