Mat-1.3656 Seminar on
numerical analysis and computational science
Monday, Feb 24, 2014, room Y347 at 14.15, Eirola & Stenberg
Antti Hannukainen, Aalto!, Department of Mathematics and Systems Analysis
Pseudospectrum based GMRES convergence analysis for the Helmholtz equation
Finite element simulation of time-harmonic wave propagation
problems leads to solution of very large indefinite linear systems. When
losses, absorbing boundary or impedance boundary conditions are
present, as often in realistic engineering applications, these linear
systems are complex valued, non-Hermitian and non-normal. The large size
of the system restricts the use of direct solvers making preconditioned
iterative solvers the method of choice, especially in the
high-frequency domain.
Analyzing the convergence properties of such
preconditioned iterative methods is difficult. This is mainly due to
indefiniteness and non-normality. Because of non-normality, the
eigenvalues alone do not give information on the convergence. Our
approach is to use convergence criteria based on the location of the
pseudospectrum of the coefficient matrix.
In this talk, we focus on convergence analysis of
the preconditioned GMRES method for Helmholtz equation with first order
absorbing boundary conditions. We show how two basic properties of the
weak problem, stability and boundedness can be used to derive inclusion
and exclusion regions for the pseudospectrum of the coefficient matrix.
As an example, we consider so-called shifted-Laplace preconditioner.