Block-circulant with circulant-block preconditioners for two-dimensional spatial fractional diffusion equations

Abstract The implicit finite difference scheme with the shifted Grünwald formula for discretizing the two-dimensional spatial fractional diffusion equations can result in discrete linear systems whose coefficient matrices are the sum of the identity matrix and a block-Toeplitz with Toeplitz-block matrix. For these coefficient matrices, we construct block-circulant with circulant-block preconditioners to further accelerate the convergence rate of the Krylov subspace iteration methods. We analyze the eigenvalue distributions for the corresponding preconditioned matrices. Theoretical results show that the Euclidean norms of the preconditioned matrices are bounded, and 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 very small. Numerical experiments demonstrate that these structured preconditioners can significantly improve the convergence behavior of the Krylov subspace iteration methods. Moreover, when the discretization grid h is refined, the preconditioned Krylov subspace iteration methods require an almost constant number of iteration steps and, hence, they show h-independent convergence behavior.