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

We present and analyze a pollution-free Petrov–Galerkin multiscale finite element method for the Helmholtz problem with large wave number κκ as a variant of Peterseim (2014). We use standard continuous Q1Q1 finite elements at a coarse discretization scale HH as trial functions, whereas the test functions are computed as the solutions of local problems at a finer scale hh. The diameter of the support of the test functions behaves like mHmH for some oversampling parameter mm. Provided mm is of the order of log(κ)log(κ) and hh is sufficiently small, the resulting method is stable and quasi-optimal in the regime where HH is proportional to κ−1κ−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.