Multiscale Petrov-Galerkin Method for High-Frequency Heterogeneous Helmholtz Equations
It addresses the pollution effect in finite element methods for high-frequency wave scattering in heterogeneous media, providing a theoretical guarantee for a specific coefficient class.
The paper develops a multiscale Petrov-Galerkin method for high-frequency Helmholtz equations with heterogeneous coefficients, proving it is pollution-free when the stability bound grows polynomially with wave number, and verifying this for a class of smooth coefficients with numerical examples.
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.