Fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations

Abstract

The discretizations of two- and three-dimensional spatial fractional diffusion equations with the shifted finite-difference formulas of the Grünwald-Letnikov type can result in discrete linear systems whose coefficient matrices are of the form D+T, where D is a nonnegative diagonal matrix and T is a block-Toeplitz with Toeplitz-block matrix or a block-Toeplitz with each block being block-Toeplitz with Toeplitz-block matrix. For these discrete spatial fractional diffusion matrices, we construct diagonal and block-circulant with circulant-block splitting preconditioner for the two-dimensional case, and diagonal and block-circulant with each block being block-circulant with circulant-block splitting preconditioner for the three-dimensional case, to further accelerate the convergence rates of Krylov subspace iteration methods, and we analyze the eigenvalue distributions for the corresponding preconditioned matrices. Theoretical results show that except for a small number of outliners the eigenvalues of the preconditioned matrices are located within a complex disk centered at 1 with the radius being exactly less than 1, and numerical experiments demonstrate that these structured preconditioners can significantly improve the convergence behavior of the Krylov subspace iteration methods. Moreover, this approach is superior to the geometric multigrid method and the preconditioned conjugate gradient methods incorporated with the approximate inverse circulant-plus-diagonal preconditioners in both iteration counts and computing times.