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.|