TY - JOUR T1 - A Block Fast Regularized Hermitian Splitting Preconditioner for Two-Dimensional Discretized Almost Isotropic Spatial Fractional Diffusion Equations AU - Liu , Yao-Ning AU - Muratova , Galina V. JO - East Asian Journal on Applied Mathematics VL - 2 SP - 213 EP - 232 PY - 2022 DA - 2022/02 SN - 12 DO - http://doi.org/10.4208/eajam.070621.300821 UR - https://global-sci.org/intro/article_detail/eajam/20251.html KW - Preconditioning, spatial fractional diffusion equation, Toeplitz matrix, two-dimensional problem. AB -

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.