A Block Fast Regularized Hermitian Splitting Preconditioner for Two-Dimensional Discretized Almost Isotropic Spatial Fractional Diffusion Equations

Abstract

Block fast regularized Hermitian splitting preconditioners for matrices arising in approximate solution of two-dimensional almost-isotropic spatial fractional diffusion equations are constructed. The matrices under consideration can be represented as the sum of two terms, each of which is a nonnegative diagonal matrix multiplied by a block Toeplitz matrix having a special structure. We prove that excluding a small number of outliers, the eigenvalues of the preconditioned matrix are located in a complex disk of radius $r<1$ and centered at the point $z_0=1$. Numerical experiments show that such structured preconditioners can significantly improve computational efficiency of the Krylov subspace iteration methods such as the generalized minimal residual and bi-conjugate gradient stabilized methods. Moreover, if the corresponding equation is almost isotropic, the methods constructed outperform many other existing preconditioners.