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
