Why it is so difficult to solve Helmholtz problems with iterative methods
In contrast to the positive definite Helmholtz equation, the
deceivingly similar looking indefinite Helmholtz equation is difficult
to solve using classical iterative methods. Applying directly a Krylov
method to the discretized equations without preconditioning leads in
general to stagnation and very large iteration counts. Using classical
incomplete LU preconditioners can even make the situation worse.
Classical domain decomposition and multigrid methods also fail to
converge when applied to such systems.
The purpose of this presentation is to investigate in each case where
the problems lie, and to explain why classical iterative methods have
such difficulties to solve indefinite Helmholtz problems. I will also
present remedies that have been proposed over the last decade, for
incomplete LU type preconditioners, domain decomposition and also