Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:350

Article

Moghaderi, Hamid¹  Dehghan, Mehdi¹  Donatelli, Marco²  Mazza, Mariarosa³ 

[1] Amirkabir Univ Technol, Fac Math & Comp Sci, Dept Appl Math, 424 Hafez Ave, Tehran 15914, Iran

[2] Univ Insubria, Dept Sci & High Technol, Via Valleggio 11, I-22100 Como, Italy

[3] Max Planck Inst Plasma Phys, Div Numer Methods Plasma Phys, Boltzmannstr 2, D-85748 Garching, Germany

关键词: Fractional diffusion equations;    CN-WSGD scheme;    Spectral analysis;    GLT theory;    Multigrid methods;   

DOI  :  10.1016/j.jcp.2017.08.064

来源: Elsevier

【 摘 要 】

Fractional diffusion equations (FDEs) are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance. In this paper, we focus on a two-dimensional space-FDE problem discretized by means of a second order finite difference scheme obtained as combination of the Crank-Nicolson scheme and the so-called weighted and shifted Grunwald formula. By fully exploiting the Toeplitz-like structure of the resulting linear system, we provide a detailed spectral analysis of the coefficient matrix at each time step, both in the case of constant and variable diffusion coefficients. Such a spectral analysis has a very crucial role, since it can be used for designing fast and robust iterative solvers. In particular, we employ the obtained spectral information to define a Galerkin multigrid method based on the classical linear interpolation as grid transfer operator and damped-Jacobi as smoother, and to prove the linear convergence rate of the corresponding two-grid method. The theoretical analysis suggests that the proposed grid transfer operator is strong enough for working also with the V-cycle method and the geometric multigrid. On this basis, we introduce two computationally favourable variants of the proposed multigrid method and we use them as preconditioners for Krylov methods. Several numerical results confirm that the resulting preconditioning strategies still keep a linear convergence rate. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free