Numerical Solution of the Fokker-Planck Equation by Finite Difference and Finite Element Methods - A Comparative Study

Abstract

Finite element and finite difference methods have been widely used, among other methods, to numerically solve the Fokker-Planck equation for investigating the time history of the probability density function of linear and nonlinear 2d and 3d problems; also the application to 4d problems has been addressed. However, due to the enormous increase in computational costs, different strategies are required for efficient application to problems of dimension ≥3. Recently, a stabilized multi-scale finite element method has been effectively applied to the Fokker-Planck equation. Also, the alternating directions implicit method shows good performance in terms of efficiency and accuracy. In this paper various finite difference and finite element methods are discussed, and the results are compared using various numerical examples.