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Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

Part of: Real functions Numerical analysis: Ordinary differential equations Numerical methods in Fourier analysis Numerical linear algebra

Published online by Cambridge University Press:  12 May 2016

Siu-Long Lei ,
Xu Chen  and
Xinhe Zhang
Siu-Long Lei*
Affiliation:
Department of Mathematics, University of Macau, Macau, China
Xu Chen*
Affiliation:
Department of Mathematics, University of Macau, Macau, China
Xinhe Zhang*
Affiliation:
Department of Mathematics, University of Macau, Macau, China
*
*Corresponding author. Email addresses:sllei@umac.mo (S.-L. Lei), 09xchen@gmail.com (X. Chen), yb57010@connect.umac.mo (X. Zhang)
*Corresponding author. Email addresses:sllei@umac.mo (S.-L. Lei), 09xchen@gmail.com (X. Chen), yb57010@connect.umac.mo (X. Zhang)
*Corresponding author. Email addresses:sllei@umac.mo (S.-L. Lei), 09xchen@gmail.com (X. Chen), yb57010@connect.umac.mo (X. Zhang)
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Abstract

High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant pre-conditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(N logN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

Keywords

High-dimensional two-sided fractional diffusion equationimplicit finite difference methodunconditionally stablemultilevel circulant preconditionerGMRES method

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65L12: Finite difference methods 65T50: Discrete and fast Fourier transforms 26A33: Fractional derivatives and integrals
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 2 , May 2016 , pp. 109 - 130
Copyright
Copyright © Global-Science Press 2016 

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