求解Laplace方程Cauchy问题的磨光化方法

A mollification method for solving Cauchy problem of Laplace equation

考虑无限带状区域x∈R,0<y<1上Laplace方程的Cauchy问题,此问题是不适定的,因为解对输入数据不具有连续依赖性,即输入数据的小扰动可能会导致数值解与精确解有很大的误差。本文将用基于Gaussian核的磨光化方法来求解Laplace方程Cauchy问题,给出了其精确解与近似解之间的误差估计。数值实验表明了该方法的有效性。

This paper is concerned with a Cauchy problem for the Laplace equation in the strip region x∈R,0<y<1,where Cauchy data are given aty=0 and the equation is solved in interval 0<y<1.As we all know,this problem is ill-posed because the solution is no longer continuously dependent on the data.That is the small disturbance of the data which may lead to a large error between the numerical solution and the exact solution.In order to obtain a stable numerical solution,the equation is solved based on a mollification method used Gaussian kernel,and the error estimation between the exact solution and the approximate solution is given.Numerical examples show the effectiveness of the method.