多联通区域中的拉普拉斯方程柯西问题的一种数值计算方法

多联通区域中的Laplace方程柯西问题的一种数值解法——基本解和边界控制技术相结合的方法,其主要思想是先通过边界控制技术来获得部分边界上的未知的Dirichlet数据的一个逼近,然后再用基本解方法去求解一个带有第二类边值条件的Laplace方程.这种方法在求解拉普拉斯方程柯西问题时与通常所用的基本解方法不同,本文主要是用基本解方法求解了一系列正问题而不是直接用基本解方法去求解拉普拉斯方程柯西问题这样一个反问题.这里由于Laplace方程柯西问题的高度不适定性,为了确保数值解的精度和稳定性,本文采用了Tikhonov正则化方法,在正则化参数的选取上采用了GCV准则.最后用数值算例证明了这种方法不论是在数值解的精度上还是数值解的稳定性上都是非常有效的.

A Numerical Method for the Cauchy Problem of the Laplace Equation in a Multi-connected Domain

A numerical method, the combination of fundamental solution and boundary control technique, is used to solve the cauchy problem of Laplace equation in muhiconneeted domain. The main idea of this method is that an approximation of the missing Dirichet boundary data is obtained by the boundry techanique, and then the Laplace equation with the Dirichlet boundary conditions is solved by the method of fundamental solution. The difference between the present method and the method of fundamental solution is that the method of fundamental solution is used to solve a sequence of direct problems instead of the inverse problem. Because of the ill-posedness of the caucby problem the Tikhonov regularization method is used to ensure the accuracy and the stabilizes of the solution, and the GCV method is used to determine a suitable regularization parameter. The effectiveness of the proposed approaches to solve Cauchv oroblem are illustrated by several numerical examoles.