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
Q^{p,p} x Q^{p,p}
2. The second Nédélec family
Q^{p-1,p} x Q^{p,p-1}
3. The ABF (Arnold-Boffi-Falk) family
Q^{p-1,p+1} x Q^{p+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. |