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In this paper we focus on high order finite element approximations of theelectric field combined with suitable preconditioners, to solve thetime-harmonic Maxwell's equations in waveguide configurations. Theimplementation of high order curl-conforming finite elements is quite delicate,especially in the three-dimensional case. Here, we explicitly describe animplementation strategy, which has been embedded in the open source finiteelement software FreeFem++ (http://www.freefem.org/ff++/). In particular, weuse the inverse of a generalized Vandermonde matrix to build a basis ofgenerators in duality with the degrees of freedom, resulting in an easy-to-usebut powerful interpolation operator. We carefully address the problem ofapplying the same Vandermonde matrix to possibly differently orientedtetrahedra of the mesh over the computational domain. We investigate thepreconditioning for Maxwell's equations in the time-harmonic regime, which isan underdeveloped issue in the literature, particularly for high orderdiscretizations. In the numerical experiments, we study the effect of varyingseveral parameters on the spectrum of the matrix preconditioned withoverlapping Schwarz methods, both for 2d and 3d waveguide configurations.

Explicit implementation strategy of high order edge finite elements and Schwarz preconditioning for the time-harmonic Maxwell's equations