Convergence analysis of GMRES for the Helmholtz equation via pseudospectrum

Most finite element methods for solving time-harmonic wave-propagation problems lead to a linear system with a non-normal coefficient matrix. The non-normality is due to boundary conditions and losses. One way to solve these systems is to use a preconditioned iterative method. Detailed mathematical analysis of the convergence properties of these methods is important for developing new and understanding old preconditioners. Due to non-normality, there is currently very little existing literature in this direction. In this paper, we study the convergence of GMRES for such systems by deriving inclusion and exclusion regions for the pseudospectrum of the coefficient matrix. All analysis is done a priori by relating the properties of the weak problem to the coefficient matrix. The inclusion is derived from the stability properties of the problem and the exclusion is established via field of values and boundedness of the weak form. The derived tools are applied to estimate the pseudospectrum of time-harmonic Helmholtz equation with first-order absorbing boundary conditions, with and without a shifted-Laplace preconditioner.