The Analysis of Matched Layers

TL;DR

This paper provides a comprehensive mathematical analysis of matched layers, especially Bérenger's method, establishing conditions for well-posedness, perfect matching, and nonamplification, with implications for Maxwell's equations.

Contribution

It offers a general proof of well-posedness and perfect matching for Bérenger's matched layers, and introduces an alternative method with zero reflection coefficient at all angles.

Findings

Bérenger's layers are well posed under general conditions

Perfect matching is achieved when layers are well posed

An alternative method with zero reflection coefficient is constructed

Abstract

A systematic analysis of matched layers is undertaken with special attention to better understand the remarkable method of B\'erenger. We prove that the B\'erenger and closely related layers define well posed transmission problems in great generality. When the B\'erenger method or one of its close relatives is well posed, perfect matching is proved. The proofs use the energy method, Fourier-Laplace transform, and real coordinate changes for Laplace transformed equations. It is proved that the loss of derivatives associated with the B\'erenger method does not occur for elliptic generators. More generally, an essentially necessary and sufficient condition for loss of derivatives in B\'erenger's method is proved. The sufficiency relies on the energy method with pseudodifferential multiplier. Amplifying and nonamplifying layers are identified by a geometric optics computation. Among the various flavors of B\'erenger's algorithm for Maxwell's equations our favorite choice leads to a strongly well posed augmented system and is both perfect and nonamplifying in great generality. 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 leading reflection coefficient equal to zero at all angles of incidence.