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A weak Galerkin finite element method for solving the asymptotic lower bound of Maxwell eigenvalue problem

Shusheng Li, Qilong Zhai

Abstract

In this paper, we propose a weak Galerkin (WG) finite element method for the Maxwell eigenvalue problem. By restricting subspaces, we transform the mixed form of Maxwell eigenvalue problem into simple elliptic equation. Then we give the WG numerical scheme for the Maxwell eigenvalue problem. Furthermore, we obtain the optimal error estimates of arbitrarily high convergence order and prove the lower bound property of numerical solutions for eigenvalues. Numerical experiments show the accuracy of theoretical analysis and the property of lower bound.

A weak Galerkin finite element method for solving the asymptotic lower bound of Maxwell eigenvalue problem

Abstract

In this paper, we propose a weak Galerkin (WG) finite element method for the Maxwell eigenvalue problem. By restricting subspaces, we transform the mixed form of Maxwell eigenvalue problem into simple elliptic equation. Then we give the WG numerical scheme for the Maxwell eigenvalue problem. Furthermore, we obtain the optimal error estimates of arbitrarily high convergence order and prove the lower bound property of numerical solutions for eigenvalues. Numerical experiments show the accuracy of theoretical analysis and the property of lower bound.
Paper Structure (7 sections, 15 theorems, 63 equations, 3 figures, 1 table)

This paper contains 7 sections, 15 theorems, 63 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Maxwell eigenvalue problem
  3. The weak Galerkin finite element method
  4. Error analysis
  5. Asymptotic lower bound
  6. Numerical Experiments
  7. Square domain

Key Result

Lemma 3.3

\newlabelprojectionMR3394450 For any $\mathbf{v}\in H({\rm curl}; \Omega)$, we have and for any $q\in H^1(\Omega)$, there holds

Figures (3)

  • Figure 6.1: the first component of the third eigenfunction.
  • Figure 6.2: the second component of the third eigenfunction.
  • Figure 6.3: the vectorgraph of the third eigenfunction.

Theorems & Definitions (24)

  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 14 more