Adaptive FEM with explicit time integration for the wave equation
Marcus J. Grote, Omar Lakkis, Carina S. Santos
TL;DR
This work develops a space-time adaptive finite element framework for the wave equation that combines computable a posteriori error estimates with compatible mesh changes and leapfrog-based local time stepping to overcome CFL constraints in explicit time integration. The method adaptively refines and coarsens the mesh while tracking error contributions from space, time, and mesh changes, achieving optimal convergence rates and substantial reductions in memory and computational cost. Numerical experiments in 1D and 2D demonstrate robust performance across forced, traveling, splitting waves, and complex geometries, with indications of dimension-independent applicability. The approach offers a scalable, explicit-time-stepping pathway for accurate wave simulations on evolving space-time meshes.
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.