Superconvergence analysis of a low order expanded conforming mixed finite element method for the extended Fisher–Kolmogorov equation

Abstract

This paper is devoted to the superconvergence analysis of a low order expanded conforming mixed finite element method(MFEM) for the extended Fisher–Kolmogorov equation. Both semi-discrete and linearized backward Euler fully-discrete schemes are analyzed with bilinear and Nédéleć elements. The present work has two main contributions: One is that some a priori bounds of the approximations are derived, which lead to the unique solvability of the semi-discrete scheme and make the nonlinear term can be dealt with efficiently. By such a way, the superclose properties and superconvergence results can be derived through the high accuracy results of the above two elements and the interpolated postprocessing technique, respectively. The other one is that a novel splitting technique is utilized to deal with the nonlinear term, which can avoid the use of popular error splitting technique and the restrictions between the time step size and mesh size. Besides, the combination technique of interpolation and projection operators also plays an important role in the superclose and superconvergence analysis. Finally, some numerical results are provided to show the validity of the theoretical analysis.