Spectral analysis and structure preserving preconditioners for fractional diffusion equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:307

Article

Donatelli, Marco¹  Mazza, Mariarosa¹  Serra-Capizzano, Stefano^1,2 

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

[2] Uppsala Univ, Div Comp Sci, Dept Informat Technol, Box 337, SE-75105 Uppsala, Sweden

关键词: Fractional diffusion equations;    Toeplitz matrix;    Locally Toeplitz sequence of matrices;    Singular value/eigenvalue distribution Preconditioning;   

DOI  :  10.1016/j.jcp.2015.11.061

来源: Elsevier

【 摘 要 】

Fractional partial order diffusion equations are a generalization of classical partial differential equations, used to model anomalous diffusion phenomena. When using the implicit Euler formula and the shifted Grnwald formu, it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a Toeplitz-like structure. In this paper we focus our attention on the case of variable diffusion coefficients. Under appropriate conditions, we show that the sequence of the coefficient matrices belongs to the Generalized Locally Toeplitz class and we compute the symbol describing its asymptotic eigenvalue/singular value distribution, as the matrix size diverges. We employ the spectral information for analyzing known methods of preconditioned Krylov and multigrid type, with both positive and negative results and with a look forward to the multidimensional setting. We also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system, with Krylov methods such as CGNR and GMRES. A number of numerical examples showthat our proposal is more effective than recently used circulant preconditioners. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free