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The Random Walk on Fixed Spheres (RWFS) introduced in our paper [25], and further developed in [26], 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 intersection 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. In [25] we have concluded that in the case of classical potential theory, the Random Walk on Fixed Spheres considerably improves the convergence rate of the standard Random Walk on Spheres method. More interesting, we succeeded there to extend the algorithm to the system of Lamé equations which cannot be solved by the conventional Random Walk on Spheres method. We present here a series of numerical experiments for 2D domains consisting of 5, 10, and 17 discs, and analyze the dependence of the variance on the number of discs and elastic constants. Further generalizations to Neumann and Dirichlet-Neumann boundary conditions are possible, see [23]. [ABSTRACT FROM AUTHOR]

A new general stochastic-deterministic approach for a numerical solution of boundary value problems of potential and elasticity theories is suggested. It is based on the use of the Poisson-like integral formulae for overlapping spheres. An equivalent system of integral equations is derived and then approximated by a system of linear algebraic equations. We develop two classes of special Monte Carlo iterative methods for solving these systems of equations which are a kind of stochastic versions of the Chebyshev iteration method and successive overrelaxation method (SOR). In the case of classical potential theory this approach accelerates the convergence of the well known Random Walk on Spheres method (RWS). What is however much more important, this approach suggests a first construction of a fast convergent finite-variance Monte Carlo method for the system of Lamé equations. [ABSTRACT FROM AUTHOR]