Residual-based a posteriori error estimation for the Maxwell's eigenvalue problem

TL;DR

This paper develops a residual-based a posteriori error estimator for Maxwell's eigenvalue problem using Nédélec finite elements, leveraging Helmholtz decomposition and superconvergence to ensure reliability and efficiency.

Contribution

It introduces a new a posteriori error estimator for Maxwell's eigenvalue problem that is proven reliable and efficient, based on Helmholtz decomposition and superconvergence analysis.

Findings

Estimator is reliable up to higher order terms.

Local efficiency is demonstrated using bubble functions.

Numerical tests confirm estimator effectiveness.

Abstract

We present an a posteriori estimator of the error in the L^2-norm for the numerical approximation of the Maxwell's eigenvalue problem by means of N\'ed\'elec finite elements. Our analysis is based on a Helmholtz decomposition of the error and on a superconvergence result between the L^2-orthogonal projection of the exact eigenfunction onto the curl of the N\'ed\'elec finite element space and the eigenfunction approximation. Reliability of the a posteriori error estimator is proved up to higher order terms and local efficiency of the error indicators is shown by using a standard bubble functions technique. The behavior of the a posteriori error estimator is illustrated on a numerical test.