Perfectly matched layers in the discretized space: An analysis and optimization

The perfectly matched layer (PML) has recently been introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. Recently, it has been pointed out that this absorbing boundary condition is the same as coordinate stretching in the complex space. In this paper, the corresponding coordinate stretching is analyzed in the discretized space of Maxwell's equations as described by the Yee algorithm. The corresponding dispersion relationship is derived for a PML medium and then the problem of reflection from a single interface is solved. A perfectly matched interface is shown not to exist in the discretized space, even though it exists in the continuum space. Numerical simulations both using finite difference method and finite element method confirm that such discretization error exists. A numerical scheme using the finite element method is then developed to optimize the PML with respect to its parameters. Examples are given to demonstrate the performance of the optimized PML and its application to the finite element solution of scattering problems.