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Error Estimate of Finite Element Approximation for Two-Sided Space-Fractional Evolution Equation with Variable Coefficient

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Abstract

In this paper, we develop and analyze a finite element method (FEM) for a one-dimensional two-sided time-dependent space-fractional diffusion equation (sFDE) with variable diffusivity. We first prove the regularity of the solution to the steady-state sFDE equipped with variable diffusivity and then utilize the regularity estimate to analyze the property of an elliptic projection, which is of great importance to derive the error estimate for the time-dependent evolution problems. It has been well-known that the bilinear form corresponding to the finite element weak formulation for variable coefficient problems may not be coercive, and thus the well-posedness of the weak formulation can not be guaranteed. To perform the error estimate, we reformulate the steady-state equation to its equivalent form and then prove the weak coercivity of the corresponding bilinear form via the Gårding’s inequality, which leads to optimal-order error estimates of the FEM to the steady-state equation and thus to the interested evolution equation. Compared with some existing works on the FEM to variable-coefficient space-fractional problems, the main advantages of the developed numerical analysis techniques lie in the relaxed assumptions on the variable coefficient and its potential extensions to high-dimensional problems. Numerical experiments are performed to verify the theoretical findings.

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Acknowledgements

The authors would like to express their most sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper. This work was supported in part by the National Natural Science Foundation of China (Nos. 11971482, 12131014), by the National Science Foundation (No. DMS-1620194), by the China Postdoctoral Science Foundation (Nos. 2020M681136, 2021M700244, 2021TQ0017, 2021T140129), and by the International Postdoctoral Exchange Fellowship Program (Talent-Introduction Program) (No. YJ20210019).

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Huan Liu

  2. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Xiangcheng Zheng

  3. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Hong Wang

  4. School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, Shandong, China

    Hongfei Fu

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  1. Huan Liu
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Liu, H., Zheng, X., Wang, H. et al. Error Estimate of Finite Element Approximation for Two-Sided Space-Fractional Evolution Equation with Variable Coefficient. J Sci Comput 90, 15 (2022). https://doi.org/10.1007/s10915-021-01698-3

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  • DOI: https://doi.org/10.1007/s10915-021-01698-3

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