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

[Thomas Witkowski]

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?

 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?

Thomas