Towards Stable Second-Kind Boundary Integral Equations for Transient Wave Problems

TL;DR

This work delivers the first standalone, stable second-kind boundary integral formulation for the 1D transient wave equation and provides rigorous $L^2$-ellipticity and inf-sup stability results for the second-kind operator $-rac{1}{2} ext{Id} + rson{ ext{K}}$. By exploiting the 1D mapping properties of the double-layer operator, it proves isomorphisms in energy spaces $H^{1}_{;0,}( ext{Σ})$ and $H^{1/2}_{0,}( ext{Σ})$, enabling unique solvability. It introduces two space-time Galerkin discretisations (piecewise constants and piecewise linear with zero initial data) and establishes discrete inf-sup stability and error estimates, all supported by comprehensive numerical experiments. The results pave the way for robust, scalable space-time BEM for transient waves and provide a template for extending second-kind formulations to higher dimensions.

Abstract

In this paper, we discuss the stable discretisation of the double layer boundary integral operator for the wave equation in $1d$. For this, we show that the boundary integral formulation is $L^2$-elliptic and also inf-sup stable in standard energy spaces. This turns out to be a particular case of a recent result on the inf-sup stability of boundary integral operators for the wave equation and contributes to its further understanding. Moreover, we present the first BEM discretisations of second-kind operators for the wave equation for which stability is guaranteed and a complete numerical analysis is offered. We validate our theoretical findings with numerical experiments.

Figures (4)

• Figure 1: Example of non-uniform mesh satisfying Assumption \ref{['Assumption:mesh']}. On each time-slice $T_j$, the degrees of freedom (DoFs) of each boundary agree with the DoFs on the opposite boundary shifted in time by $L$.
• Figure 2: Initial non-uniform mesh which does not satisfy assumption $T=nL$ for some $n\in\mathbb{N}$.
• Figure 3: Discrete inf sup constant for different mesh refinements and two types of meshes: the uniform case with $T=6$; the non-uniform case with $T=2\pi$ and initial mesh given by Figure \ref{['fig:initial_mesh']}.
• Figure 4: Discrete inf sup constant for different final times: the uniform case with $T=nL$ and $h_0=h_L=3/16$; the non-uniform case with $T=n \pi$ and $h_0 = \pi/20$ and $h_L = \pi/12$. Here $n$ is the number of timeslices (and thus refinement independent).

Theorems & Definitions (21)

• Lemma 2.1: StU22
• Remark 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Definition 4.1
• Remark 4.2