一类Dirichlet外问题数值解的概率方法

Probabilistic methods for numerical solutions of the exterior Dirichlet problem

Dirichlet外问题的定解区域是个无界区域,一般的数值方法需要对定解区域进行剖分,因而无法解决外问题.现在提出一种新的有效的概率数值方法,它从解的随机表达式出发,将无界区域上的问题转化成区域边界上的问题,此时,只要在边界上进行剖分,将问题离散化,然后在无界区域外的有界区域内构作一个辅助球,并且利用布朗运动、漂移布朗运动从球外一点出发,首中球面的位置和时间的分布,就可以获得Dirichlet外问题的数值解.

Domains for the exterior Dirichlet problem are unbounded.General numerical methods need to subdivide the domain, and so can not solve the exterior problem. Now a new efficient probabilistic numerical method is proposed. It bases on the stochastic representations of solutions and turns the problem over unbounded domains into a problem over boundary. Subdivision over boundary is only needed to make the problem discretized. An auxiliary ball is constructed in the bounded domain outside the unbounded. The distributions of the time and place of hitting spheres for Brownian motion or Brownian motion with drift from outside spheres are explored to gain the numerical solution.