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 (L2(Ω))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 L2 and H1-norm for both the scalar unknown u and the diffusion term w=−Δu and a priori error estimates in (L2)2-norm for its gradient χ=∇u for both semi-discrete and fully discrete schemes.