Weak Galerkin finite element methods for semilinear Klein–Gordon equation on polygonal meshes

The article presents the development of the weak Galerkin finite element method (WG-FEM) for semilinear hyperbolic problems. Semidiscrete error estimate in \(L^2\) $L^2$ -norm as well as \(H^1\) $H^1$ -norm have been executed for the weak Galerkin space \(({\textbf{P}}_k ({\mathcal {K}}), {\textbf{P}}_{k} (\partial {\mathcal {K}}), [{\textbf{P}}_{k-1} ({\mathcal {K}})]^2),\) $({\mathbf{P}}_k(\mathcal{K}),{\mathbf{P}}_k(\partial \mathcal{K}),{[{\mathbf{P}}_{k-1}(\mathcal{K})]}^2),$ where \(k \ge 1\) $k\ge 1$ is an integer. For a fully discrete scheme, we employ the Newmark scheme for temporal discretization. Finally, a few numerical results are provided to validate theoretical results.