10.1007/s40096-023-00518-z

An adaptive finite element method for Riesz fractional partial integro-differential equations

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  • E. Adel1,
  • I. L. El-Kalla1,
  • A. Elsaid2,
  • M. Sameeh*1,
  1. Mathematics and Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura, EG
  2. Mathematics and Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura, EG Department of Mathematics, Institute of Basic and Applied Sciences, Egypt-Japan University of Science and Technology (E-JUST), New Borg El-Arab City, Alexandria, EG
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Published 2023-08-09

How to Cite

Adel, E., El-Kalla, I. L., Elsaid, A., & Sameeh, M. (2023). An adaptive finite element method for Riesz fractional partial integro-differential equations. Mathematical Sciences, 18(4 (December 2024). https://doi.org/10.1007/s40096-023-00518-z
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Abstract

Abstract The Riesz fractional derivative has been employed to describe the spatial derivative in a variety of mathematical models. In this work, the accuracy of the finite element method (FEM) approximations to Riesz fractional derivative was enhanced by using adaptive refinement. This was accomplished by deducing the Riesz derivatives of the FEM bases to work on non-uniform meshes. We utilized these derivatives to recover the gradient in a space fractional partial integro-differential equation in the Riesz sense. The recovered gradient was used as an a posteriori error estimator to control the adaptive refinement scheme. The stability and the error estimate for the proposed scheme are introduced. The results of some numerical examples that we carried out illustrate the improvement in the performance of the adaptive technique.

Keywords

  • Adaptive finite element method,
  • Fractional partial integro-differential equation,
  • Gradient recovery techniques,
  • Riesz fractional derivative,
  • Polynomial preserving recovery
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