<div class="gmail_quote">On Thu, Feb 16, 2012 at 06:49, Thomas Witkowski <span dir="ltr"><<a href="mailto:thomas.witkowski@tu-dresden.de">thomas.witkowski@tu-dresden.de</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div id=":3dv">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?<br>
</div></blockquote><div><br></div><div>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).</div>
<div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div id=":3dv">
<br>
>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?</div>
</blockquote></div><br><div>Make the right hand side consistent by removing the null space.</div>