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

TL;DR

This paper introduces a weak Galerkin finite element method for Maxwell eigenvalue problems, achieving high convergence accuracy and establishing the lower bound property of eigenvalue approximations.

Contribution

It develops a novel WG finite element scheme that transforms the problem into a simpler elliptic form and proves the lower bound property of the eigenvalues.

Findings

Optimal error estimates with high convergence order

Numerical results confirm theoretical accuracy

Lower bound property of eigenvalues proven

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.