[petsc-users] What do with singular blocks in block matrix preconditioning?

[Jed Brown]

On Thu, Feb 16, 2012 at 06:49, Thomas Witkowski <
thomas.witkowski at tu-dresden.de> wrote:

> I consider a 2x2 block matrix (saddle point) with the left upper block
> being singular due to Neumann boundary conditions. The whole block matrix
> is still non-singular. I worked on some ideas for block preconditioning,
> but there is always some problem with the singular block. All publications
> I know assume the block to be definite. There is also some work on highly
>  singular blocks, but this is here not the case. Does some of you know
> papers about block preconditioners for some class of 2x2 saddle point
> problems, where the left upper block is assumed to be positive
> semi-definite?
>

I could search, but I don't recall a paper specifically addressing this
issue. In practice, you should remove the constant null space and use a
preconditioner that is stable even on the singular operator (as with any
singular operator).


>
> From a more practical point of view, I have the problem that,
> independently of a special kind of block preconditioner, one has always to
> solve (or to approximate the solution) a system with the singular block
> with an arbitrary right hand side. But in general the right hand side does
> not fulfill the compatibility condition of having zero mean. Is there a way
> out of this problem?
>

Make the right hand side consistent by removing the null space.
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