Unconditional superconvergence analysis for the nonlinear Bi-flux diffusion equation

Abstract

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.