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

We derive a fully computable a posteriori error estimator for a Galerkin finite element solution of the wave equation with explicit leapfrog time-stepping. Our discrete formulation accomodates 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 incorporate adaptivity into fully explicit time integration with adaptive mesh change while retaining 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 a posteriori error estimator.