Unconditional superconvergence analysis of two-grid nonconforming FEMs for the fourth order nonlinear extend Fisher-Kolmogorov equation

In this paper, an efficient nonconforming finite element method (FEM) is developed for solving the fourth order nonlinear extended Fisher-Kolmogorov equation. We firstly construct a backward Euler fully discrete scheme with a non-C0 nonconforming double set parameter rectangular Morley element, and prove that this scheme is uniquely solvable and preserves the discrete energy dissipation law. Then based on some a priori bounds of the discrete solution, the superclose estimate in energy norm is obtained unconditionally, and the global superconvergence result is deduced with the help of the interpolated postprocessing technique. Moreover, a two-grid method (TGM) is presented, which can maintain the superconvergence result and save about half of the computing cost. Finally, numerical results are provided to show that the proposed two-grid nonconforming FEMs have a good performance. Here we mention that the above analysis is also valid for the Crank-Nicolson fully discrete scheme.