Combination of the meshless finite difference approach with the Monte Carlo random walk technique for solution of elliptic problems

Abstract

This paper proposes a stochastic approach for the fast and effective numerical analysis of the second order elliptic differential equations. It is based upon the well-known Monte Carlo (MC) method with a random walk (RW) technique, carried out on the grid of points. This method allows for accurate estimation of the solution of the differential equation at selected point(s) of the domain and/or its boundary. It extends the standard formulation of the Monte Carlo–random walk (MC–RW) approach by means of its appropriate combination with the meshless version of the finite difference method. In this manner, the proposed approach may deal with elliptic equations in more general non-homogeneous form as well as boundary conditions of both essential and natural types. Moreover, arbitrarily irregular clouds of nodes may be used, with no a-priori imposed nodes structure. Therefore, the meshless MC/RW approach may be applied to the significantly wider class of problems with more complex geometry.This concept was examined on variety of 2D boundary value problems. Selected numerical results are presented and discussed. A simple Matlab code is included as well.