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

Samir Karaa, Kassem Mustapha, Amiya K. Pani

Research output: Contribution to journalArticle

8 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

Fingerprint

Fractional Diffusion
Diffusion Problem
Error Analysis
Error analysis
Finite Element Method
Finite element method
Energy
Optimal Bound
Derivatives
Leibniz' formula
Norm
Fractional Diffusion Equation
Continuous Solution
Convex Domain
Fractional Derivative
Integral Operator
Piecewise Linear
Experiments
Error Bounds
Bounded Domain

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

Cite this

Optimal Error Analysis of a FEM for Fractional Diffusion Problems by Energy Arguments. / Karaa, Samir; Mustapha, Kassem; Pani, Amiya K.

In: Journal of Scientific Computing, 19.05.2017, p. 1-17.

Research output: Contribution to journalArticle

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