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Volume 40, Issue 5
Analysis of a Multi-Term Variable-Order Time-Fractional Diffusion Equation and Its Galerkin Finite Element Approximation

Huan Liu, Xiangcheng Zheng & Hongfei Fu

J. Comp. Math., 40 (2022), pp. 814-834.

Published online: 2022-05

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  • Abstract

In this paper, we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation, and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution. We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time $t = 0$. More precisely, we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to $C^2([0,T])$ in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness, otherwise the solution exhibits the same singular behavior like its constant-order counterpart. Based on these regularity results, we prove optimal-order convergence rate of the Galerkin finite element scheme. Furthermore, we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives. Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.

  • AMS Subject Headings

35B65, 35R11, 65M12, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

liu_h@fudan.edu.cn (Huan Liu)

zhengxch@math.pku.edu.cn (Xiangcheng Zheng)

fhf@ouc.edu.cn (Hongfei Fu)

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@Article{JCM-40-814, author = {Liu , HuanZheng , Xiangcheng and Fu , Hongfei}, title = {Analysis of a Multi-Term Variable-Order Time-Fractional Diffusion Equation and Its Galerkin Finite Element Approximation}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {5}, pages = {814--834}, abstract = {

In this paper, we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation, and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution. We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time $t = 0$. More precisely, we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to $C^2([0,T])$ in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness, otherwise the solution exhibits the same singular behavior like its constant-order counterpart. Based on these regularity results, we prove optimal-order convergence rate of the Galerkin finite element scheme. Furthermore, we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives. Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2102-m2020-0211}, url = {http://global-sci.org/intro/article_detail/jcm/20549.html} }
TY - JOUR T1 - Analysis of a Multi-Term Variable-Order Time-Fractional Diffusion Equation and Its Galerkin Finite Element Approximation AU - Liu , Huan AU - Zheng , Xiangcheng AU - Fu , Hongfei JO - Journal of Computational Mathematics VL - 5 SP - 814 EP - 834 PY - 2022 DA - 2022/05 SN - 40 DO - http://doi.org/10.4208/jcm.2102-m2020-0211 UR - https://global-sci.org/intro/article_detail/jcm/20549.html KW - Variable-order, Multi-term time-fractional diffusion equation, Solution regularity, Galerkin finite element, Parareal method. AB -

In this paper, we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation, and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution. We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time $t = 0$. More precisely, we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to $C^2([0,T])$ in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness, otherwise the solution exhibits the same singular behavior like its constant-order counterpart. Based on these regularity results, we prove optimal-order convergence rate of the Galerkin finite element scheme. Furthermore, we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives. Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.

Huan Liu, Xiangcheng Zheng & Hongfei Fu. (2022). Analysis of a Multi-Term Variable-Order Time-Fractional Diffusion Equation and Its Galerkin Finite Element Approximation. Journal of Computational Mathematics. 40 (5). 814-834. doi:10.4208/jcm.2102-m2020-0211
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