# Convergence analysis of energy conserving explicit local time-stepping methods for the wave equation

## TL;DR

This paper provides a rigorous convergence analysis of an explicit local time-stepping method based on leap-frog for the wave equation, enabling efficient simulations on locally refined meshes with complex geometries.

## Contribution

It offers the first rigorous proof of convergence for the fully-discrete LTS-LF method combined with FEM, demonstrating its effectiveness in handling corner singularities.

## Key findings

- Proves convergence of the LTS-LF Galerkin FEM method.
- Shows effectiveness in complex geometries with corner singularities.
- Enables efficient wave simulations on locally refined meshes.

## Abstract

Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in heterogeneous media or complex geometry. Locally refined meshes, however, dictate a small time-step everywhere with a crippling effect on any explicit time-marching method. In [18] a leap-frog (LF) based explicit local time-stepping (LTS) method was proposed, which overcomes the severe bottleneck due to a few small elements by taking small time-steps in the locally refined region and larger steps elsewhere. Here a rigorous convergence proof is presented for the fully-discrete LTS-LF method when combined with a standard conforming finite element method (FEM) in space. Numerical results further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of corner singularities.