NUMERICAL SOLUTIONS OF LINEAR MIXED BOUNDARY VALUE PROBLEMS USING STOCHASTIC REPRESENTATIONS

The application of It^oʹs formula induces some probabilistic representations of solutions of deterministic linear problems with boundary conditions of Dirichlet, Neumann, Fourier and mixed types. These representations are used to establish some easily implementable algorithms which compute an approximate solution by means of simulation of re ected random walks. They do not require selected con gurations at the neighbourhood of the domain boundary, nor a discretization mesh. The associated simulation methods are obtained and applied to problems for each class of boundary conditions. Numerical experiments with distributed source in two- or three-dimensional geometries, and computational results with estimation of error, are reported.