Multiscale Methods for wave propagation in materials with sign-changing coefficients

TL;DR

This paper develops a multiscale finite element method tailored for electromagnetic wave problems in materials with sign-changing coefficients, ensuring stability and accuracy despite the challenges posed by metamaterials.

Contribution

It introduces a specialized CEM-GMsFEM framework for sign-changing coefficients, with theoretical stability analysis and validated numerical results.

Findings

The method achieves stable solutions for sign-changing coefficients.

Error estimates confirm the method's accuracy.

Numerical tests demonstrate robustness in complex coefficient profiles.

Abstract

From a mathematical perspective, the extraordinary properties of metamaterials are often reflected in the coefficients of the governing partial differential equations (PDEs). These coefficients may fall outside the assumptions of classical theory, particularly when the effective dielectric permittivity and/or magnetic permeability are negative. This situation can transform a coercive operator into a non-coercive one, potentially leading to ill-posedness. In this paper, we utilize the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM), specifically designed for time-harmonic electromagnetic wave problems, where the construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. Based on the framework of \texttt{T}-coercivity theory and resolution conditions, we establish the inf-sup stability and provide an a priori error estimate for the proposed method. The numerical results demonstrate the effectiveness and robustness of our approach in handling such sophisticated coefficient profiles.