  In this paper we focus on high order finite element approximations of the
electric field combined with suitable preconditioners, to solve the
time-harmonic Maxwell's equations in waveguide configurations.The
implementation of high order curl-conforming finite elements is quite delicate,
especially in the three-dimensional case. Here, we explicitly describe an
implementation strategy, which has been embedded in the open source finite
element software FreeFem++ (http://www.freefem.org/ff++/). In particular, we
use the inverse of a generalized Vandermonde matrix to build basis functions in
duality with the degrees of freedom, resulting in an easy-to-use but powerful
interpolation operator. We carefully address the problem of applying the same
Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over
the computational domain. We investigate the preconditioning for Maxwell's
equations in the time-harmonic regime, which is an underdeveloped issue in the
literature, particularly for high order discretizations. In the numerical
experiments, we study the effect of varying several parameters on the spectrum
of the matrix preconditioned with overlapping Schwarz methods, both for 2d and
3d waveguide configurations.
