Adaptive FEM with explicit time integration for the wave equation

TL;DR

This paper introduces an adaptive finite element method with explicit time integration for the wave equation, combining local mesh refinement, coarsening, and local time-stepping to improve efficiency and accuracy.

Contribution

It develops a space-time adaptive strategy incorporating local mesh refinement and local time-stepping for explicit wave equation solutions, overcoming CFL restrictions.

Findings

Optimal convergence rates demonstrated

Effective local mesh refinement and coarsening

Stable and efficient explicit time integration

Abstract

Starting from a recent a posteriori error estimator for the finite element solution of the wave equation with explicit time-stepping [Grote, Lakkis, Santos, 2024], we devise a space-time adaptive strategy which includes both time evolving meshes and local time-stepping [Diaz, Grote, 2009] to overcome any overly stringent CFL stability restriction on the time-step due to local mesh refinement. Moreover, at each time-step the adaptive algorithm monitors the accuracy thanks to the error indicators and recomputes the current step on a refined mesh until the desired tolerance is met; meanwhile, the mesh is coarsened in regions of smaller errors. Leapfrog based local time-stepping is applied in all regions of local mesh refinement to incorporate adaptivity into fully explicit time integration with mesh change while retaining efficiency. Numerical results illustrate the optimal rate of convergence of the a posteriori error estimators on time evolving meshes.