An Interior Penalty Method with $C^0$ Finite Elements for the Approximation of the Maxwell Equations in Heterogeneous Media: Convergence Analysis with Minimal Regularity

TL;DR

This paper introduces an interior penalty method using $C^0$ finite elements for Maxwell equations in heterogeneous media, providing convergence analysis under minimal regularity assumptions and demonstrating spectral correctness across polynomial degrees.

Contribution

It presents a novel interior penalty technique with convergence proof for Maxwell equations in Lipschitz domains with minimal regularity.

Findings

Method converges for all polynomial degrees.

Ensures spectral correctness of the approximation.

Applicable to domains with minimal regularity.

Abstract

The present paper proposes and analyzes an interior penalty technique using $C^0$-finite elements to solve the Maxwell equations in domains with heterogeneous properties. The convergence analysis for the boundary value problem and the eigenvalue problem is done assuming only minimal regularity in Lipschitz domains. The method is shown to converge for any polynomial degrees and to be spectrally correct.