The two-grid hybrid high-order method for the nonlinear strongly damped wave equation on polygonal mesh and its reduced-order model

This paper introduces the hybrid high-order (HHO) method for solving the nonlinear strongly damped wave equation. We comprehensively analyze the semi-discrete and fully-discrete implicit schemes, including energy and $L^2$ norm, with convergence rates of $m+1$ and $m+2$ in space ($m\ge 0$), respectively. In addition, we combine the two-grid algorithm (TGA) with the HHO method (TGA-HHO) to improve computational efficiency and analyze the TGA-HHO method. To improve the computational efficiency further, we combine the proper orthogonal decomposition (POD) technique with the TGA-HHO method (POD-TGA-HHO). Finally, we provide numerical examples to validate the effectiveness of the HHO, TGA-HHO, and POD-TGA-HHO algorithms.