Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Analysis of the PML equations in general convex geometry

Matti Lassasa1 and Erkki Somersaloa2

a1 Rolf Nevanlinna Institute, University of Helsinki, Helsinki, PO Box 4, FIN-00014, Finland

a2 Department of Mathematics, Helsinki University of Technology, Helsinki, PO Box 1100, FIN-02015, Finland


In this work, we study a mesh termination scheme in acoustic scattering, known as the perfectly matched layer (PML) method. The main result of the paper is the following. Assume that the scatterer is contained in a bounded and strictly convex artificial domain. We surround this domain by a PML of constant thickness. On the peripheral boundary of this layer, a homogenous Dirichlet condition is imposed. We show in this paper that the resulting boundary-value problem for the scattered field is uniquely solvable for all wavenumbers and the solution within the artificial domain converges exponentially fast toward the full-space scattering solution when the layer thickness is increased. The proof is based on the idea of interpreting the PML medium as a complex stretching of the coordinates in Rn and on the use of complexified layer potential techniques.

(Received February 25 2000)

(Accepted October 24 2000)