Preconditioning of GMRES for Helmholtz problems with quasimodes

Finite element methods are effective for Helmholtz problems involving complex geometries and heterogeneous media. However, the resulting linear systems are often large, indefinite, and challenging for iterative solvers, particularly at high wave numbers or near resonant conditions. We derive a GMRES convergence bound that incorporates the nonlinear behavior of the relative residual and relates convergence to harmonic Ritz values. This perspective reveals how small eigenvalues associated with quasimodes can hinder convergence, and when they cease to have an effect. These phenomena occur in domain decomposition, and we illustrate them through numerical experiments. We also combine domain decomposition methods with deflation techniques using (approximate) eigenvectors tailored to resonant regimes. Their impact on GMRES performance is evaluated.