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\(\beta \)-Robust Superconvergent Analysis of a Finite Element Method for the Distributed Order Time-Fractional Diffusion Equation

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Abstract

A distributed order time fractional diffusion equation whose solution has a weak singularity near the initial time \(t = 0\) is considered. The numerical method of the paper uses the well-known L1 scheme on a graded mesh to discretize the time Caputo fractional derivative and a standard finite element method in space. A \(\beta \)-robust discrete fractional Grönwall inequality is investigated. By this inequality, the \(\beta \)-robust optimal-rate convergence and a superconvergence bound \(\Vert \nabla R_hu^n-\nabla u_h^n\Vert \) are proved. This superconvergence bound is also used to show that a simple postprocessing of the computed solution will yield a higher order of convergence in the spatial direction. The final convergence result reveals the optimal grading that one should use for the temporal graded mesh. Numerical results show that our analysis is sharp.

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Data Availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The research of Chaobao Huang is supported in part by the National Natural Science Foundation of China under grants Nos. 12101360 and 12171278 and the Natural Science Foundation of Shandong Province under grant ZR2020QA031. The research of Hu Chen is supported in part by the National Natural Science Foundation of China under grant 11801026, sponsored by OUC Scientific Research Starting Fund of Introduced Talent. The research of Na An is supported in part by the National Natural Science Foundation of China under grant 11801332.

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Authors and Affiliations

  1. School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250014, China

    Chaobao Huang

  2. School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, China

    Hu Chen

  3. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Na An

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  1. Chaobao Huang
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  2. Hu Chen
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Correspondence to Hu Chen.

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Huang, C., Chen, H. & An, N. \(\beta \)-Robust Superconvergent Analysis of a Finite Element Method for the Distributed Order Time-Fractional Diffusion Equation. J Sci Comput 90, 44 (2022). https://doi.org/10.1007/s10915-021-01726-2

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  • DOI: https://doi.org/10.1007/s10915-021-01726-2

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