Éliane Bécache (INRIA. France)

Stability and instability results for perfectly matched layers

Abstract

In this talk we investigate the question of stability of the Perfectly Matched Layers (PML) introduced by Bérenger in order to design efficient numerical absorbing layers for the computation of time dependent solutions of Maxwell's equations in unbounded domains. Previous works aimed essentially at proving the well-posedness of such models. The corresponding results do not exclude a possible exponential blow up of the solutions for large time. We think that for the applications, a more pertinent concept is the one of stability: one wishes to get uniform estimates in time. We first show that such a stability result is valid for isotropic Maxwell and acoustic equations, for which the result can be obtained by Fourier analysis and energy estimates. Then we show that the stability properties can be lost in case of anisotropy. We illustrate in particular this on the case of anisotropic elastic waves. We show that a necessarly condition of stability can be interpreted in terms of group velocities and geometrical properties of the slowness curves. This applies for showing instability results for the classical PML models for linearized Euler equations and anisotropic Maxwell's equations.