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An Effective Finite Element Method with Shifted Fractional Powers Bases for Fractional Boundary Value Problems

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Abstract

In this paper, an effective finite element method with shifted fractional powers bases is developed for fractional convection diffusion equations involving a Riemann–Liouville derivative of order \(\alpha \in (3/2,2)\). A Petrov-Galerkin variational formulation is constructed on the domain \(\tilde{H}^{\alpha -1}(\Omega )\times \tilde{H}^{1}(\Omega )\), based on which the finite element approximation scheme is developed by employing shifted fractional power functions and continuous piecewise polynomials of degree up to \(m~(m\in \mathbb {N}^+)\) for trial and test finite element spaces, respectively. The approximation property of trial finite element space and \(\inf \)-\(\sup \) condition for discrete variational form are derived, which enables us to derive the error estimates in \(L^2(\Omega )\) and \(H^{\alpha -1}(\Omega )\) norms. Numerical examples are included to verify the theoretical findings and demonstrate an actual convergence rate of order \(\alpha -1+m\), where m equals to 1 or 2.

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Acknowledgements

The authors would like to express their sincere appreciation to the Editor and two anonymous reviewers for the suggestions and comments which have significantly improved the quality of this article.

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Authors and Affiliations

  1. Powder Metallurgy Research Institute, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Taibai Fu

  2. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Changfa Du & Yufeng Xu

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  1. Taibai Fu
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  2. Changfa Du
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Correspondence to Yufeng Xu.

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T. Fu is supported by the National Natural Science Foundation of China (No. 52101027) and the Science and Technology Innovation Program of Hunan Province (No. 2021RC2001). Y. Xu and C. Du are supported by the Natural Science Foundation of Hunan Provice (No. 2019JJ50755) and Natural Science Foundation of China (No. 51974377).

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Fu, T., Du, C. & Xu, Y. An Effective Finite Element Method with Shifted Fractional Powers Bases for Fractional Boundary Value Problems. J Sci Comput 92, 4 (2022). https://doi.org/10.1007/s10915-022-01854-3

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  • DOI: https://doi.org/10.1007/s10915-022-01854-3

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