R Leis 1986 Chichester: John Wiley viii + 266 pp price £2450 ISBN 0 471 90863 0 This book is an account of some recent developments in the theory of partial differential equations for readers thoroughly familiar with functional analysis It deals with various types of problem for the wave equation, Maxwell's equations, Schrodinger's equation, the plate equation etc, but dissipative systems are hardly discussed

https://iopscience.iop.org/article/10.1088/0031-9112/37/10/027/pdf

Initial Boundary Value Problems in Mathematical Physics

Direct and inverse elastic scattering from anisotropic media

The perfectly matched layer (PML) method is well-studied for acoustic scattering problems, electromagnetic scattering problems, and, more recently, elastic scattering problems, with homogeneous background media. The purpose of this paper is to present the stability and exponential convergence of the PML method for a three-dimensional electromagnetic scattering problem in a two-layer medium. The main contributions of this paper are threefold. First, we establish the well-posedness of the original scattering problem for any Dirichlet boundary value in $\boldsymbol{H}^{-1/2}({\rm Div},\Gamma_D),$ where $\Gamma_D$ stands for the boundary of the scatterer. Second, we propose a new weak formulation for the original problem, where the Dirichlet-to-Neumann operator is proposed on a truncation boundary inside the PML. This argument is favorable in the analysis of the PML Dirichlet-to-Neumann operator. The inf-sup condition is proved for the bilinear form. Third, we establish the well-posedness of the PML problem a...

PML Method for Electromagnetic Scattering Problem in a Two-Layer Medium

Consider an exterior problem of the three-dimensional elastic wave equation, which models the scattering of a time-harmonic plane wave by a rigid obstacle. The scattering problem is reformulated into a boundary value problem by introducing a transparent boundary condition. Given the incident field, the direct problem is to determine the displacement of the wave field from the known obstacle; the inverse problem is to determine the obstacle's surface from the measurement of the displacement on an artificial boundary enclosing the obstacle. In this paper, we consider both the direct and inverse problems. The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is studied and a frequency continuation method is developed for the inverse problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Inverse obstacle scattering for elastic waves in three dimensions

In this paper we study the convergence of the perfectly matched layer (PML) method for solving the time harmonic elastic wave scattering problems. We introduce a simple condition on the PML complex coordinate stretching function to guarantee the ellipticity of the PML operator. We also introduce a new boundary condition at the outer boundary of the PML layer which allows us to extend the reflection argument of Bramble and Pasciak to prove the stability of the PML problem in the truncated domain. The exponential convergence of the PML method in terms of the thickness of the PML layer and the strength of PML medium property is proved. Numerical results are included.

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Xueshuang Xiang

Papers

Convergence of the PML method for elastic wave scattering problems