The Crank-Nicolson weak Galerkin finite element methods for the sine-Gordon equation

This article proposes an efficient second-order weak Galerkin (WG) finite element scheme for solving the 2D damped and undamped sine-Gordon problem with Dirichlet boundary conditions and initial conditions. We also construct and study a fully discrete WG finite element method for solving the sine-Gordon equation with a damping term using the Crank–Nicolson (CN) and Euler schemes. Stability and error analyses are established on a triangular grid for the constructed schemes in L ² and H ¹ norms for the fully discrete and semi-discrete formulation. Our formulation is accurate in space and time. Finally, numerical experiments are performed to validate the theoretical conclusions.