[petsc-users] Solving/creating SPD systems

[Justin Chang]

Jed,

1) What exactly are the PETSc options for CGNE? Also, would LSQR be worth
trying? I am doing all of this through Firedrake, so I hope these things
can be done directly through simply providing command line PETSc options :)

2) So i spoke with Matt the other day, and the primary issue I am having
with LSFEM is finding a suitable preconditioner for the problematic penalty
term in Darcy's equation (i.e., the div-div term). So if I had this:

(u, v) + (u, grad(q)) + (grad(p), v) + (grad(p), grad(q)) + (div(u),
(div(v)) = (rho*b, v + grad(q))

If I remove the div-div term, I have a very nice SPD system which could
simply be solved with CG/HYPRE. Do you know of any good preconditioning
strategies for this type of problem?

Thanks,
Justin

On Thu, Dec 10, 2015 at 9:13 PM, Jed Brown <jed at jedbrown.org> wrote:

> Justin Chang <jychang48 at gmail.com> writes:
>
> > 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).
>
> LSFEM ensures an SPD system because it's effectively solving the normal
> equations.  You can use CGNE if you want, but it's not likely to work
> well.  Note that LS methods give up local conservation, which was the
> reason to choose a H(div) formulation in the first place.  You can use
> compatible spaces/Lagrange multiplies with LS techniques to maintain
> conservation, but you'll return to a saddle point problem (with more
> degrees of freedom).  There's no free lunch.
>
> > 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).
>
> (Symmetric) saddle point problems have an SPD Schur complement.  Schur
> complements are dense in general, but some discretization choices (e.g.,
> quadrature in BDM) can make the primal block diagonal or block-diagonal,
> resulting in a sparse Schur complement.  If the Schur complement is
> dense, you might be able to approximate it by a sparse matrix.  The
> quality of such an approximation depends on the physics and the
> discretization.
>
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