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Journal of Scientific Computing

, Volume 81, Issue 3, pp 1577–1601 | Cite as

Superconvergence Points for the Spectral Interpolation of Riesz Fractional Derivatives

  • Beichuan Deng
  • Zhimin ZhangEmail author
  • Xuan Zhao
Article
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Abstract

In this paper, superconvergence points are located for the approximation of the Riesz derivative of order \(\alpha \) using classical Lobatto-type polynomials when \(\alpha \in (0,1)\) and generalized Jacobi functions (GJF) for arbitrary \(\alpha > 0\), respectively. For the former, superconvergence points are zeros of the Riesz fractional derivative of the leading term in the truncated Legendre–Lobatto expansion. It is observed that the convergence rate for different \(\alpha \) at the superconvergence points is at least \(O(N^{-2})\) better than the optimal global convergence rate. Furthermore, the interpolation is generalized to the Riesz derivative of order \(\alpha > 1\) with the help of GJF, which deal well with the singularities. The well-posedness, convergence and superconvergence properties are theoretically analyzed. The gain of the convergence rate at the superconvergence points is analyzed to be \(O(N^{-(\alpha +3)/2})\) for \(\alpha \in (0,1)\) and \(O(N^{-2})\) for \(\alpha > 1\). Finally, we apply our findings in solving model FDEs and observe that the convergence rates are indeed much better at the predicted superconvergence points.

Keywords

Superconvergence Riesz fractional derivative Spectral interpolation Generalized Jacobi functions 

Mathematics Subject Classification

65N35 65M15 26A33 41A05 41A10 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA
  2. 2.Beijing Computational Science Research CenterBeijingChina
  3. 3.School of MathematicsSoutheast UniversityNanjingChina

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