Partially motivated by the observation that the curl-curl operator
behaves differently when it is applied to the divergence-free field and
the gradient field in the Hodge decomposition of a vector field, we
introduce the reduced time-harmonic Maxwell (RTHM) equations which solve the divergence-free component of the solution to the time-harmonic Maxwell equations. Three numerical schemes are formulated for solving the RTHM system. Two of them use the classical nonconforming finite element approximations, and the other is based on the interior penalty type discontinuous Galerkin methods. To weakly impose the divergence-free condition satisfied by the solutions, the schemes either work with the locally divergence-free trial spaces, or contain a weighted divergence term in the formula. With the properly chosen stabilizing terms, the optimal error estimates are established on graded meshes. These schemes and the error estimate results are further extended for solving reduced curl-curl problems.

The discrete operators in these schemes naturally define three Maxwell
eigensolvers which are free of spurious eigenmodes and do not contain any
to-be-tuned parameter. The analysis for these solvers is closely related
to the reduced curl-curl problems and their numerical approximations. Not like those Maxwell eigensolvers based on the full curl-curl problems, the compactness of the involved operator and the uniform error estimates for the source problems greatly simplify the analysis of our proposed
eigensolvers. Numerical examples for both Maxwell source and eigenproblems will be presented to demonstrate the performance of the
proposed schemes as well as their relative advantages.