What do with singular blocks in block matrix preconditioning?
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
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? Thomas 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
On Feb 17, 2012, at 12:54 PM, Thomas Witkowski wrote:
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?
You might find these things in "Mixed and Hybrid Finite Element Methods" by F. Brezzi and M. Fortin. By the way, it seems your question is more suitably posed on http://scicomp.stackexchange.com
Thomas
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
On Fri, Feb 17, 2012 at 06:54, Thomas Witkowski < [email protected]> wrote:
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?
It's common to project the null space out of the right hand side (yielding a "consistent" right hand side).
On Thu, Feb 16, 2012 at 06:49, Thomas Witkowski < [email protected]> 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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Thomas Witkowski