Toeplitz-like preconditioner for linear systems from spatial fractional diffusion equations

Document Type : Research Article

Author

• N. Akhoundi

School of mathematics and computer science, Damghan university, Damghan, Iran.

10.22067/ijnao.2021.11325.0

Abstract

‎The article deals with constructing Toeplitz-like preconditioner for linear systems arising from finite difference discretization of the spatial fractional diffusion equations‎. ‎The coefficient matrices of these linear systems have an $S+L$ structure‎, ‎where $S$ is a symmetric positive definite (SPD) matrix and $L$ satisfies $\mbox{rank}(L)\leq 2$‎. ‎We introduce an approximation for the SPD part $S$‎, ‎which is called $P_S$‎, ‎and then we show that the preconditioner $P=P_S+L$ has the Toeplitz-like structure and its displacement rank is 6‎.  ‎The analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Numerical experiments exhibit that the Toeplitz-like preconditioner can significantly improve the convergence properties of the applied iteration method.

Keywords

• Fractional diffusion equation
• Toeplitz-like matrix
• Krylov subspace methods
• PGMRES

20.1001.1.24236977.2021.11.1.5.2

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