The Analysis of Matched Layers

Laurence Halpern (1) , Sabrina Petit-Bergez (1) , Jeffrey Rauch (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Department of Mathematics - University of Michigan

A systematic analysis of matched layers is undertaken with special attention to better understanding the remarkable method of Bérenger. We prove that the Bérenger and closely related layers define well-posed transmission problem in great generality, and are perfectly matched when there is only one nonconstant absorption coefficient. The methods include energy methods and the Fourier-Laplace transform. Amplifying and nonamplifying layers are identified by a geometric optics computation. It is proved that the loss of derivative associated with the Berenger method does not occur for elliptic generators. The proof uses the energy method with pseudodifferential multiplier. We construct by an extrapolation argument an alternative matched layer method which preserves the strong hyperbolicity of the original problem and though not perfectly matched has {\it leading} reflection coefficient equal to zero at all angles of incidence. Open problems are indicated throughout.

Mots clés: amplification; PML; WKB; hyperbolic operators; weak well-posedness; geometric optics; extrapolation; reflection; amplification.

Domaines: Analyse numérique [math.NA]

Laurence Halpern, Sabrina Petit-Bergez, Jeffrey Rauch. The Analysis of Matched Layers. 2011. ⟨hal-00551855v1⟩