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

TL;DR

This paper develops a fully computable a posteriori error estimator for the wave equation solved with the leapfrog method, accommodating adaptive mesh changes and local time-stepping while maintaining explicit time integration efficiency.

Contribution

It introduces a novel a posteriori error estimator that handles mesh changes and local time-stepping in the leapfrog method for the wave equation, enhancing adaptive computational techniques.

Findings

Error estimator achieves optimal convergence rates

Numerical results validate estimator accuracy

Method supports adaptive mesh and time-stepping

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.