Stochastic methods for Dirichlet problems
It is well-known that solutions of second order Dirichlet problems possess probabilistic representations in terms of Ito type stochastic differential equations (SDEs). The authors present a numerical solution method for Dirichlet boundary value problems based on those probabilistic representations. Note that, with this idea, it is possible to find solution values of a partial differential equation at isolated points without having to construct any kind of space-mesh and without knowing approximations for the solution at any other points. Their method is similar to the approach of \textit{F. M. Buchmann} and \textit{W. P. Petersen} [BIT 43, No.~3, 519--540 (2003; Zbl 1050.65001)]. However, their approach differs primarily in the handling of the boundary. Some results of numerical examples are presented, especially for applying these techniques to Laplace and Poisson equations on the unit disk. Euler- and trapezoidal-type methods are used to approximate related SDEs and involved functionals.
scientific article; zbMATH DE number 5000193