  We shall propose and analyze some new preconditioners for the saddle-point
systems arising from the edge element discretization of the time-harmonic
Maxwell equations in three dimensions. We will first consider the saddle-point
systems with vanishing wave number, for which we present an important relation
between the solutions of the singular curl-curl system and the non-singular
saddle-point system, then demonstrate that the saddle-point system can be
efficiently solved by the Hiptmair-Xu solver. For the saddle-point systems with
non-vanishing wave numbers, we will show that the PCG with a new preconditioner
can apply for the non-singular system when wave numbers are small, while the
methods like preconditioned MINRES may apply for some existing and new
preconditioners when wave numbers are large. The spectral behaviors of the
resulting preconditioned systems for the existing and new preconditioners are
analyzed and compared, and numerical experiments are presented to demonstrate
and compare the efficiencies of these preconditioners.
