High fidelity numerical solutions of the Fokker-Planck equation

The finite element method has been employed to numerically solve Fokker-Planck equations for systems of dimension two and three (Spencer and Bergman [1994]; Wojtkiewicz, et al. [1995], Bergman, et al. [1996]). Those results provided a satisfactory level of accuracy (i. e., O(10(-4))) over the entire computational domain. However, this level may not be deemed acceptable for analyzing the reliability of certain critical systems, where failure probabilities on the order of 10(-10) or smaller are commonplace. Consequently, higher order discretization schemes have been developed and implemented to ascertain the limits of accuracy of several standard finite difference techniques, particularly in the tails of the response distribution. Herein, results for several representative systems of dimension two are discussed.