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Continuous Finite Element Method For Maxwell Eigenvalue Problems With Regular Decomposition Technique

Feiyi Liao, Haochen Liu, Hehu Xie

TL;DR

This work develops a robust continuous finite element method for Maxwell eigenvalue problems on polyhedral domains by introducing a high-regularity regular decomposition for $\mathbf{H}_0^s(\mathbf{curl};\Omega)$ with $s>\tfrac12$, allowing a representation $\boldsymbol{\xi}=\mathbf{u}+\nabla p$ with $\mathbf{u}\in\mathbf{H}^{s+1}(\Omega)$ and $p\in H^{s+1}(\Omega)$. The method discretizes with standard continuous Lagrange elements, using a mixed formulation that leverages a discrete exact sequence and smoothed projections to ensure stability and elimination of spurious zero modes. A spectral-convergence framework based on compact operators is developed, proving collective compactness and pointwise convergence of the discrete operators, and yielding optimal rates that depend on eigenfunction regularity and polynomial degree. Numerical experiments on unit cube, thick L-shaped, and Fichera corner domains confirm the theoretical rates and demonstrate the approach’s feasibility with common FE packages, without requiring specialized meshes or stabilization terms.

Abstract

With the regular decomposition technique, we decompose the space $\mathbf{H}_0^s(\mathbf{curl}; Ω)$ into the sum of a vector potential space and the gradient of a scalar space, both possessing higher regularity. Based on this new high order regular decomposition, a novel numerical method using standard high order Lagrange finite elements is designed for solving Maxwell eigenvalue problems. Specifically, the full convergence orders of the eigenpair approximations are proved for the proposed numerical method. Finally, numerical examples are provided to validate the proposed scheme and confirm the theoretical convergence results.

Continuous Finite Element Method For Maxwell Eigenvalue Problems With Regular Decomposition Technique

TL;DR

This work develops a robust continuous finite element method for Maxwell eigenvalue problems on polyhedral domains by introducing a high-regularity regular decomposition for with , allowing a representation with and . The method discretizes with standard continuous Lagrange elements, using a mixed formulation that leverages a discrete exact sequence and smoothed projections to ensure stability and elimination of spurious zero modes. A spectral-convergence framework based on compact operators is developed, proving collective compactness and pointwise convergence of the discrete operators, and yielding optimal rates that depend on eigenfunction regularity and polynomial degree. Numerical experiments on unit cube, thick L-shaped, and Fichera corner domains confirm the theoretical rates and demonstrate the approach’s feasibility with common FE packages, without requiring specialized meshes or stabilization terms.

Abstract

With the regular decomposition technique, we decompose the space into the sum of a vector potential space and the gradient of a scalar space, both possessing higher regularity. Based on this new high order regular decomposition, a novel numerical method using standard high order Lagrange finite elements is designed for solving Maxwell eigenvalue problems. Specifically, the full convergence orders of the eigenpair approximations are proved for the proposed numerical method. Finally, numerical examples are provided to validate the proposed scheme and confirm the theoretical convergence results.

Paper Structure

This paper contains 13 sections, 26 theorems, 127 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discretization for Maxwell eigenvalue problem
  4. Spectral convergence results for operators
  5. Operators and their properties
  6. Collective compactness
  7. Pointwise convergence
  8. Error estimates for eigenpair approximations
  9. Numerical results
  10. Unit cube domain
  11. Thick L-shaped domain
  12. Fichera corner
  13. Conclusions

Key Result

Lemma 2.1

(key3) Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Then the space $X_N(\Omega)$ and $X_T(\Omega)$ are compactly embedded into $\mathbf{L}^2(\Omega)$. In particular, when $\Omega$ is convex, $X_N(\Omega)$ and $X_T(\Omega)$ are continuously embedded into $\mathbf{H}^1(\Omega)$. When

Figures (8)

  • Figure 1: The unit cube domain with the mesh size $h=0.25$
  • Figure 2: Errors for the smallest positive eigenvalue and associated eigenfunction for the Maxwell eigenvalue problem on the unit cube domain
  • Figure 3: Errors for the smallest positive eigenvalue and associated eigenfunction for the Maxwell eigenvalue problem on the unit cube domain, when $\mathbf{X}_{h, \tau0}$ is chosen as $\mathscr{V}_{h, \tau0}^r\cap \mathbf{H}_0^1(\Omega)$
  • Figure 4: The Thick L-shaped domain with the mesh size $h=0.25$
  • Figure 5: Errors of the smallest eigenvalue approximation for the Maxwell eigenvalue problem on the thick L-shaped domain
  • ...and 3 more figures

Theorems & Definitions (45)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Definition 2.1
  • Lemma 2.5
  • ...and 35 more