Probabilistic Methods for Numerical Solutions to Higher-Dimensional Dirichlet Problems

In this paper, numerical solutions to the higher-dimensional Dirichlet interior problems and exterior problems over ordinary domains by probabilistic methods are obtained. Higher-dimensional Brownian family with drift has the strong Markov property and its distributions of the time and place of hitting spheres are known. By these and the stochastic representations of the solution we get the numerical solutions to the higher-dimensional Dirichlet problems.