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.