  In this paper, we develop a fast numerical method for solving the
time-dependent Riesz space fractional diffusion equations with a nonlinear
source term in the convex domain. An implicit finite difference method is
employed to discretize the Riesz space fractional diffusion equations with a
penalty term in a rectangular region by the volume-penalization approach. The
stability and the convergence of the proposed method are studied. As the
coefficient matrix is with the Toeplitz-like structure, the generalized minimum
residual method with a preconditioner based on the sine transform is exploited
to solve the discretized linear system, where the preconditioner is constructed
in view of the combination of two approximate inverse ${\tau}$ matrices, which
can be diagonalized by the sine transform. The spectrum of the preconditioned
matrix is also investigated. Numerical experiments are carried out to
demonstrate the efficiency of the proposed method.
