ANALYSIS OF A MULTI-TERM VARIABLE-ORDER TIME-FRACTIONAL DIFFUSION EQUATION AND ITS GALERKIN FINITE ELEMENT APPROXIMATION  

作　　者：Huan Liu Xiangcheng Zheng Hongfei Fu 

机构地区：[1]School of Mathematical Sciences,Fudan University,Shanghai 200433,China [2]School of Mathematical Sciences,Peking University,Beijing 100871,China [3]School of Mathematical Sciences,Ocean University of China,Qingdao 266100,China

出　　处：《Journal of Computational Mathematics》2022年第5期814-834,共21页计算数学（英文）

基　　金：the National Natural Science Foundation of China(No.11971482);the Natural Science Foundation of Shandong Province(No.ZR2017MA006);the National Science Foundation(No.DMS-1620194);the China Postdoctoral Science Foundation(Nos.2020M681136,2021TQ0017,2021T140129);the International Postdoctoral Exchange Fellowship Program(Talent-Introduction Program)(No.YJ20210019).

摘　　要：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 optimalorder 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.

关 键 词：Variable-order Multi-term time-fractional diffusion equation Solution regularity Galerkin finite element Parareal method. 

分 类 号：O17[理学—数学]