Multiscale Modeling for Time-harmonic Maxwell equations with impedance boundary conditions in highly heterogeneous media

TL;DR

This paper develops a multiscale method for solving time-harmonic Maxwell equations in complex media, overcoming numerical instability issues without requiring divergence-free basis functions, and proves its effectiveness through analysis and experiments.

Contribution

The study introduces a novel multiscale framework that avoids explicit divergence-free constraints by constructing an auxiliary space with spectral problems, ensuring coercivity and stability.

Findings

Method achieves $O(H)$ convergence independent of media contrast.

The approach maintains stability and accuracy at high wave numbers.

Numerical experiments confirm the theoretical predictions.

Abstract

Modeling time-harmonic Maxwell problems in heterogeneous media presents significant mathematical and computational challenges. Due to the inherent non-elliptic structure and non-coercive nature of Maxwell equations, conventional methods face severe numerical instabilities, particularly in high-contrast media and at high wave numbers. These challenges often lead to ill-conditioned discrete systems and prohibitively high computational costs, limiting their practical applicability. To overcome these challenges, we introduce an efficient multiscale framework for time-harmonic Maxwell equations with impedance boundary conditions in high-contrast media. A major novelty of this study lies in circumventing the need for an explicit divergence-free constraint on multiscale basis functions. To achieve this, an auxiliary space is constructed via local spectral problems incorporating a mass term and a Silver-M\"uller-type boundary penalty. This novel design guarantees the coercivity of the corresponding bilinear form and automatically excludes the kernel of the curl operator from the leading eigenspaces. Building upon the auxiliary space, we then construct the multiscale space by using a distinct bilinear form. By exploiting a resolution condition and establishing key norm relationships, we rigorously prove the coercivity of this modified bilinear form a crucial property that underpins the whole analysis. Theoretical analysis shows that, with appropriate oversampling, the method achieves $O (H)$ convergence independent of the local contrast and the approximation error increases with the wave number $k$. Extensive numerical experiments are reported to validate the effectiveness of the proposed approach.