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

Abstract

Abstract. We consider the approximation of the frequency domain threedimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the time-harmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius R t . We also show exponential (in the parameter R t ) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer.Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.

Analysis of the spectrum of a Cartesian Perfectly Matched Layer (PML) approximation to acoustic scattering problems

“…Such stretched operators have been studied by other authors when the transformation was based on polar or spherical coordinates [2,3,5] and coordinates associated with a smooth convex surface [6]. We note that a complete analysis of the discrete problem involves stability of the infinite and truncated domain source problems at the continuous level and the analysis of the truncated finite element approximation.…”

“…The complex stretching can be thought of as highly absorbing fictitious layer which attenuates outgoing radiation [1,2]. In fact, in many cases, this leads to a new problem, still on the unbounded domain, which preserves the solution inside the layer while decaying rapidly outside [2][3][4]. Because of this decay, it is feasible to develop numerical approximations by domain truncation and the application of the finite element method.…”

“…In [3], an analysis of the source problem on the infinite domain with spherical PML was given by first showing that the resulting form was coercive up to a lower order perturbation on a bounded domain. A standard argument by compact perturbation [9,10] then shows stability of the source problem once uniqueness has been established.…”

Stability analysis of FDTD to UPML for time dependent Maxwell equations

“…Bramble and Pasciak [17] have shown the existence and uniqueness of solutions to the truncated time harmonic PML problem provided that the truncated domain is sufficiently large. Bao and Wu [18] have given the convergence analysis of the same problem in spherical coordinates for three-dimensional electromagnetic scattering and established an explicit error estimate between the solutions of the scattering problem and the truncated PML problem.…”

Numerical analysis for the scattering by obstacles in a homogeneous chiral environment

“…We refer to Bao and Wu [10], Bramble and Pasciak [11], Chen and Liu [12] and Chen and Wu [13] for the convergence analysis of the PML problems for 3D maxwell's equations and 2D helmholtz equation.…”