[petsc-users] KSP convergence problem

[Jed Brown]

On Wed, Mar 20, 2013 at 4:51 PM, John Mousel <john.mousel at gmail.com> wrote:

> Right now, Mehrdad and I are just passing the constant vector. The problem
> is that the null space is extremely expensive to compute. Something like
> 5-20 times the cost of solving the Poisson equation itself depending on the
> problem size. What we have tried in the past is to find a single solution
> to Atrans*n = 0 and pass this as the nullspace. It's had success at making
> the true residual drop in unison with the preconditioned residual. However,
> because we are working with moving boundary problems, the null space is
> changing each time step. In order to get around this, we have decided to
> try to avoid giving the null space, and see if we get an accurate answer,
> and we do get pretty much the same answer when we only require
> preconditioned residual convergence. This is obviously less than robust,
> but we've yet to find a way to get the null space in an efficient manner. I
> tried programming up a GASM type algorithm where BiCG/ILU is used near the
> interface where the solution is not smooth, and GAMG is used far away where
> the changes in the null vector are very very small, but that didn't have
> much success.


It's not usually a good idea to choose a spatial discretization that is
singular with a complicated null space. Proving that an iteration remains
in the benign space is one of the first things demanded from such
discretizations. If you can't find a way to iterate in the null space or
otherwise project it out, then I would seriously reconsider your choice of
this discretization.
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