A systematic study on weak Galerkin finite-element method for second-order wave equation

In this article, we present a systematic numerical study for second-order linear wave equation using weak Galerkin finite-element methods (WG-FEMs). Various degrees of polynomials are used to construct weak Galerkin finite-element spaces. Error estimates in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} norm as well as in discrete 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} norm have been established for general weak Galerkin space (Pk(K),Pj(∂K),[Pl(K)]2),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\textbf {P}}}_k ({\mathcal {K}}), {{\textbf {P}}}_j (\partial {\mathcal {K}}), [{{\textbf {P}}}_l ({\mathcal {K}})]^2),$$\end{document} where k,j&l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k, j \& l$$\end{document} are non-negative integers with k≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 1$$\end{document}. Time discretization for fully discrete scheme is based on second order in time Newmark scheme. Finally, we provide several numerical results to confirm theoretical findings.