The implicit finite difference scheme with the shifted Gr \(\ddot{\textrm{u}}\) nwald 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 symmetric positive definite coefficient matrices, we construct block-circulant with circulant-block preconditioners to further accelerate the convergence rate of the CG method. We analyze the eigenvalue distributions for the corresponding preconditioned matrices. Theoretical results show that the eigenvalues of the preconditioned matrices are weakly clustered around 1 and the convergence of the preconditioned CG method with the block-circulant with circulant-block preconditioner is weakly dependent on the mesh size. Numerical experiments demonstrate that this structured preconditioner can improve the convergence behavior of the CG method. Moreover, this approach is superior to the preconditioned CG methods incorporated with the block diagonal and block tridiagonal preconditioners in both iteration counts and computing times.
