  In this paper, we analyze the spectra of the preconditioned matrices arising
from discretized multi-dimensional Riesz spatial fractional diffusion
equations. The finite difference method is employed to approximate the
multi-dimensional Riesz fractional derivatives, which will generate symmetric
positive definite ill-conditioned multi-level Toeplitz matrices. The
preconditioned conjugate gradient method with a preconditioner based on the
sine transform is employed to solve the resulting linear system. Theoretically,
we prove that the spectra of the preconditioned matrices are uniformly bounded
in the open interval (1/2,3/2) and thus the preconditioned conjugate gradient
method converges linearly. The proposed method can be extended to multi-level
Toeplitz matrices generated by functions with zeros of fractional order. Our
theoretical results fill in a vacancy in the literature. Numerical examples are
presented to demonstrate our new theoretical results in the literature and show
the convergence performance of the proposed preconditioner that is better than
other existing preconditioners.
