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

Arif Masud, Lawrence Bergman

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1513-1526
Number of pages14
JournalComputer Methods in Applied Mechanics and Engineering
Volume194
Issue number12-16
DOIs
StatePublished - Apr 8 2005
Externally publishedYes

Keywords

  • Fokker-Planck equation
  • Multi-scale finite element methods

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications

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