Superconvergence analysis and extrapolation of a BDF2 fully discrete scheme for nonlinear reaction–diffusion equations

The main aim of this paper is to propose a 2-step backward differential formula (BDF2) fully discrete scheme with the bilinear $Q_{11}$ finite element method (FEM) for the nonlinear reaction–diffusion equation. By use of the combination technique of the element’s interpolation and Ritz projection, and through the interpolation post-processing approach, the superclose and global superconvergence estimates with order $O(h^2+{\tau }^2)$ in $H^1$-norm are deduced rigorously. Furthermore, with the help of the asymptotic error expansion of the $Q_{11}$ element, a new suitable fully discrete scheme is developed, and the extrapolation result of order $O(h^3+{\tau }^2)$ in $H^1$-norm is derived, which is one order higher than that of the above traditional superconvergence estimate with respect to $h$. Here $h$ is the mesh size and $\tau$ is the time step. Finally, some numerical results are provided to verify the theoretical analysis. It seems that the extrapolation of the fully discrete finite element scheme has never been seen in the previous studies.