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

We 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.