Random walk method for the two- and three-dimensional Laplace, Poisson and Helmholtz's equations

@article{citeulike:888478, abstract = {The random walk method (RWM) is developed here for solving the Laplace, Poisson, and Helmholtz equations in two and three dimensions. The RWM is a local method, i.e. the solution at an arbitrary point can be determined without having to obtain the complete field solution. The method is based on the properties of diffusion processes, the It\^{o} formula, the Dynkin formula, the Feynman-Kac functional, and Monte Carlo simulation. Simplicity, stability, accuracy, and generality are the main features of the proposed method. The RWK is inherently parallel and this fact has been fully exploited in this paper. Extensive numerical results have been presented in order to understand the various parameters involved in the method. Copyright {\copyright} 2001 John Wiley \& Sons, Ltd.}, address = {Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, U.S.A.; Department of Environmental and Civil Engineering, Cornell University, Ithaca, NY 14853, U.S.A.}, author = {Chati, Mandar K. and Grigoriu, Mircea D. and Kulkarni, Salil S. and Mukherjee, Subrata }, citeulike-article-id = {888478}, doi = {10.1002/nme.178}, journal = {International Journal for Numerical Methods in Engineering}, keywords = {elasticity, random}, number = {10}, pages = {1133--1156}, posted-at = {2006-10-07 15:47:31}, priority = {2}, title = {Random walk method for the two- and three-dimensional Laplace, Poisson and Helmholtz's equations}, url = {http://dx.doi.org/10.1002/nme.178}, volume = {51}, year = {2001} }

TY - JOUR ID - citeulike:888478 L3 - citeulike-article-id:888478 TI - Random walk method for the two- and three-dimensional Laplace, Poisson and Helmholtz's equations JF - International Journal for Numerical Methods in Engineering VL - 51 IS - 10 SP - 1133 EP - 1156 AD - Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, U.S.A.; Department of Environmental and Civil Engineering, Cornell University, Ithaca, NY 14853, U.S.A. N2 - The random walk method (RWM) is developed here for solving the Laplace, Poisson, and Helmholtz equations in two and three dimensions. The RWM is a local method, i.e. the solution at an arbitrary point can be determined without having to obtain the complete field solution. The method is based on the properties of diffusion processes, the Itô formula, the Dynkin formula, the Feynman-Kac functional, and Monte Carlo simulation. Simplicity, stability, accuracy, and generality are the main features of the proposed method. The RWK is inherently parallel and this fact has been fully exploited in this paper. Extensive numerical results have been presented in order to understand the various parameters involved in the method. Copyright © 2001 John Wiley & Sons, Ltd. KW - elasticity KW - random AU - Chati, Mandar AU - Grigoriu, Mircea AU - Kulkarni, Salil AU - Mukherjee, Subrata PY - 2001/// UR - http://dx.doi.org/10.1002/nme.178 ER -