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A new low order two-grid mixed finite element method (FEM) is developed for the nonlinear Benjamin-Bona-Mahoney (BBM) equation, in which the famous nonconforming rectangular C N Q 1 r o t element and Q 0 × Q 0 constant element are used to approximate the exact solution u and the variable p → = ∇ u t , respectively. Then, based on the special properties of these two elements and interpolation post-processing technique, the superconvergence results for u in broken H 1-norm and p → in L 2-norm are obtained for the semi-discrete and Crank-Nicolson fully-discrete schemes without the restriction between the time step τ and coarse mesh size H or the fine mesh size h , which improve the results of the existing literature. Finally, some numerical results are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]