Approximate parameterized splitting preconditioning for anisotropic space-fractional diffusion equation

We develop a new approximate parameterized splitting preconditioner for the Toeplitz-like discretized systems of the unsteady-state space-fractional diffusion equation to overcome the preconditioning challenge in the anisotropic case that one of the coefficients is significantly larger than the other. The construction of preconditioner is based on parameterized matrix splitting technique along with pre- and post-circulant-like approximation, which facilitates very economic implementation. Theoretical analysis shows that the preconditioner with a suitable parameter could lead to the corresponding preconditioned matrix being expressed as the sum of a matrix whose spectrum is clustered around one, a low rank matrix and a small norm matrix, which suggests that the preconditioner is effective. Numerical results of the preconditioned GMRES method indicate that the preconditioner's practical efficiency is satisfactory.