Local Solutions in Potential Theory and Linear Elasticity Using Monte Carlo Methods

Abstract

A numerical method called the boundary walk method is described in this paper. The boundary walk method is a local method in the sense that it directly gives the solution at the point of interest. It is based on a global integral representation of the unknown solution in the form of potentials, followed by evaluating the integrals in the resulting series solutions using Monte Carlo simulation. The boundary walk method has been applied to solve interior problems in potential theory with either Dirichlet or Neumann boundary conditions. It has also been applied to solve interior problems in linear elasticity with either displacement or traction boundary conditions. Weakly singular integral formulations in linear elasticity, to which the boundary walk method has been applied, are also derived. Finally, numerical results, which are computed by applying the boundary walk method to solve some two-dimensional problems over convex domains in potential theory and linear elasticity, are presented. These solutions are compared with the known analytical solutions (when available) or with solutions from the standard boundary element method.