Abstract: The purpose of this paper is to study the mixed finite element methods with a type of lower order nonconforming finite elements for the extended Fisher-Kolmogorov(EFK) equation. Firstly, a nonconforming mixed finite element scheme is established by splitting the EFK equation into two second order equations through a intermediate variable v=-△u. Some a priori bounds are derived by use of Lyapunov functional, existence and uniqueness for the approximation solutions are also proved. The optimal error estimates for both the primitive solution u and the intermediate variable v in H¹-norm are deduced for semi-discrete scheme by use of the above priori bounds and properties of the elements. Furthermore, the superclose properties with order O(h²) are obtained through high accuracy results of the elements. Secondly, a new linearized backward Euler full-discrete scheme is established. The optimal error estimates and superclose results for u and v in H¹-norm with orders O(h+τ) and O(h² +τ) are obtained respectively through the new splitting techniques for consistency errors and nonlinear terms. Here, h, τ are parameter of the subdivision in space and time step. Finally, numerical results are provided to confirm the theoretical analysis. Our analysis provides a new understanding with nonconforming mixed finite element methods to analyze other fourth order initial boundary value problems.

Key words: EFK equation, Nonconforming mixed finite element methods, Semi-discrete and linearized backward Euler full-discrete schemes, Superclose