A kind of new mixed element method for time-fractional partial differential equations is studied. The Caputo-fractional derivative of time direction is approximated by two-step difference method and the spatial direction is discretized by a new mixed element method, whose gradient belongs to the simple ( L 2(Ω) 2) space replacing the complex H(div; Ω) space. Some a priori error estimates in L 2-norm for the scalar unknown u and in ( L 2) 2-norm for its gradient σ. Moreover, we also discuss a priori error estimates in H 1-norm for the scalar unknown u.