Unconditional superconvergence analysis for the nonlinear Bi-flux diffusion equation

•A novel important property of high accuracy of the linear finite element is proved by the B-H lemma, which is essential to the superconvergence analysis.•The stabilities of the discrete solutions in the H1 -norm are proved for the B-E and C-N fully discrete schemes.•The unique solvabilities of the two fully discrete schemes are demonstrated by the Brouwer fixed point theorem.•A splitting argument is applied to dealing with the nonlinear term and to eliminate the loss of the order with respect to τ.•The unconditional superconvergence in the H1 -norm is derived without any time step restriction for the two fully discrete schemes.•The presented FEM in this paper is also valid for other popular elements.