An unstructured mesh finite difference/finite element method for the three-dimensional time-space fractional Bloch-Torrey equations on irregular domains

Zongze Yang; Fawang Liu; Yufeng Nie; Ian Turner

doi:10.1016/j.jcp.2020.109284

An unstructured mesh finite difference/finite element method for the three-dimensional time-space fractional Bloch-Torrey equations on irregular domains

Zongze Yang, Fawang Liu, Yufeng Nie, Ian Turner

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

24 Scopus citations

Abstract

In this paper, we use the finite element method (FEM) to solve the time-space fractional Bloch-Torrey equation on irregular domains in R³. Based on linear Lagrange basis functions, a space semi-discrete FEM scheme is given. By adopting the L2−1_σ approximation for the Caputo fractional derivative, a fully discrete scheme is presented. Furthermore, we provide the details on how to implement our FEM for the space fractional Bloch-Torrey equation. Also, the stability and convergence of the fully discrete scheme is investigated. The error estimations with respect to the L² and energy norms are given. In addition, some numerical examples are presented to verify the efficiency of our method.

Original language	English
Article number	109284
Journal	Journal of Computational Physics
Volume	408
DOIs	https://doi.org/10.1016/j.jcp.2020.109284
State	Published - 1 May 2020

Keywords

Bloch-Torrey equations
Finite element method
Irregular domains
Riesz fractional derivatives
Three dimensions

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10.1016/j.jcp.2020.109284

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title = "An unstructured mesh finite difference/finite element method for the three-dimensional time-space fractional Bloch-Torrey equations on irregular domains",

abstract = "In this paper, we use the finite element method (FEM) to solve the time-space fractional Bloch-Torrey equation on irregular domains in R3. Based on linear Lagrange basis functions, a space semi-discrete FEM scheme is given. By adopting the L2−1σ approximation for the Caputo fractional derivative, a fully discrete scheme is presented. Furthermore, we provide the details on how to implement our FEM for the space fractional Bloch-Torrey equation. Also, the stability and convergence of the fully discrete scheme is investigated. The error estimations with respect to the L2 and energy norms are given. In addition, some numerical examples are presented to verify the efficiency of our method.",

keywords = "Bloch-Torrey equations, Finite element method, Irregular domains, Riesz fractional derivatives, Three dimensions",

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AU - Yang, Zongze

AU - Liu, Fawang

AU - Nie, Yufeng

AU - Turner, Ian

PY - 2020/5/1

Y1 - 2020/5/1

N2 - In this paper, we use the finite element method (FEM) to solve the time-space fractional Bloch-Torrey equation on irregular domains in R3. Based on linear Lagrange basis functions, a space semi-discrete FEM scheme is given. By adopting the L2−1σ approximation for the Caputo fractional derivative, a fully discrete scheme is presented. Furthermore, we provide the details on how to implement our FEM for the space fractional Bloch-Torrey equation. Also, the stability and convergence of the fully discrete scheme is investigated. The error estimations with respect to the L2 and energy norms are given. In addition, some numerical examples are presented to verify the efficiency of our method.

AB - In this paper, we use the finite element method (FEM) to solve the time-space fractional Bloch-Torrey equation on irregular domains in R3. Based on linear Lagrange basis functions, a space semi-discrete FEM scheme is given. By adopting the L2−1σ approximation for the Caputo fractional derivative, a fully discrete scheme is presented. Furthermore, we provide the details on how to implement our FEM for the space fractional Bloch-Torrey equation. Also, the stability and convergence of the fully discrete scheme is investigated. The error estimations with respect to the L2 and energy norms are given. In addition, some numerical examples are presented to verify the efficiency of our method.

KW - Bloch-Torrey equations

KW - Finite element method

KW - Irregular domains

KW - Riesz fractional derivatives

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M3 - 文章

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An unstructured mesh finite difference/finite element method for the three-dimensional time-space fractional Bloch-Torrey equations on irregular domains

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