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Abstract An H 1 -Galerkin mixed finite element method (MFEM) is developed for the Benjamin–Bona–Mahony (BBM) equation with the bilinear element and zero order Raviart–Thomas element pair (Q 11 ∕ Q 10 × Q 01). Then based on the special interpolation properties of the above two elements and the mean-value technique, the supercloseness and superconvergence results of order O (h 2) for the semi-discrete scheme and of order O (h 2 + (Δ t) 2) for the Crank–Nicolson fully-discrete scheme are deduced for the original variable u in H 1 -norm and the flux p → in H (d i v , Ω) -norm, respectively. Finally, a numerical example is carried out to demonstrate the theoretical analysis. [ABSTRACT FROM AUTHOR]