Unconditionally optimal error estimates of linearized virtual element methods for a class of nonlinear wave equations

Abstract

In this paper, we analyze the unconditionally optimal error estimates of the linearized virtual element schemes for a class of nonlinear wave equations. For the general nonlinear term with non-global Lipschitz continuity, we consider a modified Crank–Nicolson scheme for the time discretization and a conforming virtual element method for the spatial discretization. Using the mathematical induction and the Sobolev embedding inequality, we derive the optimal $H^1$-norm error estimates without any ratio restrictions between the time step $\tau$ and the space mesh size $h$. The key point of our approach is the boundedness of the numerical solution in the $H^1$-norm rather than in the $L^{\infty }$-norm. For the cubic nonlinear term, we develop another linearized scheme using a modified leapfrog scheme in the time direction. We show that this scheme can maintain the energy stability, which directly ensures the boundedness of the numerical solution in the $H^1$-norm. And then the unconditionally optimal $H^1$ error estimate is also established. Finally, some numerical examples are given to demonstrate the validity of our methods.