[petsc-users] move from KSPSetNullSpace to MatSetNullSpace

[Stephan Kramer]

Dear petsc devs

I've been trying to move our code from using KSPSetNullSpace to use 
MatSetNullSpace instead. Although I appreciate the convenience of the 
nullspace being propagated automatically through the solver hierarchy, 
I'm still a little confused on how to deal with the case that mat/=pmat 
in a ksp.

If I read the code correctly I need to call MatSetNullSpace on the pmat 
in order for a ksp to project that nullspace out of the residual during 
the krylov iteration. However the nullspaces of mat and pmat are not 
necessarily the same. For instance, what should I do if the system that 
I want to solve has a nullspace (i.e. the `mat` has a null vector), but 
the preconditioner matrix does not.

As an example of existing setups that we have for which this is the 
case: take a standard Stokes velocity, pressure system - where we want 
to solve for pressure through a Schur complement. Suppose the boundary 
conditions are Dirichlet for velocity on all boundaries, then the 
pressure equation has the standard, constant nullspace. A frequently 
used preconditioner is the "scaled pressure mass matrix" where G^T 
K^{-1} G is approximated by a pressure mass matrix that is scaled by the 
value of the viscosity. So in this case the actual system has a 
nullspace, but the preconditioner matrix does not.

The way we previously used to solve this is by setting the nullspace on 
the ksp of the Schur system, where the mat comes from 
MatCreateSchurComplement and the pmat is the scaled mass matrix. We then 
set the pctype to PCKSP and do not set a nullspace on its associated 
ksp. I don't really see how I can do this using only MatSetNullSpace - 
unless I create a copy of the mass matrix and have one copy with and one 
copy without a nullspace so that I could use the one with the nullspace 
for the pmat of the Schur ksp (and thus the mat of the pcksp) and the 
copy without the nullspace as the pmat for the pcksp.

We would very much appreciate some guidance on what the correct way to 
deal with this kind of situation is

Cheers
Stephan