Mat-1.3656 Seminar on numerical analysis and computational science

Monday, March 16, 2009, room U322 at 14.15, Eirola & Stenberg

Antti Hannukainen, TKK Mat

On preconditioning of iterative solution methods for the Helmholtz equation

In this talk, we will consider preconditioned iterative methods for the linear system arising from finite element discretizations of the Helmholtz equation, in the case of real and complex valued wave numbers. The real valued case yields a real symmetric indefinite linear system. Schwarz type preconditioners for these systems are based on a sufficiently fine coarse mesh, which takes care the negative eigenvalues. The remaining part is handled with a preconditioner for the Laplace operator.

In the complex valued case, the resulting linear system will be non-normal. In this case, the convergence analysis cannot be found in the literature. We will discuss, how one can analyze convergence of GMRES for these linear system by using the field of values.

As the classical Schwarz methods are all effectively two-level methods requiring a sufficiently fine coarse grid, we will also discuss decomposition methods for the Helmholtz equation. Domain decomposition methods offer a good alternative for the Schwartz type methods and GMRES.