Convergence of Lagrange Finite Element Methods for Maxwell Eigenvalue Problem in 3D

TL;DR

This paper proves the convergence of Lagrange finite element methods for the 3D Maxwell eigenvalue problem using advanced mesh techniques and constructs specialized operators to ensure uniform convergence, supported by numerical validation.

Contribution

It introduces a novel convergence proof for quadratic and higher Lagrange finite elements on complex mesh splits in 3D Maxwell problems, with new Fortin-like operators.

Findings

Proves convergence of finite element methods for Maxwell eigenvalues in 3D.

Constructs Fortin-like operators for uniform convergence.

Numerical experiments confirm theoretical results.

Abstract

We prove convergence of the Maxwell eigenvalue problem using quadratic or higher Lagrange finite elements on Worsey-Farin splits in three dimensions. To do this, we construct two Fortin-like operators to prove uniform convergence of the corresponding source problem. We present numerical experiments to illustrate the theoretical results.