An extended Lagrange FEM for the Maxwell eigenvalue problem

TL;DR

This paper introduces an extended Lagrange finite element method for solving Maxwell's eigenvalue problem, achieving optimal convergence and improved degrees of freedom compared to traditional edge element methods.

Contribution

The paper develops a novel extended Lagrange finite element space for Maxwell problems, proving convergence and super-convergence properties, with fewer degrees of freedom than existing methods.

Findings

Optimal convergence order verified numerically

Upper bound property of eigenvalues demonstrated

Lower bound property investigated using curl recovery

Abstract

We construct an extended Lagrange FE space to solve the Maxwell equation and its eigenvalue problem in $\mathbb R^d$ $(d=2,3)$, which is the sum of the vectorial $p-$order Lagrange FE space ($p\ge1$) and the gradient of the $p+1-$order Lagrange FE space. The two lowest-order methods in 3D adopt slightly less degrees of freedom than the second family of the same order edge element methods in 3D. We construct a Cl\'{e}ment interpolant operator to prove the discrete compactness of the FE space and the convergence of the new methods for both Maxwell equation and its eigenvalue problem. For the extended linear Lagrange element method, an average-type curl recovery approach is designed to obtain numerical solution of super-convergence. In the numerical part, we verify the optimal convergence order for the two lowest-order methods, discuss the upper bound property of numerical eigenvalues and investigate the lower bound property by the average-type curl recovery approach.