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Recent progress in the analysis of the
convergence of FEM for Maxwell eigenvalue problems.

*Martin Costabel*
Slides of an invited lecture at the
*22nd Chemnitz FEM Symposium*
Oberwiesenthal, September 29, 2009

Abstract:

Finite element approximation of the Maxwell eigenvalue problem can notoriously
go wrong, even if the standard "stability plus consistency" conditions are
satisfied. The reason is that - depending on the choice of the variational
formulation - one has to deal with a problem with a non-compact resolvent, or
with a non-standard eigenvalue problem for a mixed formulation.
Recently, D.N. Arnold, R. Falk and R. Winther have shown a general framework
in which the spectrally correct approximation of the Maxwell eigenvalue
problem is a reward for the obedience of the finite element spaces to some
algebraic structure, in the presence of uniform estimates of compatible
interpolation operators in the appropriate norms (keywords ``discrete
subcomplexes of the de Rham complex'' and ``uniformly bounded cochain
projectors''). These arguments cover the spectrally correct convergence for
the h version FEM using edge elements of arbitrary order on simplicial meshes.
For the p and hp versions of the edge element method, the available known
interpolants do not quite fit into this scheme. In particular, it is hard to
prove L^2 interpolation error estimates uniformly in p. A recently found tool,
the regularized Poincar\'e integral operator, can help here. This tool answers
a variety of questions in the regularity theory of vector analysis on
Lipschitz domains. In the analysis of computational electromagnetics, it can
be used to complete the proof of the discrete compactness property of the p
version edge element methods on various 2 and 3 dimensional meshes, thereby
showing spectrally correct convergence of these methods. These results have
been obtained in joint work with D. Boffi (Pavia), M. Dauge (Rennes), L.
Demkowicz (Austin), R. Hiptmair (Z\"urich), A. McIntosh (Canberra).

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