Antti Hannukainen: Convergence analysis of preconditioned GMRES for the Helmholtz equation

Finite element discretization of time-harmonic wave propagation problems leads to solution of a linear system Ax = b. When  losses or absorbing boundary conditions are present, as often in realistic engineering applications, the matrix A is complex valued, non-Hermitian and non-normal. These properties and the large number of unknowns make the linear system difficult to solve. The use of direct methods is restricted by the amount of memory, where as iterative methods suffer from the lack of good preconditioners. 

In order to develop more efficient preconditioners, one should be able to analyze the existing ones. However, convergence analysis of preconditioned iterative methods for wave-propagation problems is still under development. This is mainly due to difficulties in handling indefiniteness and non-normality. For example, due to non-normality,  eigenvalues alone do not completely characterize GMRES convergence properties.  

In this talk, we discuss different  tools for convergence analysis of  preconditioned GMRES method when applied to the Helmholtz equation. We focus on convergence criteria based on the field of values (FOV) or the pseudospectrum  and show how these criteria can be used to study two-level and shifted-laplace preconditioners when absorbing boundary conditions or losses are present in the system.