The Perfectly Matched Layer in Curvilinear Coordinates

Abstract

In 1994 B erenger showed how to construct a perfectly matched absorbing layer for the Maxwell system in rectilinear coordinates. This layer absorbs waves of any wavelength and any frequency without re ection and thus can be used to arti cially terminate the domain of scattering calculations. In this paper we show how to derive and implement the B erenger layer in curvilinear coordinates (in two space dimensions). We prove that an in nite layer of this type can be used to solve time harmonic scattering problems. We also show that the truncated B erenger problem has a solution except at a discrete set of exceptional frequencies (which might be empty). Finally numerical results show that the curvilinear layer can produce accurate solutions in the time and frequency domain.

Parametric finite elements, exact sequences and perfectly matched layers

“…In [2], Chew and Weedon proposed an alternative formulation of the PML through a complex coordinate stretching of the spatial variables of the original differential equations posed in the frequency domain. The approach was later reinterpreted in terms of an analytic continuation [3,4], and motivated the extension of the PML to curvilinear coordinates, conformal mesh terminations, and more general media. Simultaneously, the new technique was successfully applied to acoustics [5], elastic wave propagation [6], and poroelasticity [7].…”

Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems

“…As discussed in [5], the PML problem can be viewed as a complex coordinate transformation. Following [10], a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition.…”

“…The observation that a PML method could be considered as a complex change of variable was made by Chew and Weedon [4]. Using this technique, Collino and Monk [5] derived PML equations based on rectangular and polar coordinates. There, they also showed the existence and uniqueness of solutions of the truncated acoustic PML except for a countable number of wave numbers.…”

“…This stems from the fact that the acoustic problem is strongly elliptic (up to perturbation) while the electromagnetic operator has an infinite dimensional kernel consisting of functions which are gradients. For example, Collino and Monk [5] use a perturbation analysis to derive their existence result. Carrying this argument over to the electromagnetic PML poses significant analytical difficulties requiring the analysis of vector decompositions involving the complexvalued PML coefficient.…”

Additive Schwarz Method for Scattering Problems Using the PML Method at Interfaces

“…Our derivation of the PML method, described in detail in [8] is based on an analytic continuation, as in [9,2] Details of the implementation in 2D can by found in [13]. The basic idea is an analytic continuation of the solution in the exterior along a distance variable.…”