Maxwell eigenvalues and discrete compactness in two dimensions

Abstract

We present an elementary proof of the discrete compactness result for a general class of hp finite elements introduced in [1,2]. We discuss h-convergence of 2D elements only, and in this context, the results are not new as the analysis of H(curl-conforming elements for Maxwell's equations can be reduced to the long-known results for Raviart-Thomas elements [3]. The work is based on the result of Kikuchi [4,5] for Nedelec's edge triangular elements of the lowest order and presents an alternative to techniques presented in [3,6]. In particular, the present version does not use an inverse inequality argument, and therefore, is valid for h-adaptive meshes. We conclude the presentation with a number of 2D computational experiments, including nonconvex domains.