On regularized Hermitian splitting iteration methods for solving discretized almost‐isotropic spatial fractional diffusion equations

The shifted finite-difference discretization of the one-dimensional almost-isotropic spatial fractional diffusion equation results in a discrete linear system whose coefficient matrix is a sum of two diagonal-times-Toeplitz matrices. For this kind of linear systems, we propose a class of regularized Hermitian splitting iteration methods and prove its asymptotic convergence under mild conditions. For appropriate circulant-based approximation to the corresponding regularized Hermitian splitting preconditioner, we demonstrate that the induced fast regularized Hermitian splitting preconditioner possesses a favorable preconditioning property. Numerical results show that, when used to precondition Krylov sub-space iteration methods such as generalized minimal residual and biconjugate gradient stabilized methods, the fast preconditioner significantly outperforms several existing ones.