Perfectly matched layers (PML) are a recent technique for simulating the absorption of waves in open domains. They have been introduced for electromagnetic waves and extended, since then, to other models of wave propagation, including waves in elastic anisotropic media. In this last case, some numerical experiments have shown that the PMLs are not always stable. In this paper, we investigate this question from a theoretical point of view. In the first part, we derive a necessary condition for the stability of the PML model for a general hyperbolic system. This condition can be interpreted in terms of geometrical properties of the slowness diagrams and used for explaining instabilities observed with elastic waves but also with other propagation models (anisotropic MaxwellÕs equations, linearized Euler equations). In the second part, we specialize our analysis to orthotropic elastic waves and obtain separately a necessary stability condition and a sufficient stability condition that can be expressed in terms of inequalities on the elasticity coefficients of the model.

We construct and analyze a new family of rectangular (two-dimensional) or cubic (three-dimensional) mixed finite elements for the approximation of the acoustic wave equations. The main advantage of this element is that it permits us to obtain through mass lumping an explicit scheme even in an anisotropic medium. Nonclassical error estimates are given for this new element.

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.

Abstract.In this work, we investigate the Perfectly Matched Layers (PML) introduced by Bérenger [3] for designing efficient numerical absorbing layers in electromagnetism. We make a mathematical analysis of this model, first via a modal analysis with standard Fourier techniques, then via energy techniques. We obtain uniform in time stability results (that make precise some results known in the literature) and state some energy decay results that illustrate the absorbing properties of the model. This last technique allows us to prove the stability of the Yee's scheme for discretizing PML's.Résumé. Dans ce travail, nous considérons le modèle de couches parfaitement adaptées, dit PML (Perfectly Matched Layers), introduit par Bérenger [3] pour la modélisation de frontières absorbantes enélectromagnétisme. Nous menons une analyse mathématique de ce modèle, d'une part par une analyse modale par transformation de Fourier, d'autre part par des techniquesénergétiques. Nous obtenons ainsi des résultats de stabilité uniforme en temps (qui précisent des résultats déjà connus dans la littérature) etétablissons des résultats de décroissance d'énergie qui illustrent les propriétés d'absorption du modèle. Cette dernière technique permet aussi de démontrer la stabilité du schéma de Yee pour discrétiser les couches absorbantes. Mathematics Subject Classification.

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Abstract. We construct and analyze a new family of quadrangular (in two dimensions) or cubic (in three dimensions) mixed finite elements for the approximation of elastic wave equations. Our elements lead to explicit schemes (via mass lumping), after time discretization, including in the case of anisotropic media. Error estimates are given for these new elements.

In this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical Lagrange finite elements. The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasi-reversibility method. Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations.