We consider the problem of solving the discrete Maxwell eigenvalue problem in a closed simply connected cavity. For related eigenvalue problems for symmetric second order elliptic operators, efficient iterative schemes for the computation of a couple of the smallest eigenvalues/eigenvectors have been proposed [1]. They are based on a preconditioned inverse iteration and a comprehensive analysis has been presented by the second author [3]. In the case of the Maxwell eigenvalue problem the large kernel of the curl-operator thwarts the straightforward application of these algorithms. However, when the discretization is based on edge elements, we have an explicit representation of Kern(curl) through gradients of linear finite element functions. This paves the way for a fast approximate projection onto the orhtogonal complement of Kern(curl), which can be coupled with the edge element multigrid scheme developed by the first author [2]. Numerical experiments confirm the excellent performance of various algorithmic variants of this approach.

[1]: J. Bramble, A. Knyazev, and J. Pasciak, A subspace preconditioning  algorithm for eigenvector/eigenvalue computation, Advances Comp.
      Math., 6  (1996), pp. 159--189
[2]: R. Hiptmair, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal., 36 (1999), pp. 204--225
[3]: K. Neymeyr, A geometric theory for preconditioned inverse iteration applied to a subspace, Report 130, SFB 382, Universitaet Tuebingen,
      Tuebingen, November 1999