On the absorbing boundary condition for the time-dependent Maxwell equations

The numerical solution of the time-dependent Maxwell equations in an unbounded domain requires the introduction of artificial absorbing boundary conditions (ABCs) designed to minimize the amplitude of the parasitic waves reflected by the artificial frontier of the domain of computation. In order to construct ABCs which lead to a wellposed problem (from a mathematical point of view), and to a stable algorithm (from a numerical point of view), it is often necessary to perform a rigourous mathematical and numerical analysis. In a previous study, Joly et. al[1] have proposed a new second order ABC for the Maxwell's equation in dimension 3, that is particularly well-adapted to a variational approach. In the framework of the Finite element method, and in particular for the transient problems, this can be viewed as an alternative to the famous Bérenger condition (PML)[2, 3]. In this paper, different methods to handle the absorbing boundary condition are first reviewed. Then, we present how to apply the second-order ABC proposed in[1] in the framework of a finite element method. We propose a stable variational formulation, and an efficient way to approximate it. The question of the implementation in a finite element 3D code, based on a Taylor-Hood method of approximation, is adressed. Concluding remarks follow.