Some new error estimates for finite element methods for second order hyperbolic equations using the Newmark method

Abstract

We consider a family of conforming finite element schemes with piecewise polynomial space of degree $k$ in space for solving the wave equation, as a model for second order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is $h^{k}+\tau ^{2}$ in the discrete norms of $\mathcal {L}^{\infty }(0,T;\mathcal {H}^1(\Omega ))$ and $\mathcal {W}^{1,\infty }(0,T;\mathcal {L}^2(\Omega ))$, where $h$ and $\tau $ are the mesh size of the spatial and temporal discretization, respectively. These error estimates are useful since they allow us to get second order time accurate approximations for not only the exact solution of the wave equation but also for its first derivatives (both spatial and temporal). Even though the proof presented in this note is in some sense standard, the stated error estimates seem not to be present in the existing literature on the finite element methods which use the Newmark method for the wave equation (or general second order hyperbolic equations).