Space-Time Petrov-Galerkin FEM for Fractional Diffusion Problems

Beiping Duan, Bangti Jin, Raytcho Lazarov, Joseph Pasciak, Zhi Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

11 Citations (Scopus)

Abstract

We present and analyze a space-time Petrov-Galerkin finite element method for a time-fractional diffusion equation involving a Riemann-Liouville fractional derivative of order α (0,1) α (0,1)} in time and zero initial data. We derive a proper weak formulation involving different solution and test spaces and show the inf-sup condition for the bilinear form and thus its well-posedness. Further, we develop a novel finite element formulation, show the well-posedness of the discrete problem, and derive error bounds in both energy and L 2 {L^{2}} norms for the finite element solution. In the proof of the discrete inf-sup condition, a certain nonstandard L 2 {L^{2}} stability property of the L 2 {L^{2}} projection operator plays a key role. We provide extensive numerical examples to verify the convergence analysis.
Original languageEnglish
Pages (from-to)1-20
Number of pages20
JournalComputational Methods in Applied Mathematics
Volume18
Issue number1
DOIs
Publication statusPublished - 1 Jan 2018

Keywords

  • Error Estimates
  • Fractional Diffusion
  • Petrov-Galerkin Method
  • Space-Time Finite Element Method

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this