[petsc-users] Solving/creating SPD systems

[Jed Brown]

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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