The weak Galerkin finite element method with or without stabilizer for the damped sine-Gordon equation and its reduced-order simulation

In this work, an efficient scheme is proposed for the 2D damped sine-Gordon equation. The new scheme utilizes the unified weak Galerkin finite element method with or without stabilizer (UWG) in the spatial direction and the second-order explicit differentiation formula in the temporal direction. Both semi discrete schemes and fully discrete schemes are analyzed. First, we demonstrate the stability of the schemes. Subsequently, optimal error estimates are derived in the energy norm and the $L^2$ norm. Furthermore, by combining the proper orthogonal decomposition (POD) technique with the WG method, we develop a reduced-order algorithm, significantly improving computational efficiency. Finally, the theoretical results are validated by numerical experiments.