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Abstract. The aim of this paper is to introduce an "exact" bounded perfectly matched layer (PML) for the scalar Helmholtz equation. This PML is based on using a non integrable absorbing function. "Exactness" must be understood in the sense that this technique allows exact recovering of the solution to time-harmonic scattering problems in unbounded domains. In spite of the singularity of the absorbing function, the coupled fluid/PML problem is well posed when the solution is sought in an adequate weighted Sobolev space. The resulting variational formulation can be numerically dealt with standard finite elements. The high accuracy of this approach is numerically demonstrated as compared with a classical PML technique. 1. Introduction. The typical first step for the numerical solution by finite elements or finite differences of any scattering problem in an unbounded domain is to truncate the computational domain, which entails an inherent difficulty: to choose boundary conditions to replace the Sommerfeld radiation condition at infinity (see for instance [21]).There are several techniques to deal with this: boundary element methods, infinite element methods, Dirichlet-to-Neumann methods based on Fourier expansions, or the use of absorbing boundary conditions. The potential advantages of each of them have been widely studied in the literature [25,35,18].If the domain of the original problem is truncated with a sphere, then the Dirichletto-Neumann (DtN) boundary condition is exactly known (see [21,32]). However, this boundary condition involves an infinite series which must be truncated for numerical use, thus introducing errors. Moreover, the exact DtN condition is non local, leading to dense blocks in the linear system to be solved, when a finite element method is used.As a partial solution to these drawbacks, local absorbing boundary conditions (ABCs) can be introduced to preserve the computational efficiency of the numerical method. Those of Bayliss and Turkel [5], Engquist and Majda [19], and Feng [20] are among the most widely used. However, in spite of the simple implementation of lowest order ABCs, good accuracy is only achieved for higher order ones [37], because these conditions are not fully non-reflecting on the truncated boundary of

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.

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