Time and space adaptivity of the wave equation discretized in time by a second order scheme

The aim of this paper is to obtain a posteriori error bounds of optimal order in time and space for the linear second-order wave equation discretized by the Newmark scheme in time and the finite element method in space. Error estimates are derived in the $L^{\mathrm{\infty }}$-in-time/energy-in-space norm. Numerical experiments are reported for several test cases and confirm equivalence of the proposed estimators and the true error.