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. The fourth-order Bi-flux diffusion equation with a nonlinear reaction term is studied by the linear finite element method (FEM). First, a novel important property of high accuracy of this element is proved by the Bramble-Hilbert (B-H) lemma, which is essential to the superconvergence analysis. Then, the Backward-Euler (B-E) and Crank-Nicolson (C-N) fully discrete schemes are developed, and the stabilities of their numerical solutions and the unique solvabilities are demonstrated. Furthermore, by applying a splitting argument to dealing with the nonlinear term, the superconvergence results in H1-norm are derived without any restriction between the mesh size h and the time step τ. Finally, numerical results are presented to verify the rationality of the theoretical analysis.