Stable multiscale Petrov–Galerkin finite element method for high frequency acoustic scattering

Abstract

Abstract We present and analyze a pollution-free Petrov–Galerkin multiscale finite element method for the Helmholtz problem with large wave number κ as a variant of Peterseim (2014). We use standard continuous Q 1 finite elements at a coarse discretization scale H as trial functions, whereas the test functions are computed as the solutions of local problems at a finer scale h . The diameter of the support of the test functions behaves like m H for some oversampling parameter m . Provided m is of the order of log ( κ ) and h is sufficiently small, the resulting method is stable and quasi-optimal in the regime where H is proportional to κ − 1 . In homogeneous (or more general periodic) media, the fine scale test functions depend only on local mesh-configurations. Therefore, the seemingly high cost for the computation of the test functions can be drastically reduced on structured meshes. We present numerical experiments in two and three space dimensions.