| The time-domain scattering problems have a wide range of applications in many scientific and engineering fields such as seismology,oceanography,material science and biomedical science,and therefore their rigorous mathematical theoretical analysis is significant.These analyses include both the establishment of the well-posedness and stability of various scattering problems,and the theoretical analysis of mathematical methods for truncating various unbounded physical regions into bounded computable regions.In this thesis,we study the well-posedness and stability of the scattering problem and the convergence of the PML truncation method by using the three-dimensional time-domain acoustic scattering problem as a model.For the analysis of the well-posedness and stability of the three-dimentional time-domain acoustic scattering problem,we first transform the original scattering problem equivalently into an initial-boundary problem in a bounded domain over a finite time interval by introducing the transparent boundary condition to truncate the unbounded domain;then the bridge between the time domain and the complex frequency domain is established by using the Laplace transform;finally,using the variational method and the energy estimate method,we prove the well-posedness and stability of the scattering problem.For the convergence analysis of the PML method for time-domain acoustic scattering problem,we consider the well-posedness and stability of the PML problem and the convergence of the PML method.Transform the Helmholtz equation in spherical coordinate in the complex frequency domain into the time-domain PML equation by using the real coordinate extension and the inverse Laplace transform.The time-domain PML problem is proved to be well-posed and stable,based on the variational method and the energy estimate method.By analyzing the error between the DtN operators of the original scattering problem and the PML problem,the error estimate between the solution of the original scattering problem and the solution of the PML problem is reduced to a boundary value problem within the PML layer.The exponential convergence of the PML method with respect to the thickness of the PML layer and the PML absorbing parameter is proved by using the stability of the PML layer system and the decay of the PML extension. |
