Analysis of preconditioned iterative methods for the Helmholtz equation Antti Hannukainen1 The nite element discretization of time-harmonic wave propagation problems, such as the Helmholtz equation or the Maxwell equations, leads a to solution of large, com- plex valued, inde nite, non-hermitian and non-normal linear systems. These systems are typically solved with a suitable preconditioned iterative method. Due to inde niteness and non-normality of the linear system, a rigorous convergence analysis of such iterative solution methods is still a challenge. Because of the non-normality, the eigenvalues alone do not give information of the con- vergence. In addition, due to inde nteness, the interesting eigenvalue is the one closest to the origin. This eigenvalue cannot be bounded using techniques familiar from convergence analysis of elliptic problems. In this talk we discuss these di culties and possible solu- tions to them. As a model problem, we use the Helmholtz equation discretized with the standard rst order nite element method, and solved using the preconditioned GMRES method. The non-normality can be handeled in analysis by using a suitable convergence crite- rion. For GMRES, two possiblities exist , estimating the location of the eld of values or the pseudospectrum. The FOV based convergence criterion has been used to study Laplace preconditioners for Helmholtz equation with losses. The major shortcomings of FOV based convergence criterion are in handling the inde niteness of the linear sys- tem. These problems with FOV based criterion motivate us to study a pseudospectrum based criterion. The pseudospectrum is well suited for handling inde nite problems. We demonstrate this for the Helmholtz equation with absorbing bounary conditions. 1 T