In this paper we prove the discrete compactness property for a wide class of $p$ finite element approximations of nonelliptic variational eigenvalue problems in two and three space dimensions. In a very general framework, we find sufficient conditions for the $p$-version of a generalized discrete compactness property, which is formulated in the setting of discrete differential forms of order $\ell$ on a polyhedral domain in $\mathbb{R}^d$ ($0<\ell<d$). One of the main tools for the analysis is a recently introduced smoothed Poincare lifting operator [M. Costabel and A. McIntosh, Math. Z., 265 (2010), pp. 297-320]. In the case $\ell=1$ our analysis shows that several widely used families of edge finite elements satisfy the discrete compactness property in $p$ and hence provide convergent solutions to the Maxwell eigenvalue problem. In particular, Nedelec elements on triangles and tetrahedra (first and second kind) and on parallelograms and parallelepipeds (first kind) are covered by our theory.
['Daniele Boffi', 'Martin Costabel', 'Monique Dauge', 'L. Demkowicz', 'Ralf Hiptmair']
Discrete Compactness for the $p$-Version of Discrete Differential Forms