In this paper, a local discontinuous Galerkin (LDG) scheme for the time-fractional diffusion equation is proposed and analyzed. The Caputo time-fractional derivative (of order $\alpha$, with $0< \alpha <1$) is approximated by a finite difference method with an accuracy of order $3-\alpha$, and the space discretization is based on the LDG method. For the finite difference method, we summarize and supplement some previous work by others, and apply it to the analysis of the convergence and stability of the proposed scheme. The optimal error estimate is obtained in the $L^2$ norm, indicating that the scheme has temporal $(3 -\alpha)$ th-order accuracy and spatial $(k+1)$ th-order accuracy, where $k$ denotes the highest degree of a piecewise polynomial in discontinuous finite element space. The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.
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