Multiscale Petrov-Galerkin Method for High-Frequency Heterogeneous Helmholtz Equations

TL;DR

This paper introduces a multiscale Petrov-Galerkin finite element method for high-frequency acoustic scattering in heterogeneous media, demonstrating pollution-free performance under certain stability conditions and validating through numerical experiments.

Contribution

The paper develops a novel multiscale Petrov-Galerkin method that is pollution-free for high-frequency heterogeneous Helmholtz problems under polynomial stability bounds.

Findings

Method is pollution-free under polynomial stability bounds.

Numerical experiments verify stability estimates.

Applicable to both smooth and discontinuous coefficients.

Abstract

This paper presents a multiscale Petrov-Galerkin finite element method for time-harmonic acoustic scattering problems with heterogeneous coefficients in the high-frequency regime. We show that the method is pollution- free also in the case of heterogeneous media provided that the stability bound of the continuous problem grows at most polynomially with the wave number k. By generalizing classical estimates of [Melenk, Ph.D. Thesis 1995] and [Hetmaniuk, Commun. Math. Sci. 5 (2007)] for homogeneous medium, we show that this assumption of polynomially wave number growth holds true for a particular class of smooth heterogeneous material coefficients. Further, we present numerical examples to verify our stability estimates and implement an example in the wider class of discontinuous coefficients to show computational applicability beyond our limited class of coefficients.