Application of multi-scale finite element methods to the solution of the Fokker-Planck equation

摘要

This paper presents an application of multi-scale finite element methods to the solution of the multi-dimensional Fokker-Planck equation. The Fokker-Planck, or forward Kolmogorov, equation is a degenerate convective-diffusion equation arising in Markov-Process theory. It governs the evolution of the transition probability density function of the response of a broad class of dynamical systems driven by Gaussian noise, and completely describes the response process. Analytical solutions for the Fokker-Planck equation have been developed for only a limited number of low-dimensional systems, leading to a large body of approximation theory. One such approach successfully applied to the solution of these problems in the past is the finite element method, though-for systems of dimension three or less. In this paper, a multi-scale finite element method is applied to the Fokker-Planck equation in an effort to develop a formulation that can yield higher accuracy on cruder spatial discretizations, thus reducing the computational overhead associated with large scale problems that arise in higher dimensions.