[petsc-users] Solving indefinite systems with iterative solvers

[Thomas Witkowski]

Can any of you give me some advise, e.g. some link to literature, how  
to solve indefinite systems with iterative solvers in an efficient  
way? I have to solve systems coming from FEM discretization of a  
6th-order time dependent PDE on structured grids. The time is  
discretized with a standard backward euler scheme that introduces the  
current timestep in one of the submatrices. This leads to some  
problems. When the timestep is "small", the overall system has only  
positive (but complex) eigenvalues. Using an apropriate krylov method  
this can be solved within a reasonable number of iterations. When the  
timestep is increased, the smallest eigenvalues become negative and  
much larger in their magnitude. The number of iterations required for  
solving this systems rise dramatically. It may be possible to increase  
the timestep by just 1%, but the number of iterations of solving the  
system rise up to three order of magnitude. Does any of you know some  
methods/literature which can be used to deal with such systems?

If it is of interest, the overall system looks like:

A  0  I
0  A  tI
fI+2A  I  A

A is the discretization of the Laplace operator, I the identity  
operator, t is the timestep and f: R^n -> R is some function defined  
on the domain.

Thanks for any hint,

Thomas