Optimal Error Analysis of a FEM for Fractional Diffusion Problems by Energy Arguments

Samir Karaa, Kassem Mustapha*, Amiya K. Pani

*Corresponding author for this work

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, a priori optimal error bounds in (Formula presented.)-, (Formula presented.)-norms, and a quasi-optimal bound in (Formula presented.)-norm are derived for the semidiscrete FEM for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a (Formula presented.) type of weights to take care of the singular behavior of the continuous solution at (Formula presented.). The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.

Original languageEnglish
Pages (from-to)1-17
Number of pages17
JournalJournal of Scientific Computing
DOIs
Publication statusAccepted/In press - May 19 2017

Keywords

  • Energy argument error analysis
  • Finite elements
  • Fractional diffusion equation
  • Nonsmooth data

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Software
  • Engineering(all)
  • Computational Theory and Mathematics

Fingerprint Dive into the research topics of 'Optimal Error Analysis of a FEM for Fractional Diffusion Problems by Energy Arguments'. Together they form a unique fingerprint.

  • Cite this