Monte Carlo Methods andAppi, Vol. 8, No. 2, pp. 171 - 202 (2002) © KSP2002 Random Walk on Spheres methods for iterative solution of elasticity problems * Karl K. Sabelfeld 1 ' 2 and Irina A. Shalimova 2 1 Weierstrass Institute for Applied Analysis and Stochastics Mohrenstrasse 39, D - 10117 Berlin, Germany E-Mail: sabelfeld@wias-berlin.de 2 Institute of Computational Mathematics and Mathematical Geophysics, Russian Acad. Sei. Lavrentieva str., 6, 630090 Novosibirsk, Russia E-mail: ias@osmf.sscc.ru Abstract — Random Walk on Spheres methods for solving some 2D and 3D boundary value problems of elasticity theory are developed. The boundary value problems studied include the elastic thin plate problems with simply supported boundary, rigid fixing of the boundary, and general 2D and 3D problems for the Lame equation. Unbiased estimators for some special classes of domains based on the generalized Mean Value Theorem which relates the solution at an arbitrary point inside the sphere with the integral of the solution over the sphere are constructed. We study a variance reduction technique based on the explicit Simulation of the first passage of a sphere for the Wiener process starting at an arbitary point inside this sphere. Along with the conventional random walk methods we apply another type of iteration method, the Schwarz iterative procedure whose convergence for the Lame equation was proved in 1936 by S.L. Sobolev. We construct also different types of iterative procedures which combine the probabilistic and conventional deterministic methods of Solutions. Keywords: Biharmonic equation, Lame equation, Random Walk on Spheres, Schwarz iterations l Introduction It is well known that probabilistic representations of Solutions to classical boundary value problems of parabolic and elliptic types in the form of expectations over diffusion stochas- tic processes can be used for a numerical solution by the Monte Carlo Simulation. For the numerical solution of the relevant stochastic differential equation governing the diffusion process, one needs usually a simple finite-difference scheme, e.g., the Euler scheme inside the domain, but considerable difficulties arise when approximating the random process near the boundary: one should take care that in each step, the process is inside the *The support by the European Grant INTAS-99-1501, and NATO Linkage Grant 978912 are kindly acknowledged Brought to you by | Purdue University Libraries Authenticated Download Date | 5/29/15 3:28 AM