WIAS Preprint No. 704, (2001)

Random walk on spheres methods for iterative solution of elasticity problems


  • Sabelfeld, Karl K.
  • Shalimova, Irina

2010 Mathematics Subject Classification

  • 65C05 76N20


  • Random Walk on Spheres Process, Schwarz iterations, Lame equation, elastic plates


Random Walk on Spheres method 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 Lamé 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. 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 Lamé 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.

Appeared in

  • Monte Carlo Methods Appl., 8 (2002) pp. 171--202.

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