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

Abstract 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.

[1]  Y. Wen.  Method for Random Vibration of Hysteretic Systems , 1976 .

[2]  Muhammad Ayub,et al.  A NEW STABILIZED FORMULATION FOR CONVECTIVE-DIFFUSIVE HEAT TRANSFER , 2003 .

[3]  Y. Wen.  Approximate Method for Nonlinear Random Vibration , 1975 .

[4]  Walter V. Wedig,et al.  On the Calculation of Stationary Solutions of Multi-Dimensional Fokker–Planck Equations by Orthogonal Functions , 2000 .

[5]  T. Hughes.  Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[6]  Nonlinear Stochastic Mechanics , 1992 .

[7]  J. D. Atkinson.  Eigenfunction expansions for randomly excited non-linear systems , 1973 .

[8]  T. Hughes,et al.  Streamline-upwind/Petrov-Galerkin methods for advection dominated flows. , 1980 .

[9]  Billie F. Spencer,et al.  Robust Numerical Solution of the Transient Fokker-Planck Equation for Nonlinear Dynamical Systems , 1992 .

[10]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems , 1987 .

[11]  B. Spencer,et al.  On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems , 1993 .

[12]  G. Schuëller.  A state-of-the-art report on computational stochastic mechanics , 1997 .

[13]  Thomas J. R. Hughes,et al.  A stabilized mixed finite element method for Darcy flow , 2002 .

[14]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: II. Beyond SUPG , 1986 .

[15]  Lawrence A. Bergman,et al.  On the reliability of the linear oscillator and systems of coupled oscillators , 1982 .

[16]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[17]  T. Hughes,et al.  The Galerkin/least-squares method for advective-diffusive equations , 1988 .

[18]  Arif Masud,et al.  A multiscale/stabilized finite element method for the advection–diffusion equation , 2004 .