On the Absorbing Boundary Condition for the Time-Dependent Maxwell Equations

F. Assous (Israel)


Maxwell equations; Absorbing Boundary Conditions; Fi nite Element methods


The numerical solution of the time-dependent Maxwell equations in an unbounded domain requires the introduc tion of artificial absorbing boundary conditions (ABCs) de signed to minimize the amplitude of the parasitic waves re flected by the artificial frontier of the domain of compu tation. In order to construct ABCs which lead to a well posed 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 equa tion in dimension 3, that is particularly well-adapted to a variational approach. In the framework of the Finite ele ment method, and in particular for the transient problems, this can be viewed as an alternative to the famous B´erenger 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 pro posed in [1] in the framework of a finite element method. We propose a stable variational formulation, and an effi cient way to approximate it. The question of the implemen tation in a finite element 3D code, based on a Taylor-Hood method of approximation, is adressed. Concluding remarks follow.

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