For bounded, simply-connected domains, an eigenvalue problem for the curl operator with divergence-free eigensolutions is investigated. Two different weak formulations are presented and discretized by finite elements.
The first one directly delivers the eigenvalues and eigensolutions of the original problem, but its discretization leads to a degenerate generalized eigenvalue problem with both matrices being non-definite, so that an application of standard solvers is not possible.
The second formulation leads to an eigenvalue problem, whose eigenvalues are the square of the eigenvalues of the curl operator and the corresponding eigenspaces are invariant subspaces of the original problem. The finite element discretization results in a generalized eigenvalue problem with Hermitian matrices, the one on the right hand side being positive definite. For this approach, spectral convergence, optimal-order error estimates and absence of spurious modes are established.
In the case of eigenspaces of dimension larger than 1, the eigensolutions of the curl operator cannot be reconstructed from the eigensolutions of the second formulation. In this case, one can resort to the first variational method.
Finally, some numerical experiments confirm the theoretical results.