Numerical Analysis of a Fast Finite Element Method for a Hidden-Memory Variable-Order Time-Fractional Diffusion Equation

被引：10

作者：

Jia, Jinhong ^{[1
]}

Wang, Hong ^{[2
]}

Zheng, Xiangcheng ^{[3
]}

机构：

[1] Shandong Normal Univ, Sch Math & Stat, Jinan 250358, Shandong, Peoples R China

[2] Univ South Carolina, Dept Math, Columbia, SC 29208 USA

[3] Peking Univ, Sch Math Sci, Beijing 100871, Peoples R China

来源：

JOURNAL OF SCIENTIFIC COMPUTING | 2022年 / 91卷 / 02期

基金：

中国博士后科学基金; 美国国家科学基金会; 中国国家自然科学基金;

关键词：

Variable-order time-fractional diffusion equation; Hidden-memory; Finite element method; Optimal-order error estimate; Divide and conquer; DIFFERENCE METHOD; REGULARITY; APPROXIMATION; ALGORITHMS; MODELS; SCHEME;

D O I：

10.1007/s10915-022-01820-z

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We investigate a fast finite element scheme to a hidden-memory variable-order time-fractional diffusion equation. Different from the traditional LI methods, a fast approximation to the hidden-memory variable-order fractional derivative is derived to reduce the computational cost of generating coefficients from O(N-2) to O(N log N), where N refers to the number of time steps. We then develop different techniques from the analysis of L1 methods to prove error estimates for the corresponding fast fully-discrete finite element scheme. Furthermore, a fast divide and conquer algorithm is proposed to reduce the complexity of solving the linear systems from O(M N-2) to O(M N log(2 )N) where M stands for the spatial degree of freedom. Numerical experiments are presented to substantiate the theoretical results.

引用

页数：17

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