[petsc-users] Solving/creating SPD systems

[Justin Chang]

Pardon me for my apparent lack of understanding over what may be simple
concepts, but why is div[u]*div[v] singular in the context of LSFEM?

On Fri, Dec 11, 2015 at 12:15 PM, Jed Brown <jed at jedbrown.org> wrote:

> Justin Chang <jychang48 at gmail.com> writes:
>
> > Jed,
> >
> > 1) What exactly are the PETSc options for CGNE?
>
> -ksp_type cgne
>
> (Conjugate Gradients on the Normal Equations)
>
> > 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 :)
>
> You can try, but I think this line of thinking is getting off in the weeds.
>
> > 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?
>
> That term is singular, so if the penalty is strong, it will be a bear to
> solve.
>
> Penalties suck.
>
> Sometimes you can add more variables to get better compatibility.  See
> FOSLL*, for example.
>
> My opinion is that least squares methods are riddled with lame
> compromises and tradeoffs that you shouldn't have to make.  If you want
> something robust, use compatible spaces and (possibly) deal with the
> fact that you are solving a saddle point problem.
>
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