Heterogeneous Multiscale Method for the Maxwell equations with high contrast

Angewandte Mathematik: Institut für Analysis und Numerik, Westfälische Wilhelms-Universität Münster; current address: Institut für Mathematik, Universität Augsburg, Universitätstr. 14, 86159 Augsburg, Germany

^* Corresponding author: barbara.verfuerth@uni-muenster.de

Received: 26 October 2017
Accepted: 25 October 2018

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 Bouchitté, Bourel and Felbacq [C.R. Math. Acad. Sci. Paris 347 (2009) 571–576], 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.