Analysis of a Cartesian Pml Approximation to the Three Dimensional Electromagnetic Wave Scattering Problem

We consider the application of a perfectly matched layer (PML) technique applied in Cartesian geometry to approximate solutions of the electromagnetic wave (Maxwell) scattering problem in the frequency domain. The PML is viewed as a complex coordinate shift (“stretching”) and leads to a variable complex coefficient equation for the electric field posed on an infinite domain, the complement of a bounded scatterer. The use of Cartesian geometry leads to a PML operator with simple coefficients, although, still complex symmetric (non-Hermitian). The PML reformulation results in a problem which preserves the original solution inside the PML layer while decaying exponentially outside. The rapid decay of the PML solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation (for the truncated problem). For suitably defined Cartesian PML layers, we prove existence and uniqueness of the solutions to the infinite domain and truncated domain PML equations provided that the truncated domain is sufficiently large. We show that the PML reformulation preserves the solution in the layer while decaying exponentially outside of the layer. Our approach is to develop variational stability for the infinite domain electromagnetic wave scattering PML problem from that for the acoustic wave (Helmholtz) scattering PML problem given in [12]. The stability and exponential convergence of the truncated PML problem is then proved using the decay properties of solutions of the infinite domain problem. Although, we do not provide a complete analysis of the resulting finite element approximation, we believe that such an analysis should be possible using the techniques in [6].