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

Thomas Witkowski thomas.witkowski at tu-dresden.de
Fri Feb 17 05:54:17 CST 2012

Maybe some related question: Most textbooks write that the compatibility 
condition to solve a system with constant null space is that the right 
hand side has zero mean value. Today I read part of the Multigrid-book 
written by Trottenberg, and there the condition is written in a 
different form (eq 5.6.22 on page 185): the integral of the right hand 
side must be equal on the whole domain and on the boundary. Does any of 
you have an explanation for this condition? Is there a book/paper that 
considers the compatibility condition in more details?


Am 16.02.2012 12:49, schrieb 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

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