Discrete compactness for the hp version of rectangular edge finite elements

Daniele Boffi, Martin Costabel, Monique Dauge, Leszek Demkowicz

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 the other, thus allowing for a real hp adaptivity. As a consequence of our result, for the first time a rigorous proof of convergence for the p version of edge element approximation of Maxwell's eigenproblem is presented.

We also present a full description of the Maxwell spectrum on the reference element with three different families of polynomial spaces:

1. The full tensor product family Qp,p x Qp,p
2. The second Nédélec family Qp-1,p x Qp,p-1
3. The ABF (Arnold-Boffi-Falk) family Qp-1,p+1 x Qp+1,p-1.

September 2004 (original version)
ICES Report (Austin, Texas) N. 04-29

July 2005 (shorter version, without ABF)
SINUM paper Siam J. Numer. Anal. Vol. 44, 2006, pp 979-1004.