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

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 H¹-norm error estimates without any ratio restrictions between the time step τ and the space mesh size h. The key point of our approach is the boundedness of the numerical solution in the H¹-norm rather than in the L^∞-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 H¹-norm. And then the unconditionally optimal H¹ error estimate is also established. Finally, some numerical examples are given to demonstrate the validity of our methods.