Convergence analysis of an energy-stable linearized virtual element method for the strongly damped Klein-Gordon equation

Abstract

In this paper, we propose and analyze an efficient, linearized, fully discrete scheme for the nonlinear, strongly damped Klein-Gordon equation on polygonal meshes. The numerical scheme uses a conforming virtual element method for spatial discretization and a modified leapfrog (central finite difference) scheme for time discretization, with the nonlinear term $|u{|}^{p-1}u$ is treated semi-implicitly. We first prove that the proposed scheme is energy dissipative in the sense of discrete energy, and then the stability of the numerical solution in the $H^1$-norm is established using mathematical induction, which plays an important role in handling the nonlinear term. By applying the boundedness of the numerical solution and the Sobolev embedding inequality, we derive the optimal $H^1$ error estimate of order $O(h^k+{\tau }^2)$ without imposing any ratio restrictions between the time step τ and the mesh size h. Additionally, we remark that the leapfrog virtual element scheme can be applied to some more complex nonlinear damped wave equations. Finally, some numerical examples are provided to confirm the theoretical results.