Asymptotic models for scattering problems from unbounded media with high conductivity

We analyze the accuracy and well-posedness of generalized impedance boundary value problems in the framework of scattering problems from unbounded highly absorbing media. We restrict ourselves in this first work to the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficulties in the analysis for the generalized impedance boundary conditions, since classical compactness arguments are no longer possible. Our new analysis is based on the use of Rellich-type estimates and boundedness of $L2$ solution operators. We also discuss numerical approximation of obtained GIBC (up to order 3) and numerically test theoretical convergence rates

Electromagnetic Wave Scattering from Rough Penetrable Layers

International audienceWe consider scattering of time-harmonic electromagnetic waves from an unbounded penetrable dielectric layer mounted on a perfectly conducting infinite plate. This model describes for instance propagation of monochromatic light through dielectric photonic assemblies mounted on a metal plate. We give a variational formulation for the electromagnetic scattering problem in a suitable Sobolev space of functions defined in an unbounded domain containing the dielectric structure. Further, we derive a Rellich identity for a solution to the variational formulation. For simple material configurations and under suitable non-trapping and smoothness conditions, this integral identity allows to prove an a-priori estimate for such a solution. A-priori estimates for solutions to more complicated material configurations are then shown using a perturbation approach. While the estimates derived from the Rellich identity show that the electromagnetic rough surface scattering problem has at most one solution, a limiting absorption argument finally implies existence of a solution to the problem

The limiting absorption principle and a radiation condition for the scattering by a periodic layer

Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These non-zero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply non-uniqueness of any solution to the scattering problem. In this paper, we consider a medium that is defined in the upper two-dimensional half-space by a penetrable and periodic contrast. We prove that there is a so-called limiting absorption solution to the associated scattering problem. By definition, such a solution is the limit of a sequence of unique solutions for artificial complex-valued wave numbers tending to the above-mentioned real-valued wave number. Our method of proof seems to be new: By the Floquet-Bloch transform we first reduce the scattering problem to a finite-dimensional one that is set in the linear space spanned by all surface waves. In this space, we then compute explicitly which modes propagate along the periodic structure to the left or to the right. This finally yields a representation for our limiting absorption solution which leads to a proper extension of the well known upward propagating radiation condition. Finally, we prove uniqueness of a solution under this radiation condition

Spectral Volumetric Integral Equation Methods for Acoustic Medium Scattering in a Planar Homogeneous 3D Waveguide

Scattering of acoustic waves from an inhomogeneous medium can be described by the Lippmann-Schwinger integral equation. For scattering problems in free space, Vainikko proposed a fast spectral solution method that exploits the convolution structure of this equation's integral operator by using the fast Fourier transform. In a planar 3--dimensional waveguide, the integral operator of the Lippmann-Schwinger integral equation fails to be a convolution. In this paper, we show that the separable structure of the kernel nevertheless allows to construct fast spectral collocation methods similar to Vainikko's technique. The numerical analysis of this method requires smooth material parameters; if the material parameters are, say, discontinuous, no theoretical statement on convergence is available. We show how to construct a Galerkin variant of Vainikko's method for which a rigorous convergence analysis is available even for discontinuous materials. For several distant scattering objects inside the 3--dimensional waveguide this discretization technique leads to a computational domain consisting of one large box containing all scatterers, and hence many unnecessary unknowns. However, the integral equation can be reformulated as a coupled system with unknowns defined on the different parts of the scatterer. Discretizing this coupled system by a combined spectral/multipole approach yields an efficient method for waveguide scattering from multiple objects

On Uniqueness in Electromagnetic Scattering from Biperiodic Structures

Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectric structure mounted on a perfectly conducting plate in three dimensions. Given that uniqueness of solution holds, existence of solution follows from a well-known Fredholm framework for the variational formulation of the problem in a suitable Sobolev space. In this paper, we derive a Rellich identity for a solution to this variational problem under suitable smoothness conditions on the material parameter. Under additional non-trapping assumptions on the material parameter, this identity allows us to establish uniqueness of solution for all wave numbers.Comment: 22 page