Analysis of a Multi-Term Variable-Order Time-Fractional Diffusion Equation and Its Galerkin Finite Element Approximation
Year: 2022
Author: Huan Liu, Xiangcheng Zheng, Hongfei Fu
Journal of Computational Mathematics, Vol. 40 (2022), Iss. 5 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2102-m2020-0211
Journal of Computational Mathematics, Vol. 40 (2022), Iss. 5 : pp. 814–834
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: Variable-order Multi-term time-fractional diffusion equation Solution regularity Galerkin finite element Parareal method.
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