Heterogeneous Multiscale Method for the Maxwell equations with high contrast

TL;DR

This paper introduces a novel Heterogeneous Multiscale Method for high-contrast Maxwell scattering problems, providing new homogenization results, stability analysis, and error estimates with explicit wavenumber dependence, validated by numerical experiments.

Contribution

It presents a new multiscale method tailored for high-contrast Maxwell equations, including a homogenization approach and stability analysis with explicit wavenumber dependence.

Findings

New homogenization result for high-contrast Maxwell problems

Stability analysis with explicit wavenumber dependence

Numerical experiments confirming theoretical convergence

Abstract

In this paper, we suggest a new Heterogeneous Multiscale Method (HMM) for the (time-harmonic) Maxwell scattering problem with high contrast. The method is constructed for a setting as in Bouchitt\'e, Bourel and Felbacq (C.R. Math. Acad. Sci. Paris 347(9-10):571--576, 2009), where the high contrast in the parameter leads to unusual effective parameters in the homogenized equation. We present a new homogenization result for this special setting, compare it to existing homogenization approaches and analyze the stability of the two-scale solution with respect to the wavenumber and the data. This includes a new stability result for solutions to time-harmonic Maxwell's equations with matrix-valued, spatially dependent coefficients. The HMM is defined as direct discretization of the two-scale limit equation. With this approach we are able to show quasi-optimality and a priori error estimates in energy and dual norms under a resolution condition that inherits its dependence on the wavenumber from the stability constant for the analytical problem. This is the first wavenumber-explicit resolution condition for time-harmonic Maxwell's equations. Numerical experiments confirm our theoretical convergence results.