Circulant-based approximate inverse preconditioners for a class of fractional diffusion equations

摘要

We consider fast solving a class of spatial fractional diffusion equations where the fractional differential operators are comprised of Riemann-Liouville and Caputo fractional derivatives. A circulant-based approximate inverse preconditioner is established for the discrete linear systems resulted from the finite difference discretization of this kind of fractional diffusion equations. By sufficiently exploring the Toeplitz-like structure and the rapid decay properties of the internal sub-matrices in the coefficient matrix, we show that the spectrum of the preconditioned matrix is clustered around one. Numerical experiments are performed to demonstrate the effectiveness of the proposed preconditioner.