A posteriori error estimates for the wave equation with mesh change in the leapfrog method

Abstract

We derive a fully computable aposteriori error estimator for a Galerkin finite element solution of the wave equation with explicit leapfrog time-stepping. Our discrete formulation accommodates both time evolving meshes and leapfrog based local time-stepping (Diaz & Grote, 2009), which overcomes the stringent stability restriction on the time-step due to local mesh refinement. Thus we account for adaptive time-stepping with mesh change in a fully explicit time integration while retaining its efficiency. The error analysis relies on elliptic reconstructors and abstract grid transfer operators, which allows for use-defined elliptic error estimators. Numerical results using the elliptic Babu\v{s}ka-Rheinboldt estimators illustrate the optimal rate of convergence with mesh size of the aposteriori error estimator.