Discrete compactness for the p-version of discrete differential forms

TL;DR

This paper establishes the discrete compactness property for p-version finite element methods applied to non-elliptic eigenvalue problems, ensuring convergence of solutions for Maxwell equations using various edge finite elements.

Contribution

It provides a general framework and sufficient conditions for the discrete compactness property in p-version finite element approximations of differential forms.

Findings

Nédélec elements satisfy the discrete compactness property

Convergence of Maxwell eigenvalue solutions is guaranteed

Applicable to a wide class of finite element families

Abstract

In this paper we prove the discrete compactness property for a wide class of p-version finite element approximations of non-elliptic 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 any order on a d-dimensional polyhedral domain. One of the main tools for the analysis is a recently introduced smoothed Poincar\'e lifting operator [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincar\'e integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2010)]. For forms of order 1 our analysis shows that several widely used families of edge finite elements satisfy the discrete compactness property in p-version and hence provide convergent solutions to the Maxwell eigenvalue problem. In particular, N\'ed\'elec elements on triangles and tetrahedra (first and second kind) and on parallelograms and parallelepipeds (first kind) are covered by our theory.