A Finite Element/Spectral Method for Approximating the Time-Harmonic Maxwell System in R³

We describe and analyze a method for computing an approximation to the time-harmonic electromagnetic field scattered by a bounded inhomogeneity. The method is to couple a finite element scheme on a bounded domain with a series solution outside the bounded domain. The main result of the paper is to show that the proposed numerical scheme possesses a unique solution with quasi-optimal approximation properties. We do this by verifying the conditions of the Babuška–Brezzi theory for saddle point problems. In analyzing the continuous problem, we also provide a new variational proof of the existence of solutions of the continuous Maxwell problem.