Existence, Uniqueness, and Error Analysis of a Mixed Finite Element Approximation for the Extended Fisher–Kolmogorov Equation

Abstract

In this article, we present a numerical analysis of the extended Fisher–Kolmogorov equation and use a mixed finite element method for its approximation. Our analysis begins with establishing the existence and uniqueness of the weak solution based on the Faedo–Galerkin approach. Next, for the fully discrete scheme, a Crank–Nicolson time discretization is employed. The stability estimate is then derived using an appropriate Lyapunov functional, and subsequently the existence of the fully discrete solution is proved using Brouwer's fixed‐point theorem. The error analysis demonstrates second‐order convergence for both the solution variable and . Our study also investigates the influence of the fourth‐order term in the extended Fisher–Kolmogorov equation, highlighting its stabilizing effects and its role in the dynamics of front propagation. Finally, numerical experiments are presented to confirm the theoretical results and to illustrate the effectiveness and reliability of the proposed scheme.