A Posteriori Error Estimates for Maxwell’s Eigenvalue Problem

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 Nédélec 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 Nédélec 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.

Adaptive Finite Element Method for the Maxwell Eigenvalue Problem

“…The analysis relies on the classical equivalence with a mixed variational formulation [10]. The numerical results presented in [12] confirm that the adaptive scheme driven by our error indicator converges in three dimensions with optimal rate with respect to the number of degrees of freedom.…”

“…We recall the reliability and efficiency properties proved in [12]. It turns out that in the case of the mixed formulation it is possible to obtain a local efficiency estimate.…”

“…We are going to study and analyze an adaptive finite element scheme in the framework of [26,24,17,29,11]. The scheme is based on the following local error indicator (see [12])…”

“…For the reader's convenience, we recall the reliability and efficiency properties proved in [12]. As it is common for eigenvalue problems, the efficiency property is not local in the sense that it relies on the difference between ω and ω h which is a global quantity.…”

“…Section 3 defines our error indicator and describes the adaptive scheme. Reliability and efficiency from [12] are recalled and the theory concerning the convergence of the adaptive method is described. The auxiliary results needed for the convergence proof are collected in Section 4.…”