Discrete Compactness for the hp Version of Rectangular Edge Finite Elements

Abstract

Discretization of Maxwell eigenvalue problems with edge finite elements involves a simultaneous use of two discrete subspaces of H1 and H(curl), reproducing the exact sequence condition. Kikuchi's discrete compactness property, along with appropriate approximability conditions, implies the convergence of discrete eigenpairs to the exact ones. In this paper we prove the discrete compactness property for the edge element approximation of Maxwell's eigenpairs on general hp adaptive rectangular meshes. Hanging nodes, yielding 1-irregular meshes, are covered, and the order of the used elements can vary from one rectangle to another, thus allowing for a real hp adaptivity. As a particular case, our analysis covers the convergence result for the p-method.