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A Galerkin Finite Element Method for a Class of Time–Space Fractional Differential Equation with Nonsmooth Data

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Abstract

In this article, a Galerkin finite element approximation for a class of time–space fractional differential equation is studied, under the assumption that \(u_{tt}, u_{ttt}, u_{2\alpha ,tt}\) are continuous for \(\varOmega \times (0,T]\), but discontinuous at time \(t=0\). In spatial direction, the Galerkin finite element method is presented. And in time direction, a Crank–Nicolson time-stepping is used to approximate the fractional differential term, and the product trapezoidal method is employed to treat the temporal fractional integral term. By using the properties of the fractional Ritz projection and the fractional Ritz–Volterra projection, the convergence analyses of semi-discretization scheme and full discretization scheme are derived separately. Due to the lack of smoothness of the exact solution, the numerical accuracy does not achieve second order convergence in time, which is \(O(k^{3-\beta }+k^{3}t_{n+1}^{-\beta }+k^{3}t_{n+1}^{-\beta -1})\), \(n=0,1,\ldots ,N-1\). But the convergence order in time is shown to be greater than one. Numerical examples are also included to demonstrate the effectiveness of the proposed method.

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Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Number 11301333; Innovation Program of Shanghai Municipal Education Commission under Grant Number 14YZ165; Funding Scheme for Training Young Teachers in Shanghai Colleges under Grant Number zzhg12001; Natural Science Foundation of Anhui provence under Grant Number 1408085MA1; and Funding Scheme for Training Young Teachers in Shanghai Colleges under Grant Number 14AZ17.

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Authors and Affiliations

  1. Department of Fundamental Courses, Shanghai Customs College, Shanghai, 201204, China

    Zhengang Zhao

  2. School of Mathematical Sciences, Huaibei Normal University, Huaibei, 235000, China

    Yunying Zheng

  3. Department of Mathematics and Physics, Shanghai DianJi University, Shanghai, 201306, China

    Peng Guo

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  1. Zhengang Zhao
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  2. Yunying Zheng
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  3. Peng Guo
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Correspondence to Zhengang Zhao.

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Zhao, Z., Zheng, Y. & Guo, P. A Galerkin Finite Element Method for a Class of Time–Space Fractional Differential Equation with Nonsmooth Data. J Sci Comput 70, 386–406 (2017). https://doi.org/10.1007/s10915-015-0107-3

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  • DOI: https://doi.org/10.1007/s10915-015-0107-3

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