Approximation of eigenvalues in mixed form, Discrete Compactness Property, and application to hp mixed finite elements

In this paper we discuss the Discrete Compactness Property (DCP) which is a well-known tool for the analysis of finite element approximations of Maxwell’s eigenvalues. We restrict our presentation to Maxwell’s eigenvalues, but the theory applies to more general situations and in particular to mixed finite element schemes that can be written in the framework of de Rham complex and which enjoy suitable compactness properties. We investigate the relationships between DCP and standard mixed conditions for the good approximation of eigenvalues. As a consequence of our theory, the convergence analysis of the rectangular hp version of Raviart–Thomas finite elements for the approximation of Laplace eigenvalues is presented as a corollary of the analogous result for hp edge elements applied to the approximation of Maxwell’s eigenvalues [D. Boffi, M. Costabel, M. Dauge, L. Demkowicz, Discrete compactness for the hp version of rectangular edge finite elements, SIAM J. Numer. Anal. 44 (3) (2006) 979–1004].