A numerical method based on probability theory

Abstract

By using the connections between Brownian family with drift and elliptic differential equations, an efficient probabilistic computing method is given. This method is applied to a wide-range Dirichlet problem. Detail analysis and deduction of solving the problem are offered. The stochastic representation of the solution to the problem makes a 3-dimensional problem turned into a 2-dimensional problem. And an auxiliary ball is constructed. The strong Markov property and the joint distributions of the time and place of hitting spheres for Brownian family with drift are employed. Finally, good convergence of the numerical solution to the problem over domain with arbitrary boundary is obtained.

Keywords

probabilistic method / Brownian family with drift / finite element spaces over boundaries

References

[1] TangLi, ZouJie-zhong, ZhangAi-ping. A probabilistic computing method by Brownian family with drift[J]. Journal of Central South University of Technology, 2001, 32(6): 648-650(in Chinese)

[2] KaratzasI, StevenE. S.Brownian Motion and Stochastic Calculus[M], 1988, New York, Springer-Verlag

[3] ZhuQi-ding, LinQunSuperconvergence Theory of Finite Element Methods (in Chinese)[M], 1989, Changsha, Hunan Science Press

[4] ShaoXian-xi, YinChuan-cun. The joint distributions of the hitting time and place for Brownian motion with drift[J]. Journal of Engineering Mathematics, 1997, 14(14): 123-126(in Chinese)

[5] ФFksendalBStochastic Differential Equations[M], 1995, Berlin Heidelberg, Springer-Verlag