Stable Multiscale Petrov-Galerkin Finite Element Method for High Frequency Acoustic Scattering

TL;DR

This paper introduces a pollution-free multiscale Petrov-Galerkin finite element method for high-frequency acoustic scattering problems, achieving stability and quasi-optimality with reduced computational cost through local test functions.

Contribution

The paper develops a novel multiscale Petrov-Galerkin method for the Helmholtz problem that is stable and efficient at high frequencies, with test functions computed locally and only depending on local mesh configurations.

Findings

Method is stable and quasi-optimal for large wave numbers.

Test functions depend only on local mesh configurations in periodic media.

Numerical experiments confirm effectiveness in 2D and 3D.

Abstract

We present and analyze a pollution-free Petrov-Galerkin multiscale finite element method for the Helmholtz problem with large wave number $\kappa$ as a variant of [Peterseim, ArXiv:1411.1944, 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 $mH$ for some oversampling parameter $m$. Provided $m$ is of the order of $\log(\kappa)$ and $h$ is sufficiently small, the resulting method is stable and quasi-optimal in the regime where $H$ is proportional to $\kappa^{-1}$. 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.