Abstract. In this paper, we propose and analyze perfectly matched absorbing layers for a problem of time-harmonic acoustic waves propagating in a duct in the presence of a uniform flow. The absorbing layers are designed for the pressure field, satisfying the convected scalar Helmholtz equation. A difficulty, compared to the Helmholtz equation, comes from the presence of so-called inverse upstream modes which become unstable, instead of evanescent, with the classical Bérenger's perfectly matched layers (PMLs). We investigate here a PML model, recently introduced for timedependent problems, which makes all outgoing waves evanescent. We then analyze the error due to the truncation of the domain and prove that the convergence is exponential with respect to the size of the layers for both the classical and the new PML models. Numerical validations are finally presented. 1. Introduction. Perfectly matched layers (PMLs) were introduced by Bérenger [3] in order to design efficient numerical absorbing boundary conditions (more precisely, absorbing layers) for the computation of time-dependent solutions of Maxwell's equations in unbounded domains. They have since been used for numerous applications, mostly in the time domain [4,28,5,23] but also for time-harmonic wave-like equations [27,15].In particular, PMLs have been used for the solution in the time domain of the linearized Euler equations [19,13,16,26], which model acoustic propagation in the presence of a flow. In this case, it has been observed that PMLs can lead to instabilities, due to the presence of waves whose phase and group velocities have opposite signs [26] (see [2] for a general analysis of this phenomenon). Some techniques have been developed to overcome this difficulty, making the layers stable but, unfortunately, no longer perfectly matched [16,1]. More recently, ideas for designing stable PMLs for this problem have emerged from several teams independently. These new approaches, which seem to be very closely related, have been developed for time-dependent applications in [20,11,14] and for time-harmonic applications in the present paper. These different works all deal with the case of a parallel flow, which is orthogonal to the layers.

We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real-valued negative permittivity/permeability which models an ideal metamaterial. When the interface between the two media has a corner, according to the value of the contrast (ratio) of the physical constants, this non-coercive problem can be ill-posed (not Fredholm) in H 1. This is due to the degeneration of the two dual singularities which then behave like r±iη = e±iη ln r with η ∈ ℝ*. This apparition of propagative singularities is very similar to the apparition of propagative modes in a waveguide for the classical Helmholtz equation with Dirichlet boundary condition, the contrast playing the role of the wavenumber. In this work, we derive for our problem a functional framework by adding to H 1 one of these propagative singularities. Well-posedness is then obtained by imposing a radiation condition, justified by means of a limiting absorption principle, at the corner between the two media.

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