Explicit Time Stepping for the Wave Equation using CutFEM with Discrete Extension

TL;DR

This paper introduces an explicit cut finite element method for the wave equation that employs a leap frog scheme and an extension operator, enabling fully explicit time stepping with proven stability and optimal error estimates.

Contribution

The paper develops a novel fully explicit cut finite element method for the wave equation using an extension operator and mass matrix lumping, with stability and error analysis.

Findings

The method is stable under certain conditions.

Optimal order a priori error estimates are derived.

Numerical examples demonstrate the method's effectiveness.

Abstract

In this note we develop a fully explicit cut finite element method for the wave equation. The method is based on using a standard leap frog scheme combined with an extension operator that defines the nodal values outside of the domain in terms of the nodal values inside the domain. We show that the mass matrix associated with the extended finite element space can be lumped leading to a fully explicit scheme. We derive stability estimates for the method and provide optimal order a priori error estimates. Finally, we present some illustrating numerical examples.