Asymptotically constant-free and polynomial-degree-robust a posteriori estimates for space discretizations of the wave equation

Abstract: We derive an equilibrated a posteriori error estimator for the space (semi) discretization of the scalar wave equation by finite elements. In the idealized setting where time discretization is ignored and the simulation time is large, we provide fully-guaranteed upper bounds that are asymptotically constant-free and show that the proposed estimator is efficient and polynomial-degree-robust, meaning that the efficiency constant does not deteriorate as the approximation order is increased. To the best of our knowledge, this work is the first to derive provably efficient error estimates for the wave equation. We also explain, without analysis, how the estimator is adapted to cover time discretization by an explicit time integration scheme. Numerical examples illustrate the theory and suggest that it is sharp.