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Error Estimates of Mixed Finite Element Methods for Time-Fractional Navier–Stokes Equations

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Abstract

This paper studies the Galerkin finite element approximation of time-fractional Navier–Stokes equations. The discretization in space is done by the mixed finite element method. The time Caputo-fractional derivative is discretized by a finite difference method. The stability and convergence properties related to the time discretization are discussed and theoretically proven. Under some certain conditions that the solution and initial value satisfy, we give the error estimates for both semidiscrete and fully discrete schemes. Finally, a numerical example is presented to demonstrate the effectiveness of our numerical methods.

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Acknowledgments

The authors want to thank Prof. Yang Liu, Inner Mongolia University, China, for his kindness and help with the numerical example. The authors would like to express their sincere gratitude to the anonymous reviewers for their careful reading of the manuscript, as well as their comments that lead to a considerable improvement of the original manuscript.

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Authors and Affiliations

  1. Department of Mathematics, LMIB of the Ministry of Education, Beihang University, Beijing, 100191, China

    Xiaocui Li & Xiaoyuan Yang

  2. School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    Yinghan Zhang

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  1. Xiaocui Li
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  2. Xiaoyuan Yang
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Correspondence to Xiaoyuan Yang.

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This research is supported by the National Natural Science Foundation of China under Grant 61271010 and by Beijing Natural Science Foundation under Grant 4152029.

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Li, X., Yang, X. & Zhang, Y. Error Estimates of Mixed Finite Element Methods for Time-Fractional Navier–Stokes Equations. J Sci Comput 70, 500–515 (2017). https://doi.org/10.1007/s10915-016-0252-3

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  • DOI: https://doi.org/10.1007/s10915-016-0252-3

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