Error analysis of the fully Galerkin approximations for the nonlinear extended-Fisher–Kolmogorov equation

In this article, we present a fully discrete Crank–Nicolson Galerkin finite element method for solving the two-dimensional nonlinear extended-Fisher–Kolmogorov equation: \(u_t + \gamma \Delta ^2 u -\Delta u -u +u^{3} = 0.\) $u_t+\gamma {\mathrm{\Delta }}^{2}u-\mathrm{\Delta }u-u+u^3=0.$ The boundedness of the numerical solution in the maximum norm, unique solvability, and related convergence results in \(L^2\) $L^2$ and \(L^{\infty }\) $L^{\infty }$ -norms are studied in detail. Also, a new linearized Crank–Nicolson Galerkin modification scheme is designed and error estimate without any time step restrictions is established. Finally, some computational experiments in one and two dimension cases are provided to illustrate the efficacy of our method and to confirm the theoretical results.