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.
MSC: 35P05 65D32 65M60Keywords: Mixed finite element approximation Maxwell eigenvalue problem Numerical integration a b s t r a c tThe behaviour of electromagnetic resonances in cavities is modelled by a Maxwell eigenvalue problem (EVP). In the present work, we rewrite the corresponding variational problem, as it arises with a view to the application of a finite element method, in a mixed formulation. For the modelling of realistic problems the integrals occurring in this mixed formulation often cannot be evaluated exactly. We take into account the error arising from numerical quadrature and show convergence to the approximations using exact integration. Finally, some numerical results are presented.
Cavity resonators are modelled using a Maxwell eigenvalue problem. In order to obtain a reliable finite element approximation one has to carefully use an appropriate discrete finite element space. In the present paper we extend the known conditions to assure a correct approximation of the spectrum to the case where numerical integration occurs and where curvilinear boundaries are present. We present a set of sufficient conditions which are similar to the case where those so called variational crimes are absent.
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