Generalized Impedance Boundary Conditions for Scattering by Strongly Absorbing Obstacles: The Scalar Case

Abstract

We derive different classes of generalized impedance boundary conditions for the scattering problem from strongly absorbing obstacles. Compared to existing works, our construction is based on an asymptotic development of the solution with respect to the medium absorption. Error estimates are derived to validate the accuracy of each condition.

Full Range Scattering Estimates and Their Application to Cloaking

“…This is consistent with the well-known fact that an infinitely "lossy" layer effectively behaves as a sound-soft barrier (see for example [5] or [11]). …”

“…Lemma 5 was proved and used in [5]. Similar inequalities related to the quantities div and curl were introduced in [6].…”

Asymptotic models for scattering from unbounded media with high conductivity

“…As a brief introduction to the construction of generalized impedance boundary conditions, we recall some basic ideas, definitions and computations from [9]. The formal computation of these conditions for rough layers is the same as for bounded absorbing inclusions and we can skip most of the computations.…”

“…In such a model, the moisture of soil causes absorption of the electromagnetic wave inside the ground, and thus naturally leads to a scattering problem for a rough absorbing layer. Since waves inside the absorbing part of the medium decay exponentially with respect to the distance to the layer's boundary, a lot of research has been carried out how to replace the wave scattering problem inside the absorbing layer by some easily handable absorbing boundary condition on the interface in between the absorbing layer and free space [1,8,9,14,15]. The aim of such a boundary condition is to set up an approximate scattering problem merely in the complement of the absorbing object, while still guaranteeing a reliable error bound on the solution of the approximate problem.…”

“…Since compact embeddings of Sobolev spaces do not hold in our unbounded setting, we cannot use such types of compactness arguments which are always present in earlier works on generalized impedance boundary conditions for bounded objects [9]. The L 2 solution operator does part of this job.…”