Towards coercive boundary element methods for the wave equation

TL;DR

This paper investigates the ellipticity of boundary integral operators for the wave equation in one dimension, introducing a modified Hilbert transformation to improve stability analysis and confirm error estimates through numerical validation.

Contribution

It generalizes ellipticity results for boundary integral operators using a modified Hilbert transformation, enabling standard error estimates for wave equation boundary element methods.

Findings

Ellipticity of the single layer operator is established in Sobolev spaces.

The modified Hilbert transformation facilitates stability analysis.

Numerical results confirm the theoretical error estimates.

Abstract

In this note, we discuss the ellipticity of the single layer boundary integral operator for the wave equation in one space dimension. This result not only generalizes the well-known ellipticity of the energetic boundary integral formulation in $L^2$, but it also turns out to be a particular case of a recent result on the inf-sup stability of boundary integral operators for the wave equation. Instead of the time derivative in the energetic formulation, we use a modified Hilbert transformation, which allows us to stay in Sobolev spaces of the same order. This results in the applicability of standard boundary element error estimates, which are confirmed by numerical results.