Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study

doi:10.1007/s42967-019-00042-9

Communications on Applied Mathematics and Computation ›› 2020, Vol. 2 ›› Issue (2): 163-177.doi: 10.1007/s42967-019-00042-9

• ORIGINAL PAPER • 下一篇

Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study

Bangti Jin¹, Zhi Zhou²

1 Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK;
2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

收稿日期:2018-03-10 修回日期:2019-07-15 出版日期:2020-06-20 发布日期:2020-02-19
通讯作者: Zhi Zhou, Bangti Jin E-mail:zhizhou@polyu.edu.hk;b.jin@ucl.ac.uk,bangti.jin@gmail.com

Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study

Bangti Jin¹, Zhi Zhou²

1 Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK;
2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

Received:2018-03-10 Revised:2019-07-15 Online:2020-06-20 Published:2020-02-19
Contact: Zhi Zhou, Bangti Jin E-mail:zhizhou@polyu.edu.hk;b.jin@ucl.ac.uk,bangti.jin@gmail.com

摘要/Abstract

摘要： In this work, we develop an efcient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of order α(0, 1) in time. The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear fnite elements in space and corrected high-order BDF convolution quadrature in time. At each time step, instead of solving the linear algebraic system exactly, we employ a multigrid iteration with a Gauss-Seidel smoother to approximate the solution efciently. Illustrative numerical results for nonsmooth problem data are presented to demonstrate the app

关键词: Subdifusion, Convolution quadrature, Multigrid, Incomplete iterative scheme

Abstract: In this work, we develop an efcient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of order α(0, 1) in time. The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear fnite elements in space and corrected high-order BDF convolution quadrature in time. At each time step, instead of solving the linear algebraic system exactly, we employ a multigrid iteration with a Gauss-Seidel smoother to approximate the solution efciently. Illustrative numerical results for nonsmooth problem data are presented to demonstrate the app

Key words: Subdifusion, Convolution quadrature, Multigrid, Incomplete iterative scheme

中图分类号:

Bangti Jin, Zhi Zhou. Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 163-177.

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Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study

Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study

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