Discrete Compactness for the p-Version of Discrete Differential Forms

In this paper we prove the discrete compactness property for a wide class of p 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 order ℓ on a polyhedral domain in Rd (0 < ℓ < d). One of the main tools for the analysis is a recently introduced smoothed Poincar´e lifting operator [M. Costabel and A. McIntosh, On Bogovski˘ı and regularized Poincar´e integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2009)]. In the case ℓ = 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, N´ed´elec elements on triangles and tetrahedra (first and...

• Type: Journal
• Year: 2011
• Where: SIAMNUM
• Authors: Daniele Boffi, Martin Costabel, Monique Dauge, Leszek F. Demkowicz, Ralf Hiptmair
• Keywords: Discrete Compactness Property, Eigenvalue Problem, SIAMNUM 2011, finite Elements