期刊
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
卷 38, 期 4, 页码 1580-1614出版社
SIAM PUBLICATIONS
DOI: 10.1137/17M1115447
关键词
diagonal-times-Toeplitz matrices; preconditioners; variable coefficients; space-fractional diffusion equations; Krylov subspace methods
资金
- University of Macau [MYRG2016-00063-FST]
- FDCT of Macao [054/2015/A2]
- HKRGC [GRF 12301214, 12302715, 12306616, 12200317]
- HKBU [FRG/15-16/064]
In this paper, we study Toeplitz-like linear systems arising from time-dependent onedimensional and two-dimensional Riesz space-fractional diffusion equations with variable diffusion coefficients. The coefficient matrix is a sum of a scalar identity matrix and a diagonal-times-Toeplitz matrix which allows fast matrix-vector multiplication in iterative solvers. We propose and develop a splitting preconditioner for this kind of matrix and analyze the spectra of the preconditioned matrix. Under mild conditions on variable diffusion coefficients, we show that the singular values of the preconditioned matrix are bounded above and below by positive constants which are independent of temporal and spatial discretization step-sizes. When the preconditioned conjugate gradient squared method is employed to solve such preconditioned linear systems, the method converges linearly within an iteration number independent of the discretization step-sizes. Numerical examples are given to illustrate the theoretical results and demonstrate that the performance of the proposed preconditioner is better than other tested solvers.
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