Perfectly Matched Layers for Time-Harmonic Second Order Elliptic Problems

The main goal of this work is to give a review of the Perfectly Matched Layer (PML) technique for time-harmonic problems. Precisely, we focus our attention on problems stated in unbounded domains, which involve second order elliptic equations writing in divergence form and, in particular, on the Helmholtz equation at low frequency regime. Firstly, the PML technique is introduced by means of a simple porous model in one dimension. It is emphasized that an adequate choice of the so called complex absorbing function in the PML yields to accurate numerical results. Then, in the two-dimensional case, the PML governing equation is described for second order partial differential equations by using a smooth complex change of variables. Its mathematical analysis and some particular examples are also included. Numerical drawbacks and optimal choice of the PML absorbing function are studied in detail. In fact, theoretical and numerical analysis show the advantages of using non-integrable absorbing functions. Finally, we present some relevant real life numerical simulations where the PML technique is widely and successfully used although they are not covered by the standard theoretical framework.