The PML (for Perfectly Matched Layer) technique in computational scattering theory was introduced by Bérenger, and the work presented here is based on Collino and Monk’s analysis of the technique in polar coordinates. The fundamental idea is that a scatterer is surrounded by an absorbing layer so that solutions to the equations are not contaminated by artificial reflections created by the boundary conditions.

This paper is devoted to proving that, with certain assumptions, the PML equations are solvable for all wave numbers, and these solutions converge exponentially as the PML layer is thickened.

The paper begins with a brief review of the method as described by Collino and Monk. There follows a careful, rigorous, and well-written development of a sequence of lemmas that collectively establish the main theorem. (Even a nonexpert, such as myself, should be able to at least follow the thrust of the arguments.) My one minor criticism is that, both in the abstract and elsewhere, the authors appear to stress the exponential rate of convergence, but the proof of that particular aspect emerges as a relatively minor step in the proof of Lemma 2.4. It appears to be an important aspect of the result and should have been given greater prominence.

Overall, this paper seems to be an easy-to-read and worthwhile contribution to a difficult theory.