An efficient numerical simulation of the two-dimensional semilinear wave equation

A fully discrete finite element approximations of the solution for a semilinear wave equation is considered and analyzed in this paper. The optimal H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document} error estimates for r-th order FEMs (r=1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(r=1, 2)$$\end{document} are derived without any restriction on the time step size. Numerical examples are given to support our theoretical results and demonstrate the efficiency of the methods.