A new analysis of a numerical method for the time-fractional Fokker-Planck equation with general forcing

Can Huang, Kim Ngan Le, Martin Stynes

Research output: Contribution to journalArticleResearchpeer-review

2 Citations (Scopus)

Abstract

First, a new convergence analysis is given for the semidiscrete (finite elements in space) numerical method that is used in Le et al. (2016, Numerical solution of the time-fractional Fokker-Planck equation with general forcing. SIAM J. Numer. Anal.,54 1763-1784) to solve the time-fractional Fokker-Planck equation on a domain Ω × [0,T] with general forcing, i.e., where the forcing term is a function of both space and time. Stability and convergence are proved in a fractional norm that is stronger than the L2(Ω) norm used in the above paper. Furthermore, unlike the bounds proved in Le et al., the constant multipliers in our analysis do not blow up as the order of the fractional derivative α approaches the classical value of 1. Secondly, for the semidiscrete (L1 scheme in time) method for the same Fokker-Planck problem, we present a new L2(Ω) convergence proof that avoids a flaw in the analysis of Le et al.'s paper for the semidiscrete (backward Euler scheme in time) method.

Original languageEnglish
Pages (from-to)1217-1240
Number of pages24
JournalIMA Journal of Numerical Analysis
Volume40
Issue number2
DOIs
Publication statusPublished - 24 Apr 2020

Keywords

  • finite elements
  • fractional Fokker-Planck equation
  • Gronwall inequality
  • initial-boundary value problem
  • time-dependent forcing

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