Unconditionally optimal error estimate of the Crank–Nicolson extrapolation Galerkin finite element method for Kuramoto–Tsuzuki equation

In this paper, a linearized Crank–Nicolson extrapolation Galerkin finite element is investigated for two-dimensional Kuramoto–Tsuzuki equation and the unconditionally optimal error estimate is obtained without any certain time-step restrictions dependent on the spatial mesh size. The key to the analysis is to derive the boundness of the error between finite element approximation and Ritz projection of the exact solution in energy norm in terms of mathematical induction for two cases, i.e., τ≥h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \ge h$$\end{document} and τ≤h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \le h$$\end{document}, where τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} denotes the time-step size and h is the spatial mesh size. Finally, numerical results are provided to confirm the theoretical findings.