复连通域上Laplace方程Cauchy问题的数值方法研究

The project aims to study Cauchy problem for the Laplace equation in a complex connected domain analytically and numerically. Cauchy problem for the Laplace equation is central to many areas of science and technology, such as geophysical exploration, medical surgery and non-destructive testing etc. Now the research on Cauchy problem is mainly focus on simple connected domains. Few methods are defined in complex domains, but this problem will be more significance. Based on the properties of harmonic polynomials in complex connected domains, this proposal will improve the harmonic polynomial method in a simple connected domain and apply it to the problem, which is defined in a complex connected domain. The convergence and error analysis, which is connected with 2N+1, will be established. We will derive the stability of the method by the lower bound for the smallest singular value of the operator. The proposal will enrich the research of Cauchy problem. The mathematical and computational techniques developed are applicable to Cauchy problem for other equations.

本项目的主旨是研究一般复连通区域上Laplace方程Cauchy问题的数值方法和其数学理论。Laplace方程Cauchy问题是许多科技领域的核心问题，如：地理勘探、医疗成像、无损探伤等。目前这类问题的数值算法主要集中在单连通域上，关于复连通域问题的算法还很少，但是实际问题中更多的定义在复连通区域上。本项目根据复连通域上调和函数的一些性质，将已有的单连通域上的调和多项式方法进行改进，将其应用到我们要研究的复连通域的问题上，证明此时调和多项式方法的收敛性，估计误差与多项式个数2N+1之间的关系，并通过讨论算子的最小特征值下界研究算法的稳定性。本项目将丰富Cauchy问题数值方法的研究，具有重要的应用价值；并且提出的理论和数值方法也可以扩展到复连通域其它类型方程的Cauchy问题中。

本项目针对声学和电磁学散射问题进行研究，主要研究成果包括如下内容：.(1)利用带Tikhonov正则化的Fourier-Bessel方法求解光滑、有界区域内的带有噪声数据的Helmholtz方程的Cauchy问题；通过选取适当的正则化参数来分析方法的收敛性和稳定性。.(2)考虑了一个用时谐的电磁平面波入射到外面包裹一层手性介质的良导体障碍物的散射问题.，建立了一个二维散射模型并用积分方程方法，讨论了解的存在性和唯一性。