An operator method for a numerical quadrature finite element method for a Maxwell‐eigenvalue problem

Abstract

SUMMARYWe consider a Maxwell-eigenvalue problem on a brick. As is well known, we need to pay special attention to avoiding the so-called spurious eigenmodes. We extend the results obtained in (SIAM J. Numer. Anal. 2000; 38:580-607) to include the use of numerical quadrature. For simplicity, we restrict ourselves to a Gauss-Lobatto integration scheme. The numerical quadrature variational problem can be recasted in an operator form. The main goal of the article consists of proving that a set of necessary and sufficient conditions for spurious freeness remain valid while using numerical quadrature with sufficient precision.

The Influence of Variational Crimes on the Finite Element Approximation of a Maxwell Eigenvalue Problem

“…In this paper we considered the convergence of finite element approximations of a finite element Maxwell EVP We extended known results on the occurrence of numerical integration [4] and curvilinear boundaries [5] and have shown convergence results for the case where both variational crimes are committed simultaneously.…”

“…In [4] the appfication of a numerical integration on the variational and operator formulation has been considered, while in [5] the inclusion of curvilinear boundaries was studied. In the present work we combine both results to the situation were both "variational crimes" are present and extend the results obtained in [4] and [5].…”

On the influence of numerical integration on mixed finite element approximations of a Maxwell eigenvalue problem

“…Next, from the properties of µ −1 , the uniform equivalence of the norms · and · h , (cf. [7,Lemma 4.4]), and Definitions 7 and 6, we find that…”

“…e.g. [2]), one easily obtains the coercivity ofâ(·, ·) and the inf-sup-condition forb(·, ·) (see [7,Lemma 4.4 and 4.10]). From this observation, one may readily show Proof.…”