Comparison of a Finite Difference and a Mixed Finite Element Formulation of the Uniaxial Perfectly Matched Layer

Abstract: We consider the anisotropic uniaxial formulation of the perfectly matched layer model (UPML). We prove the decay of different energies for the UPML, under certain assumptions, to demonstrate the well-posedness of this formulation. We present and analyze a mixed finite element method for the time domain discretization of the UPML to simulate wave propagation in unbounded domains in two dimensions. On rectangles the spatial discretization uses bilinear finite elements for the electric field and the lowest order RaviartThomas divergence conforming elements for the magnetic field. We use a centered finite difference method for the time discretization. We compare the finite element technique presented to the finite difference time domain method (FDTD) via a stability, dispersion, phase error and numerical reflection coefficient analysis. We derive the reflection coefficient for the case of a semi-infinite layer to show consistency between the numerical and continuous models, and in the case of a finite PML to study the effects of terminating the absorbing layer. Finally, we demonstrate the effectiveness of the mixed finite element scheme by numerical examples and provide comparisons with the split field PML discretized by the FDTD method. In conclusion, we observe that the mixed finite element scheme for the PML model has absorbing properties that are comparable to the FDTD method.