Random walk on fixed spheres for Laplace and Lamé equations

Abstract

The Random Walk on Fixed Spheres (RWFS) introduced in our previous paper is presented in details for Laplace and Lamé equations governing static elasticity problems. The approach is based on the Poisson type integral formulae written for each disc of a domain consisting of a family of overlapping discs. The original differential boundary value problem is equivalently reformulated in the form of a system of integral equations defined on the intersection surfaces (arches, in 2D, and caps, if generalized to 3D spheres). To solve the obtained system of integral equations, a Random Walk procedure is constructed where the random walks are living on the intersecting surfaces. Since the spheres are fixed, it is convenient to construct also discrete random walk methods for solving the system of linear equations approximating the system of integral equations. We develop here two classes of special Monte Carlo iterative methods for solving these systems of linear algebraic equations which are constructed as a kind of randomized versions of the Chebyshev iteration method and Successive Over Relaxation (SOR) method. It is found that in this class of randomized SOR methods, the Gauss-Seidel method has a minimal variance ...