Multigrid Computation of Maxwell Eigenvalues

We consider the problem of solving the discrete Maxwell eigenvalue problem $curl\frac{1}{\mu }curlu={\omega }^{2}u,\omega >0$, in a closed simply connected cavity $\mathrm{\Omega }\subset {\mathbb{R}}^3$. 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 in [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 $Kern(curl{)}^{\mathrm{\perp }}$, which can be coupled with the edge element multigrid scheme developed by the author [2]. Numerical experiments confirm the good performance of this approach for large scale problems.
References

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, Tech. Rep. 130, SFB 382, Universität Tübingen, Tübingen, Germany, November 1999. Submitted to Math. Comp.