A new Heterogeneous Multiscale Method for time-harmonic Maxwell's equations based on divergence-regularization

TL;DR

This paper introduces a novel heterogeneous multiscale method for time-harmonic Maxwell equations in periodic media, utilizing divergence-regularization to simplify implementation and enable rigorous error analysis.

Contribution

The paper develops a new HMM that incorporates divergence-regularization in cell problems, facilitating easier implementation and providing a rigorous theoretical foundation.

Findings

Proves a priori error estimates in $ extbf{H}( ext{curl})$ and $H^{-1}$ norms.

Establishes equivalence between the HMM and a discretized two-scale homogenized equation.

Derives reliable a posteriori error estimates for the method.

Abstract

In this paper, we suggest a new heterogeneous multiscale method (HMM) for the time-harmonic Maxwell equations in locally periodic media. The method is constructed by using a divergence-regularization in one of the cell problems. This allows us to introduce fine-scale correctors that are not subject to a cumbersome divergence-free constraint and which can hence easily be implemented. To analyze the method, we first revisit classical homogenization theory for time-harmonic Maxwell equations and derive a new homogenization result that makes use of the divergence-regularization in the two-scale homogenized equation. We then show that the HMM is equivalent to a discretization of this equation. In particular, writing both problems in a fully coupled two-scale formulation is the crucial starting point for a corresponding numerical analysis of the method. With this approach we are able to prove rigorous a priori error estimates in the $\mathbf{H}(\mbox{curl})$- and the $H^{-1}$-norm and we derive reliable and efficient localized residual-based a posteriori error estimates.