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

Abstract

We present and analyze a pollution-free Petrov-Galerkin multiscale finite element method for the Helmholtz problem with large wave number <em>κ</em> as a variant of [Peterseim, ArXiv:1411.1944, 2014]. We use standard continuous <em>Q₁</em> finite elements at a coarse discretization scale <em>H</em> as trial functions, whereas the test functions are computed as the solutions of local problems at a finer scale <em>h</em>. The diameter of the support of the test functions behaves like <em>mH</em> for some oversampling parameter <em>m</em>. Provided <em>m</em> is of the order of log(<em>κ</em>) and <em>h</em> is sufficiently small, the resulting method is stable and quasi-optimal in the regime where <em>H</em> is proportional to <em>κ−1</em>. 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.