A finite element method for Maxwell's transmission eigenvalue problem in anisotropic media

TL;DR

This work develops a curl-conforming finite element framework, based on $\mathbb T$-coercivity, to solve Maxwell's transmission eigenvalue problem in anisotropic media. By employing Nedélec edge elements, it formulates a discrete eigenproblem and proves well-posedness for the associated source problems, along with discrete compactness and optimal convergence rates for eigenpairs via a rigorous error analysis. The method is supported by three-dimensional numerical experiments across constant, variable, and anisotropic coefficients, confirming the theoretical rates and demonstrating robustness to anisotropy. The findings advance reliable numerical treatment of MTEP in complex media, with potential applications in inverse scattering and material characterization.

Abstract

In this paper, we introduce a finite element method employing the Nedéléc element space for solving the Maxwell's transmission eigenvalue problem in anisotropic media. The well-posedness of the source problems are derived using $\mathbb T$-coercivity approach. We discuss the discrete compactness property of the finite element space under the case of anisotropic coefficients and conduct a finite element error analysis for the proposed approach. Additionally, we present some numerical examples to support the theoretical result.

Figures (3)

• Figure 1: The three components of $\bm \upsilon_1$ on the cube with $\bm A=2\bm I,\bm N = 16\bm I$ (at the top), $\bm A=F_1\bm I,\bm N = F_2\bm I$ (at the middle) and $\bm A=\bm F_4,\bm N = \bm F_3$ (at the bottom) computed by the quadratic edge element method.
• Figure 2: The three components of $\bm \omega_1-\bm \upsilon_1$ on the cube with $\bm A=2\bm I,\bm N = 16\bm I$ (at the top), $\bm A=F_1\bm I,\bm N = F_2\bm I$ (at the middle) and $\bm A=\bm F_4,\bm N = \bm F_3$ (at the bottom) computed by the quadratic edge element method.
• Figure 3: The three components of $\bm\omega_1-\bm \upsilon_1$ on the thick L-shape domain with $\bm A=2\bm I,\bm N = 16\bm I$ (at the top), $\bm A=F_1\bm I,\bm N = F_2\bm I$ (at the middle) and $\bm A=\bm F_4,\bm N = \bm F_3$ (at the bottom) computed by the quadratic edge element method.

Theorems & Definitions (23)

• Lemma 2.1
• proof
• Lemma 2.2: Lemma 3.50 in monk1
• Corollary 2.1: Poincaré inequality
• proof
• Lemma 2.3
• proof
• Lemma 3.1: Theorem 5.41 in monk1
• Lemma 3.2
• proof