Numerical homogenization for indefinite time-harmonic Maxwell equations

TL;DR

This paper introduces a new numerical homogenization method for indefinite time-harmonic Maxwell equations in heterogeneous media, effectively reducing computational complexity while maintaining accuracy at high wavenumbers.

Contribution

It develops a novel edge multiscale approach with a nonstandard variational formulation that avoids explicit heterogeneity resolution and scales nearly linearly with the reciprocal of the wavenumber.

Findings

Method achieves accurate solutions without resolving heterogeneity explicitly.

Mesh size depends almost linearly on the reciprocal of the wavenumber.

Numerical experiments validate theoretical stability and approximation properties.

Abstract

We propose a novel numerical homogenization method based on the edge multiscale approach for solving indefinite time-harmonic Maxwell equations in heterogeneous media with large wavenumber. Numerical methods for these equations in homogeneous media with high wavenumber are particularly challenging due to the so-called pollution effect: the mesh size must be significantly smaller than the reciprocal of the wavenumber to achieve a desired accuracy. This challenge is amplified in heterogeneous media, which frequently occur in practical applications such as metamaterial simulations, since resolving the heterogeneity is necessary for obtaining reliable solutions. Our approach overcomes this difficulty by avoiding explicit resolution of the heterogeneity, while employing a mesh size that depends almost linearly on the reciprocal of the wavenumber. The approximation properties and stability of the method rely critically on the development and rigorous analysis of a novel, nonstandard variational formulation, which constitutes the main innovation of this work. Extensive numerical experiments are provided to validate our theoretical findings.