Robust Multiscale Methods for Helmholtz equations in high contrast heterogeneous media

TL;DR

This paper introduces a new multiscale finite element method tailored for Helmholtz equations in high-contrast heterogeneous media, effectively addressing pollution effects and operator non-hermiticity.

Contribution

The paper develops the CEM-GMsFEM, a novel multiscale approach that overcomes key challenges in solving Helmholtz equations in complex media, with proven stability and error estimates.

Findings

Method effectively captures physical phenomena in heterogeneous media

Theoretical stability and error bounds are established

Numerical tests validate the method's accuracy and robustness

Abstract

In this paper, we provide the constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) to solve Helmholtz equations in heterogeneous medium. This novel multiscale method is specifically designed to overcome problems related to pollution effect, high-contrast coefficients, and the loss of hermiticity of operators. We establish the inf-sup stability and give an a priori error estimate for this method under a number of established assumptions and resolution conditions. The theoretical results are validated by a set of numerical tests, which further show that the multiscale technique can effectively capture pertinent physical phenomena.