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

TL;DR

This paper introduces a new finite element method for Maxwell eigenvalue problems using a regular decomposition technique, achieving high-order convergence and validated through numerical experiments.

Contribution

It presents a novel high-order regular decomposition approach combined with standard finite elements for Maxwell eigenvalue problems, with proven convergence.

Findings

Proved full convergence orders for eigenpair approximations.

Validated the method with numerical examples.

Confirmed theoretical convergence results through experiments.

Abstract

With the regular decomposition technique, we decompose the space $\mathbf{H}_0^s(\mathbf{curl}; \Omega)$ 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.