A new linearized crank-nicolson mixed element scheme for the extended Fisher-Kolmogorov equation

Abstract

We present a new mixed finite element method for solving the extended Fisher-Kolmogorov (EFK) equation. We first decompose the EFK equation as the two second-order equations, then deal with a second-order equation employing finite element method, and handle the other second-order equation using a new mixed finite element method. In the new mixed finite element method, the gradient u belongs to the weaker (L 2) 2 space taking the place of the classical H (div; Ω) space. We prove some a priori bounds for the solution for semidiscrete scheme and derive a fully discrete mixed scheme based on a linearized Crank-Nicolson method. At the same time, we get the optimal a priori error estimates in L 2 and H 1 -norm for both the scalar unknown u and the diffusion term w = - Δ u and a priori error estimates in (L 2) 2 -norm for its gradient χ = u for both semi-discrete and fully discrete schemes. © 2013 Jinfeng Wang et al.