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

TL;DR

This work tackles time-harmonic wave propagation in media with sign-changing coefficients by extending CEM-GMsFEM with sign-aware auxiliary spaces built from a local eigenproblem using $|\sigma|$ and $|c|$. It proves inf-sup stability via $T$-coercivity under a resolution condition and derives an a priori error bound that combines a wavenumber dependent term and an exponential decay term governed by the oversampling level $m$, yielding optimal $O(H)$ convergence when $m$ scales appropriately with $H$. Theoretical results are complemented by numerical experiments on flat-interface, random-inclusion, and 2D negative-index metamaterial models, showing robustness and accuracy improvements over standard $Q_1$ FEM. The approach demonstrates the potential of sign-aware multiscale bases to efficiently capture complex metamaterial phenomena while maintaining stability and accuracy.

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.

Figures (11)

• Figure 1: An illustration of the two-scale mesh, a fine element $h$, a coarse element $K_i$ and its oversampling coarse element $K_i^m$ with the oversampling layer $m=1$.
• Figure 1: The relative errors of the proposed method with different numbers of oversampling layers $m$ and the $Q_1$ FEM are calculated w.r.t. the coarse mesh size $H$.
• Figure 2: (a) The solution approximated by the CEM-GMsFEM when we choose $H=1/40$ and $m=3$. (d) The solution approximated by the $Q1$ FEM when we choose $H=1/40$. (d) The solution approximated by the $Q1$ FEM when we choose $H=1/40$. (b)/(e) The exact solution. (c) The difference of the solution approximated by CEM-GMsFEM with the exact solution. (f) The difference of solution approximated by Q1-FEM with exact solution.
• Figure 3: (a) Coefficients with $(\sigma_*^+,\sigma_*^-)=(1,10^3)$; (b) Source term based on Gaussian functions.
• Figure 4: The relative errors of the proposed method with different numbers of oversampling layers $m$ and the $Q_1$ FEM are calculated w.r.t. the coarse mesh size $H$.

Theorems & Definitions (20)

• Lemma 3.1
• Lemma 3.2
• Proof 1
• Lemma 3.3
• Proof 2
• Lemma 4.1
• Proof 3
• Theorem 4.2
• Proof 4
• Lemma 4.3