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Spectral Analysis for Preconditioning of Multi-Dimensional Riesz Fractional Diffusion Equations

Spectral Analysis for Preconditioning of Multi-Dimensional Riesz Fractional Diffusion Equations
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Year:    2022

Author:    Xue-Lei Lin, Michael K. Ng, Xin Huang, Hai-Wei Sun, Xue-Lei Lin, Michael K. Ng, Hai-Wei Sun

Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 3 : pp. 565–591

Abstract

In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives, which generates symmetric positive definite ill-conditioned multi-level Toeplitz matrices. The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval $(\frac{1}{2},\frac{3}{2})$ and thus the preconditioned conjugate gradient method converges linearly within an iteration number independent of the discretization step-size. Moreover, the proposed method can be extended to handle ill-conditioned multi-level Toeplitz matrices whose blocks are generated by functions with zeros of fractional order. Our theoretical results fill in a vacancy in the literature. Numerical examples are presented to show the convergence performance of the proposed preconditioner that is better than other preconditioners.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/nmtma.OA-2022-0032

Numerical Mathematics: Theory, Methods and Applications, Vol. 15 (2022), Iss. 3 : pp. 565–591

Published online:    2022-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    27

Keywords:    Multi-dimensional Riesz fractional derivative multi-level Toeplitz matrix sine transform based preconditioner preconditioned conjugate gradient method.

Author Details

Xue-Lei Lin

Michael K. Ng

Xin Huang

Hai-Wei Sun

Xue-Lei Lin

Michael K. Ng

Hai-Wei Sun

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