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

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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 \(L^2(\varOmega )\)-, \(H^1(\varOmega )\)-norms, and a quasi-optimal bound in \(L^{\infty }(\varOmega )\)-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 \(t^m\) type of weights to take care of the singular behavior of the continuous solution at \(t=0\). 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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Sultan Qaboos University, Al-Khod, 123, Muscat, Oman

    Samir Karaa

  2. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Kassem Mustapha

  3. Department of Mathematics, Institute of Technology Bombay, Powai, Mumbai, 400076, India

    Amiya K. Pani

Authors
  1. Samir Karaa
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  2. Kassem Mustapha
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  3. Amiya K. Pani
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Corresponding author

Correspondence to Kassem Mustapha.

Additional information

The valuable comments of the referees improved the paper. The support of the Science Technology Unit at KFUPM through King Abdulaziz City for Science and Technology (KACST) under National Science, Technology and Innovation Plan (NSTIP) Project No. 13-MAT1847-04 is gratefully acknowledged.

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Karaa, S., Mustapha, K. & Pani, A.K. Optimal Error Analysis of a FEM for Fractional Diffusion Problems by Energy Arguments. J Sci Comput 74, 519–535 (2018). https://doi.org/10.1007/s10915-017-0450-7

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  • DOI: https://doi.org/10.1007/s10915-017-0450-7

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