# Towards coercive boundary element methods
for the wave equation

Olaf Steinbach1, Carolina Urzúa–Torres2, Marco Zank3

(1Institut für Angewandte Mathematik, TU Graz,
Steyrergasse 30, 8010 Graz, Austria
<EMAIL_ADDRESS>
2Delft Institute of Applied Mathematics, TU Delft,
Mekelweg 4, 2628CD Delft, The Netherlands
<EMAIL_ADDRESS>
3Fakultät für Mathematik, Universität Wien,
Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
<EMAIL_ADDRESS>
###### 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.

##  1 Introduction

Time-domain boundary integral equations and boundary element methods for the
wave equation are well established in the literature; we mention the
groundbraking works of Bamberger and Ha Duong [2], Aimi et al. [1], and the
review article [4] by Costabel and Sayas. Other works include [5, 7, 8, 9, 10,
11], to mention a few.

The main difficulties in the numerical analysis of these formulations are in
the so-called norm gap, coming from continuity and coercivity estimates in
different space-time Sobolev norms. When using the energetic boundary element
method, a complete stability and error analysis can be done in
$L^{2}(\Sigma)$, see [8], where $\Sigma$ is the lateral boundary of the space-
time domain $Q:=\Omega\times(0,T)$.

Using a generalized inf-sup stable variational formulation [17] for the wave
equation, in [13] we derived inf-sup stability conditions for all boundary
integral operators in related trace spaces. In fact, this work was motivated
by our previous result [18] on the spatially one-dimensional case. When
replacing the time derivative in the energetic boundary integral formulation
by a modified Hilbert transformation [15], the resulting composition with the
single layer boundary integral operator becomes elliptic in the natural energy
space $[H^{1/2}_{,0}(\Sigma)]^{\prime}$, similarly to what is known for
boundary integral operators for second-order elliptic partial differential
equations. Note that
$H^{1/2}_{,0}(\Sigma):=[H^{1}_{,0}(\Sigma),L^{2}(\Sigma)]_{1/2}$ is defined by
interpolation, with $H^{1}_{,0}(\Sigma)=\\{v\in H^{1}(\Sigma):v(T)=0\\}$.
Analogously, $H^{1}_{0,}(\Sigma)$ covers zero initial conditions, i.e.,
$v(0)=0$.

In this paper, we present a detailed derivation of this new approach, and we
discuss the corresponding numerical analysis of a related new boundary element
method. In Section 2, we recall the energetic space-time boundary integral
formulation [1, 8], and we provide a simplified proof of the ellipticity
result in $L^{2}(\Sigma)$. In particular, we obtain that the single layer
boundary integral operator $V:L^{2}(\Sigma)\to H^{1}_{0,}(\Sigma)$ is an
isomorphism. Using duality arguments, we obtain that
$V:[H^{1}_{,0}(\Sigma)]^{\prime}\to L^{2}(\Sigma)$ is also an isomorphism.
Finally, by an interpolation argument, we conclude that
$V:[H^{1/2}_{,0}(\Sigma)]^{\prime}\to H^{1/2}_{0,}(\Sigma)$ is an isomorphism
as well. While this implies an inf-sup stability estimate, as also discussed
in [13], in Section 3 we introduce a modified Hilbert transformation
${\mathcal{H}}_{T}:H^{1/2}_{0,}(\Sigma)\to H^{1/2}_{,0}(\Sigma)$, see [15], to
establish ellipticity of ${\mathcal{H}}_{T}V$ in
$[H^{1/2}_{,0}(\Sigma)]^{\prime}$ in Section 4. Although the main result, as
given in Lemma 4.1, still involves some unknown constant, Proposition 4.2
gives numerical evidence on the behavior of the ellipticity constant, which
agrees with the constant known from the energetic formulation. In Section 5,
we present some numerical results which confirm the a priori error estimates,
as given in Section 4. In Section 6, we finally draw some conclusions for
future work.

##  2 Energetic space-time boundary integral equation

As in [1], we consider the Dirichlet boundary value problem for the
homogeneous wave equation in the one-dimensional spatial domain $\Omega=(0,L)$
with zero initial conditions, and for a given time horizon $T>0$,

$\left.\begin{array}[]{rcll}\partial_{tt}u(x,t)-\partial_{xx}u(x,t)&=&0&\mbox{for}\;(x,t)\in
Q:=(0,L)\times(0,T),\\\\[2.84526pt]
u(x,0)=\partial_{t}u(x,t)_{|t=0}&=&0&\mbox{for}\;x\in(0,L),\\\\[2.84526pt]
u(0,t)&=&g_{0}(t)&\mbox{for}\;t\in(0,T),\\\\[2.84526pt]
u(L,t)&=&g_{L}(t)&\mbox{for}\;t\in(0,T).\end{array}\right\\}$ (2.1)

In the one-dimensional case, the fundamental solution of the wave equation is
the Heaviside function

$U^{*}(x,t)=\frac{1}{2}\,H(t-|x|),$

and we can represent the solution $u$ of (2.1) by using the single layer
potential

$u(x,t)=(\widetilde{V}w)(x,t)=\frac{1}{2}\int_{0}^{t-|x|}w_{0}(s)\,ds+\frac{1}{2}\int_{0}^{t-|x-L|}w_{L}(s)\,ds\quad\mbox{for}\;(x,t)\in
Q$

with the density functions $w=(w_{0},w_{L})$. Note that for any function
$z\colon\,(0,T)\to\mathbb{R}$, we set $z(t)=0$ for $t<0$ or $t>T$ in the
remainder of this work. To determine the yet unknown density functions
$(w_{0},w_{L})$, we consider the boundary integral equations for $x\to 0$,

$(V_{0}w)(t):=\frac{1}{2}\int_{0}^{t}w_{0}(s)\,ds+\frac{1}{2}\int_{0}^{t-L}w_{L}(s)\,ds=g_{0}(t)\quad\mbox{for}\;t\in(0,T),$
(2.2)

and for $x\to L$,

$(V_{L}w)(t):=\frac{1}{2}\int_{0}^{t-L}w_{0}(s)\,ds+\frac{1}{2}\int_{0}^{t}w_{L}(s)\,ds=g_{L}(t)\quad\mbox{for}\;t\in(0,T).$
(2.3)

We write the boundary integral equations (2.2) and (2.3) in compact form, for
$w=(w_{0},w_{L})$, as

$(Vw)(t)=\begin{pmatrix}(V_{0}w)(t)\\\
(V_{L}w)(t)\end{pmatrix}=\begin{pmatrix}V_{00}&V_{0L}\\\
V_{L0}&V_{LL}\end{pmatrix}\begin{pmatrix}w_{0}\\\
w_{L}\end{pmatrix}(t)=\begin{pmatrix}g_{0}(t)\\\
g_{L}(t)\end{pmatrix}=g(t),\quad t\in(0,T).$ (2.4)

In the energetic boundary element method [1], instead of (2.4), the time
derivative of (2.4) is considered,

$\partial_{t}(Vw)(t)=\partial_{t}g(t)\quad\mbox{for}\;t\in(0,T).$ (2.5)

We introduce the related energetic bilinear form

$\displaystyle a(w,v)$ $\displaystyle:=$ $\displaystyle\langle
v,\partial_{t}Vw\rangle_{L^{2}(\Sigma)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{T}v_{0}(t)\frac{d}{dt}\int_{0}^{t}w_{0}(s)\,ds\,dt+\frac{1}{2}\int_{0}^{T}v_{0}(t)\frac{d}{dt}\int_{0}^{t-L}w_{L}(s)\,ds\,dt$
$\displaystyle+\frac{1}{2}\int_{0}^{T}v_{L}(t)\frac{d}{dt}\int_{0}^{t-L}w_{0}(s)\,ds\,dt+\frac{1}{2}\int_{0}^{T}v_{L}(t)\frac{d}{dt}\int_{0}^{t}w_{L}(s)\,ds\,dt$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,\int_{0}^{T}v_{0}(t)\,w_{0}(t)\,dt+\frac{1}{2}\,\int_{0}^{T}v_{0}(t)\,w_{L}(t-L)\,dt$
$\displaystyle+\frac{1}{2}\,\int_{0}^{T}v_{L}(t)\,w_{0}(t-L)\,dt+\frac{1}{2}\,\int_{0}^{T}v_{L}(t)\,w_{L}(t)\,dt\,.$

When using both the Cauchy–Schwarz and Hölder inequality, we conclude

$\displaystyle|a(w,v)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\,\|v_{0}\|_{L^{2}(0,T)}\|w_{0}\|_{L^{2}(0,T)}+\frac{1}{2}\,\|v_{0}\|_{L^{2}(0,T)}\|w_{L}\|_{L^{2}(0,T-L)}$
$\displaystyle+\frac{1}{2}\,\|v_{L}\|_{L^{2}(0,T)}\|w_{0}\|_{L^{2}(0,T-L)}+\frac{1}{2}\,\|v_{L}\|_{L^{2}(0,T)}\|w_{L}\|_{L^{2}(0,T)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\,\|v_{0}\|_{L^{2}(0,T)}\Big{[}\|w_{0}\|_{L^{2}(0,T)}+\|w_{L}\|_{L^{2}(0,T)}\Big{]}$
$\displaystyle+\frac{1}{2}\,\|v_{L}\|_{L^{2}(0,T)}\Big{[}\|w_{0}\|_{L^{2}(0,T)}+\|w_{L}\|_{L^{2}(0,T)}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,\Big{[}\|v_{0}\|_{L^{2}(0,T)}+\|v_{L}\|_{L^{2}(0,T)}\Big{]}\Big{[}\|w_{0}\|_{L^{2}(0,T)}+\|w_{L}\|_{L^{2}(0,T)}\Big{]}$
$\displaystyle\leq$
$\displaystyle\sqrt{\|v_{0}\|_{L^{2}(0,T)}^{2}+\|v_{L}\|_{L^{2}(0,T)}^{2}}\sqrt{\|w_{0}\|^{2}_{L^{2}(0,T)}+\|w_{L}\|^{2}_{L^{2}(0,T)}}$
$\displaystyle=$ $\displaystyle\|v\|_{L^{2}(\Sigma)}\|w\|_{L^{2}(\Sigma)}$

for all $v=(v_{0},v_{L}),w=(w_{0},w_{L})\in L^{2}(\Sigma):=L^{2}(0,T)\times
L^{2}(0,T)$, where

$\|z\|_{L^{2}(\Sigma)}:=\left(\|z_{0}\|_{L^{2}(0,T)}^{2}+\|z_{L}\|_{L^{2}(0,T)}^{2}\right)^{1/2}\quad\text{
for }z=(z_{0},z_{L})\in L^{2}(\Sigma).$

Moreover, the energetic bilinear form $a(\cdot,\cdot)$ is also
$L^{2}(\Sigma)$-elliptic, see [1, Theorem 2.1]. For later reference, we will
give a simplified proof of this result. For this, we introduce

$n:=\min\Big{\\{}m\in{\mathbb{N}}:T\leq mL\Big{\\}},$ (2.6)

which is the number of time slices $T_{j}:=((j-1)L,jL)$ for $j=1,\ldots,n$ in
the case $T=nL$. In the case $T<nL$, we define the last time slice as
$T_{n}:=((n-1)L,T)$, while all the others remain unchanged.

###### Theorem 2.1

[1, Theorem 2.1] For all $w\in L^{2}(\Sigma)$, there holds the ellipticity
estimate

$a(w,w)=\langle
w,\partial_{t}Vw\rangle_{L^{2}(\Sigma)}\,\geq\,\sin^{2}\frac{\pi}{2(n+1)}\,\|w\|_{L^{2}(\Sigma)}^{2},$
(2.7)

where the number $n\in{\mathbb{N}}$ of time slices is defined in (2.6).

Proof. For $w=(w_{0},w_{L})\in L^{2}(\Sigma)$, we write

$\displaystyle 2\,a($ $\displaystyle w,w)$
$\displaystyle=\int_{0}^{T}[w_{0}(t)]^{2}\,dt+\int_{0}^{T}w_{0}(t)\,w_{L}(t-L)\,dt+\int_{0}^{T}w_{L}(t)\,w_{0}(t-L)\,dt+\int_{0}^{T}[w_{L}(t)]^{2}\,dt$
$\displaystyle=\sum\limits_{j=1}^{n}\left[\|w_{0}\|_{L^{2}(T_{j})}^{2}+\int_{T_{j}}w_{0}(t)w_{L}(t-L)\,dt+\int_{T_{j}}w_{L}(t)w_{0}(t-L)\,dt+\|w_{L}\|_{L^{2}(T_{j})}^{2}\right].$

For $t\in T_{1}$, we have $t-L<0$, and therefore $w_{0}(t-L)=w_{L}(t-L)=0$
follows. For $j=2,\ldots,n-1$, we have, using the Cauchy–Schwarz inequality,

$\int_{T_{j}}w_{0}(t)w_{L}(t-L)\,dt\leq\left(\int_{T_{j}}[w_{0}(t)]^{2}\,dt\right)^{1/2}\left(\int_{T_{j}}[w_{L}(t-L)]^{2}\,dt\right)^{1/2}\\\
\leq\left(\int_{T_{j}}[w_{0}(t)]^{2}\,dt\right)^{1/2}\left(\int_{T_{j-1}}[w_{L}(t)]^{2}\,dt\right)^{1/2}=\|w_{0}\|_{L^{2}(T_{j})}\|w_{L}\|_{L^{2}(T_{j-1})}\,.$

Correspondingly, for $j=n$ and $T_{n}=((j-1)L,T)$, $T\leq nL$, we have

$\displaystyle\int_{T_{n}}w_{0}(t)$ $\displaystyle
w_{L}(t-L)\,dt\leq\left(\int_{T_{n}}[w_{0}(t)]^{2}\,dt\right)^{1/2}\left(\int_{(n-1)L}^{T}[w_{L}(t-L)]^{2}\,dt\right)^{1/2}$
$\displaystyle=\left(\int_{T_{n}}[w_{0}(t)]^{2}\,dt\right)^{1/2}\left(\int_{(n-2)L}^{T-L}[w_{L}(t)]^{2}\,dt\right)^{1/2}$
$\displaystyle\leq\left(\int_{T_{n}}[w_{0}(t)]^{2}\,dt\right)^{1/2}\left(\int_{T_{n-1}}[w_{L}(t)]^{2}\,dt\right)^{1/2}=\|w_{0}\|_{L^{2}(T_{n})}\|w_{L}\|_{L^{2}(T_{n-1})}\,.$

Hence, we conclude

$\displaystyle 2\,a(w,w)\geq$
$\displaystyle\sum\limits_{j=1}^{n}\Big{[}\|w_{0}\|^{2}_{L^{2}(T_{j})}+\|w_{L}\|^{2}_{L^{2}(T_{j})}\Big{]}$
$\displaystyle-\sum\limits_{j=2}^{n}\Big{[}\|w_{0}\|_{L^{2}(T_{j})}\|w_{L}\|_{L^{2}(T_{j-1})}+\|w_{L}\|_{L^{2}(T_{j})}\|w_{0}\|_{L^{2}(T_{j-1})}\Big{]}$
$\displaystyle=$
$\displaystyle{\scriptsize\left(\begin{pmatrix}1&-\frac{1}{2}&&&&&\\\
-\frac{1}{2}&1&-\frac{1}{2}&&&&\\\ &-\frac{1}{2}&1&-\frac{1}{2}&&&\\\
&&-\frac{1}{2}&1&-\frac{1}{2}&&\\\ &&&\ddots&\ddots&\ddots&\\\
&&&&-\frac{1}{2}&1&-\frac{1}{2}\\\
&&&&&-\frac{1}{2}&1\end{pmatrix}\begin{pmatrix}\|w_{0}\|_{L^{2}(T_{1})}\\\
\|w_{L}\|_{L^{2}(T_{2})}\\\ \|w_{0}\|_{L^{2}(T_{3})}\\\
\|w_{L}\|_{L^{2}(T_{4})}\\\ \vdots\\\ \|w_{0}\|_{L^{2}(T_{n-1})}\\\
\|w_{L}\|_{L^{2}(T_{n})}\end{pmatrix},\begin{pmatrix}\|w_{0}\|_{L^{2}(T_{1})}\\\
\|w_{L}\|_{L^{2}(T_{2})}\\\ \|w_{0}\|_{L^{2}(T_{3})}\\\
\|w_{L}\|_{L^{2}(T_{4})}\\\ \vdots\\\ \|w_{0}\|_{L^{2}(T_{n-1})}\\\
\|w_{L}\|_{L^{2}(T_{n})}\end{pmatrix}\right)}$
$\displaystyle+{\scriptsize\left(\begin{pmatrix}1&-\frac{1}{2}&&&&&\\\
-\frac{1}{2}&1&-\frac{1}{2}&&&&\\\ &-\frac{1}{2}&1&-\frac{1}{2}&&&\\\
&&-\frac{1}{2}&1&-\frac{1}{2}&&\\\ &&&\ddots&\ddots&\ddots&\\\
&&&&-\frac{1}{2}&1&-\frac{1}{2}\\\
&&&&&-\frac{1}{2}&1\end{pmatrix}\begin{pmatrix}\|w_{L}\|_{L^{2}(T_{1})}\\\
\|w_{0}\|_{L^{2}(T_{2})}\\\ \|w_{L}\|_{L^{2}(T_{3})}\\\
\|w_{0}\|_{L^{2}(T_{4})}\\\ \vdots\\\ \|w_{L}\|_{L^{2}(T_{n-1})}\\\
\|w_{0}\|_{L^{2}(T_{n})}\end{pmatrix},\begin{pmatrix}\|w_{L}\|_{L^{2}(T_{1})}\\\
\|w_{0}\|_{L^{2}(T_{2})}\\\ \|w_{L}\|_{L^{2}(T_{3})}\\\
\|w_{0}\|_{L^{2}(T_{4})}\\\ \vdots\\\ \|w_{L}\|_{L^{2}(T_{n-1})}\\\
\|w_{0}\|_{L^{2}(T_{n})}\end{pmatrix}\right)}$

and further,

$\displaystyle a(w,w)$
$\displaystyle\geq\frac{\lambda_{\min}}{2}{\scriptsize\left[\left(\begin{pmatrix}\|w_{0}\|_{L^{2}(T_{1})}\\\
\|w_{L}\|_{L^{2}(T_{2})}\\\ \|w_{0}\|_{L^{2}(T_{3})}\\\
\|w_{L}\|_{L^{2}(T_{4})}\\\ \vdots\\\ \|w_{0}\|_{L^{2}(T_{n-1})}\\\
\|w_{L}\|_{L^{2}(T_{n})}\end{pmatrix},\begin{pmatrix}\|w_{0}\|_{L^{2}(T_{1})}\\\
\|w_{L}\|_{L^{2}(T_{2})}\\\ \|w_{0}\|_{L^{2}(T_{3})}\\\
\|w_{L}\|_{L^{2}(T_{4})}\\\ \vdots\\\ \|w_{0}\|_{L^{2}(T_{n-1})}\\\
\|w_{L}\|_{L^{2}(T_{n})}\end{pmatrix}\right)+\left(\begin{pmatrix}\|w_{L}\|_{L^{2}(T_{1})}\\\
\|w_{0}\|_{L^{2}(T_{2})}\\\ \|w_{L}\|_{L^{2}(T_{3})}\\\
\|w_{0}\|_{L^{2}(T_{4})}\\\ \vdots\\\ \|w_{L}\|_{L^{2}(T_{n-1})}\\\
\|w_{0}\|_{L^{2}(T_{n})}\end{pmatrix},\begin{pmatrix}\|w_{L}\|_{L^{2}(T_{1})}\\\
\|w_{0}\|_{L^{2}(T_{2})}\\\ \|w_{L}\|_{L^{2}(T_{3})}\\\
\|w_{0}\|_{L^{2}(T_{4})}\\\ \vdots\\\ \|w_{L}\|_{L^{2}(T_{n-1})}\\\
\|w_{0}\|_{L^{2}(T_{n})}\end{pmatrix}\right)\right]}$
$\displaystyle=\frac{\lambda_{\min}}{2}\,\Big{[}\|w_{0}\|^{2}_{L^{2}(0,T)}+\|w_{L}\|^{2}_{L^{2}(0,T)}\Big{]},$

where

$\lambda_{\min}=2\,\sin^{2}\frac{\pi}{2(n+1)}$

is the minimal eigenvalue of the involved matrix, which is related to the
finite difference approximation of the Laplacian in one dimension.

From the above properties, we conclude that

$\partial_{t}V:L^{2}(\Sigma)\to L^{2}(\Sigma)$

defines an isomorphism. Since the time derivative

$\partial_{t}:H^{1}_{0,}(\Sigma)\to L^{2}(\Sigma)$

is also an isomorphism, e.g., [15, Sect. 2.1], so is

$V:L^{2}(\Sigma)\to H^{1}_{0,}(\Sigma)\,.$ (2.8)

Note that, for $u=(u_{0},u_{L})\in H^{1}_{0,}(\Sigma):=H^{1}_{0,}(0,T)\times
H^{1}_{0,}(0,T)$, we have

$\|u\|^{2}_{H^{1}_{0,}(\Sigma)}:=\|\partial_{t}u_{0}\|^{2}_{L^{2}(0,T)}+\|\partial_{t}u_{L}\|^{2}_{L^{2}(0,T)}.$

For $\partial_{t}:H^{1}_{0,}(0,T)\to L^{2}(0,T)$, the inverse is given by

$u(t)=(\partial_{t}^{-1}f)(t)=\int_{0}^{t}f(s)\,ds,\quad t\in(0,T),$

with $f\in L^{2}(0,T)$, $u\in H^{1}_{0,}(0,T)$. Analogously, for
$\partial_{t}:H^{1}_{,0}(0,T)\to L^{2}(0,T)$, we find the inverse as

$u(t)=(\overline{\partial}_{t}^{-1}f)(t)=-\int_{t}^{T}f(s)\,ds,\quad
t\in(0,T).$

For $w,v\in L^{2}(\Sigma)$ and $u=Vw=(u_{0},u_{L})\in H^{1}_{0,}(\Sigma)$, we
therefore obtain

$\langle\overline{\partial}_{t}^{-1}Vw,v\rangle_{L^{2}(\Sigma)}=-\int_{0}^{T}\int_{t}^{T}u_{0}(s)\,ds\,v_{0}(t)\,dt-\int_{0}^{T}\int_{t}^{T}u_{L}(s)\,ds\,v_{L}(t)\,dt\,.$

For $\ast\in\\{0,L\\}$ we compute

$\displaystyle-\int_{0}^{T}\int_{t}^{T}u_{\ast}(s)\,ds\,v_{\ast}(t)\,dt$
$\displaystyle=-\int_{0}^{T}\int_{t}^{T}u_{\ast}(s)\,ds\,\partial_{t}\int_{0}^{t}v_{\ast}(s)\,ds\,dt$
$\displaystyle=$
$\displaystyle\left.-\int_{t}^{T}u_{\ast}(s)\,ds\int_{0}^{t}v_{\ast}(s)\,ds\right|_{0}^{T}+\int_{0}^{T}\partial_{t}\int_{t}^{T}u_{\ast}(s)\,ds\int_{0}^{t}v_{\ast}(s)\,ds\,dt$
$\displaystyle=$
$\displaystyle-\int_{0}^{T}u_{\ast}(t)\int_{0}^{t}v_{\ast}(s)\,ds\,,$

i.e.,

$\langle\overline{\partial}_{t}^{-1}Vw,v\rangle_{L^{2}(\Sigma)}=-\langle
Vw,\partial_{t}^{-1}v\rangle_{L^{2}(\Sigma)}\,.$

On the other hand, for $z_{0}=\partial_{t}^{-1}w_{0}$ we have
$w_{0}=\partial_{t}z_{0}$, and hence

$\int_{0}^{t}w_{0}(s)\,ds=\int_{0}^{t}\partial_{s}z_{0}(s)\,ds=z_{0}(t)=\partial_{t}\int_{0}^{t}z_{0}(s)\,ds.$

With this, we conclude

$\langle\overline{\partial}_{t}^{-1}Vw,v\rangle_{L^{2}(\Sigma)}=-\langle
V\partial_{t}\partial_{t}^{-1}w,\partial_{t}^{-1}v\rangle_{L^{2}(\Sigma)}=-\langle\partial_{t}V\partial_{t}^{-1}w,\partial_{t}^{-1}v\rangle_{L^{2}(\Sigma)}=-a(\partial_{t}^{-1}w,\partial_{t}^{-1}v),$

and, in particular for $v=w$, Theorem 2.1 gives

$-\langle\overline{\partial}_{t}^{-1}Vw,w\rangle_{L^{2}(\Sigma)}=\langle\partial_{t}V\partial_{t}^{-1}w,\partial_{t}^{-1}w\rangle_{L^{2}(\Sigma)}\geq\sin^{2}\frac{\pi}{2(n+1)}\,\|\partial_{t}^{-1}w\|_{L^{2}(\Sigma)}^{2}.$

For $\ast\in\\{0,L\\}$, we define

$z_{\ast}(t)=(\partial_{t}^{-1}w_{\ast})(t)=\int_{0}^{t}w_{\ast}(s)\,ds,\quad
t\in(0,T),$

to compute

$\displaystyle\|\partial_{t}^{-1}w_{\ast}\|^{2}_{L^{2}(0,T)}$
$\displaystyle=\|z_{\ast}\|^{2}_{L^{2}(0,T)}=\int_{0}^{T}z_{\ast}(t)\,z_{\ast}(t)\,dt=-\int_{0}^{T}\partial_{t}\int_{t}^{T}z_{\ast}(s)\,ds\,z_{*}(t)\,dt$
$\displaystyle=\left.-\int_{t}^{T}z_{\ast}(s)\,ds\,z_{\ast}(t)\right|_{0}^{T}+\int_{0}^{T}\int_{t}^{T}z_{*}(s)\,ds\;\partial_{t}z_{\ast}(t)\,dt$
$\displaystyle=\int_{0}^{T}v_{*}(t)\,w_{*}(t)\,dt,$

where

$v_{*}(t)=\int_{t}^{T}z_{*}(s)\,ds\quad\text{ for
}t\in(0,T),\quad\partial_{t}v_{*}=-z_{*},\quad v_{*}\in H^{1}_{,0}(0,T).$

From this, we conclude

$\|\partial_{t}^{-1}w_{*}\|_{L^{2}(0,T)}=\frac{|\langle
w_{*},v_{*}\rangle_{(0,T)}|}{\|\partial_{t}v_{*}\|_{L^{2}(0,T)}}\leq\sup\limits_{0\neq\phi\in
H^{1}_{,0}(0,T)}\frac{|\langle
w_{*},\phi\rangle_{(0,T)}|}{\|\partial_{t}\phi\|_{L^{2}(0,T)}}=\|w_{*}\|_{[H^{1}_{,0}(0,T)]^{\prime}}.$

Indeed, we have

$\|\partial_{t}^{-1}w_{*}\|_{L^{2}(0,T)}=\|w_{*}\|_{[H^{1}_{,0}(0,T)]^{\prime}},$

and therefore,

$-\langle\overline{\partial}_{t}^{-1}Vw,w\rangle_{L^{2}(\Sigma)}\geq\sin^{2}\frac{\pi}{2(n+1)}\,\|w\|_{[H^{1}_{,0}(\Sigma)]^{\prime}}^{2}.$
(2.9)

In fact, by the density of $L^{2}(\Sigma)$ in $[H^{1}_{,0}(\Sigma)]^{\prime}$,
the operator

$-\overline{\partial}_{t}^{-1}V:[H^{1}_{,0}(\Sigma)]^{\prime}\to
H^{1}_{,0}(\Sigma)$

defines an isomorphism, and so does

$V:[H^{1}_{,0}(\Sigma)]^{\prime}\to L^{2}(\Sigma).$ (2.10)

For the single layer boundary integral operator $V$, we have obtained the
mapping properties (2.8) and (2.10), respectively. When applying an
interpolation argument, this gives that

$V:[H^{1/2}_{,0}(\Sigma)]^{\prime}\to H^{1/2}_{0,}(\Sigma)$

is an isomorphism as well, where the Sobolev space
$H^{1/2}_{0,}(\Sigma)=H^{1/2}_{0,}(0,T)\times H^{1/2}_{0,}(0,T)$ is endowed
with the Hilbertian norm

$\|z\|_{H^{1/2}_{0,}(\Sigma)}:=\left(\|z_{0}\|_{H^{1/2}_{0,}(0,T)}^{2}+\|z_{L}\|_{H^{1/2}_{0,}(0,T)}^{2}\right)^{1/2}\quad\text{
for }z=(z_{0},z_{L})\in H^{1/2}_{0,}(\Sigma)$

and analogously, the Sobolev space $H^{1/2}_{,0}(\Sigma)$ is introduced.
Hence, we conclude the inf-sup stability condition

$c_{S}\,\|w\|_{[H^{1/2}_{,0}(\Sigma)]^{\prime}}\leq\sup\limits_{0\neq
v\in[H^{1/2}_{0,}(\Sigma)]^{\prime}}\frac{|\langle
Vw,v\rangle_{\Sigma}|}{\|v\|_{[H^{1/2}_{0,}(\Sigma)]^{\prime}}}\quad\mbox{for
all}\;w\in[H^{1/2}_{,0}(\Sigma)]^{\prime}$ (2.11)

with a constant $c_{S}>0.$ In fact, (2.11) corresponds to the inf-sup
condition in [13, Theorem 5.7], where the test space is slightly larger than
used in (2.11). But we will show that $V:[H^{1/2}_{,0}(\Sigma)]^{\prime}\to
H^{1/2}_{0,}(\Sigma)$ in combination with a modified Hilbert transformation
[15, 16, 19] even satisfies an ellipticity estimate similar as in (2.7).

##  3 A modified Hilbert transformation

For $u\in L^{2}(0,T)$, we consider the Fourier series

$u(t)=\sum_{k=0}^{\infty}u_{k}\,\sin\left(\left(\frac{\pi}{2}+k\pi\right)\frac{t}{T}\right),\quad
u_{k}=\frac{2}{T}\int_{0}^{T}u(t)\,\sin\left(\left(\frac{\pi}{2}+k\pi\right)\frac{t}{T}\right)\,dt,$
$u(t)=\sum_{k=0}^{\infty}\overline{u}_{k}\,\cos\left(\left(\frac{\pi}{2}+k\pi\right)\frac{t}{T}\right),\quad\overline{u}_{k}=\frac{2}{T}\int_{0}^{T}u(t)\,\cos\left(\left(\frac{\pi}{2}+k\pi\right)\frac{t}{T}\right)\,dt.$

From [15, Lemma 2.1], we have

$\|u\|^{2}_{[H^{1/2}_{,0}(0,T)]^{\prime}}=\frac{T^{2}}{2}\,\sum\limits_{k=0}^{\infty}\left(\frac{\pi}{2}+k\pi\right)^{-1}\overline{u}_{k}^{2}\,.$

As in [15], we introduce the transformation operator
${\mathcal{H}}_{T}:L^{2}(0,T)\to L^{2}(0,T)$ as

${\mathcal{H}}_{T}u(t):=\sum\limits_{k=0}^{\infty}u_{k}\,\cos\left(\left(\frac{\pi}{2}+k\pi\right)\frac{t}{T}\right),\quad
t\in(0,T),$ (3.1)

which is norm preserving and bijective. By construction, we have that the
transformation operator ${\mathcal{H}}_{T}:H^{1/2}_{0,}(0,T)\to
H^{1/2}_{,0}(0,T)$ is also an isometric isomorphism, and

$\langle\partial_{t}u,{\mathcal{H}}_{T}u\rangle_{(0,T)}=\|u\|^{2}_{H^{1/2}_{0,}(0,T)}\quad\mbox{for
all}\;u\in H^{1/2}_{0,}(0,T).$

Note that $H^{1/2}_{0,}(0,T):=[H^{1}_{0,}(0,T),L^{2}(0,T)]_{1/2}$ is
constructed by interpolation, where $H^{1}_{0,}(0,T):=\\{v\in
H^{1}(0,T):v(0)=0\\}$. In the same way, we define $H^{1/2}_{,0}(0,T)$ but with
zero condition at the final time $t=T$. It is easy to see that

$|\langle\partial_{t}u,{\mathcal{H}}_{T}z\rangle_{(0,T)}|\leq\|u\|_{H^{1/2}_{0,}(0,T)}\|z\|_{H^{1/2}_{0,}(0,T)}\quad\mbox{for
all}\;u,z\in H^{1/2}_{0,}(0,T).$ (3.2)

The transformation operator ${\mathcal{H}}_{T}$, as defined in (3.1), allows a
closed representation, see [15, Lemma 2.8], which generalizes the well-known
Hilbert transformation, e.g., [3]. Moreover, following [16, Eqn. (2.5)] we
conclude the following representation for $u,z\in H^{1}_{0,}(0,T)$,

$\langle\partial_{t}u,{\mathcal{H}}_{T}z\rangle_{(0,T)}=-\frac{1}{\pi}\int_{0}^{T}\partial_{t}u(t)\int_{0}^{T}\ln\left[\tan\frac{\pi(s+t)}{4T}\tan\frac{\pi|t-s|}{4T}\right]\,\partial_{s}z(s)\,ds\,dt\,.$

This representation also allows for an efficient evaluation of the bilinear
form $\langle\partial_{t}u,{\mathcal{H}}_{T}z\rangle_{(0,T)}$ by using
hierarchical matrices, see [16] for a more detailed discussion.

##  4 A space-time approach in energy spaces

Instead of the boundary integral equation (2.5), we may replace the
application of the time derivative by the modified Hilbert transformation
${\mathcal{H}}_{T}:H^{1/2}_{0,}(\Sigma)\to H^{1/2}_{,0}(\Sigma)$, i.e., we
consider the boundary integral equation to find
$w\in[H^{1/2}_{,0}(\Sigma)]^{\prime}$ such that

${\mathcal{H}}_{T}Vw={\mathcal{H}}_{T}g\quad\mbox{in}\;[H^{1/2}_{,0}(\Sigma)]^{\prime},$

where $g\in H^{1/2}_{0,}(\Sigma)$ is a given Dirichlet datum. The related
bilinear form is given as

$a_{{\mathcal{H}}_{T}}(w,v):=\langle
v,{\mathcal{H}}_{T}Vw\rangle_{\Sigma}\quad\mbox{for
all}\;v,w\in[H^{1/2}_{,0}(\Sigma)]^{\prime}.$

Recall that for $u=(u_{0},u_{L})\in H^{1/2}_{0,}(\Sigma)$, we have

$\partial_{t}u=(\partial_{t}u_{0},\partial_{t}u_{L})=(v_{0},v_{L})=:v\in[H^{1/2}_{,0}(\Sigma)]^{\prime}\,,$

satisfying

$\|u\|_{H^{1/2}_{0,}(\Sigma)}=\|v\|_{[H^{1/2}_{,0}(\Sigma)]^{\prime}}\,.$

For $v=\partial_{t}u$, $w=\partial_{t}z$ with $u,z\in H^{1}_{0,}(\Sigma)$, we
can write

$\displaystyle a_{{\mathcal{H}}_{T}}(w,v)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{T}v_{0}(t)\,{\mathcal{H}}_{T}\left(\int_{0}^{t}w_{0}(s)\,ds+\int_{0}^{t-L}w_{L}(s)\,ds\right)dt$
$\displaystyle+\frac{1}{2}\int_{0}^{T}v_{L}(t)\,{\mathcal{H}}_{T}\left(\int_{0}^{t-L}w_{0}(s)\,ds+\int_{0}^{t}w_{L}(s)\,ds\right)dt$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,\Big{[}\langle\partial_{t}u_{0},{\mathcal{H}}_{T}(z_{0}+z_{L}(\cdot-L))\rangle_{(0,T)}+\langle\partial_{t}u_{L},{\mathcal{H}}_{T}(z_{0}(\cdot-L)+z_{L})\rangle_{(0,T)}\Big{]}\,.$

When using (3.2), we obtain

$\displaystyle|a_{{\mathcal{H}}_{T}}($ $\displaystyle w,v)|$
$\displaystyle\leq\frac{1}{2}\,\Big{[}\|u_{0}\|_{H^{1/2}_{0,}(0,T)}\|z_{0}+z_{L}(\cdot-L)\|_{H^{1/2}_{0,}(0,T)}+\|u_{L}\|_{H^{1/2}_{0,}(0,T)}\|z_{0}(\cdot-L)+z_{L}\|_{H^{1/2}_{0,}(0,T)}\Big{]}$
$\displaystyle\leq\frac{1}{2}\,\Big{[}\|u_{0}\|_{H^{1/2}_{0,}(0,T)}+\|u_{L}\|_{H^{1/2}_{0,}(0,T)}\Big{]}\Big{[}\|z_{0}\|_{H^{1/2}_{0,}(0,T)}+\|z_{L}\|_{H^{1/2}_{0,}(0,T)}\Big{]}$
$\displaystyle\leq\sqrt{\|u_{0}\|_{H^{1/2}_{0,}(0,T)}^{2}+\|u_{L}\|^{2}_{H^{1/2}_{0,}(0,T)}}\sqrt{\|z_{0}\|^{2}_{H^{1/2}_{0,}(0,T)}+\|z_{L}\|^{2}_{H^{1/2}_{0,}(0,T)}}$
$\displaystyle=\|u\|_{H^{1/2}_{0,}(\Sigma)}\|z\|_{H^{1/2}_{0,}(\Sigma)}$
$\displaystyle=\|v\|_{[H^{1/2}_{,0}(\Sigma)]^{\prime}}\|w\|_{[H^{1/2}_{,0}(\Sigma)]^{\prime}}$

for all $v,w\in L^{2}(\Sigma)$, i.e., the density of $L^{2}(\Sigma)$ in
$[H^{1/2}_{,0}(\Sigma)]^{\prime}$ yields the boundedness of the bilinear form
$a_{{\mathcal{H}}_{T}}(\cdot,\cdot)$.

###### Lemma 4.1

For $w\in[H^{1/2}_{,0}(\Sigma)]^{\prime}$, there holds

$a_{{\mathcal{H}}_{T}}(w,w)=\langle{\mathcal{H}}_{T}Vw,w\rangle_{\Sigma}\geq\frac{1}{2}\left(1-\frac{1}{2}\,\sup\limits_{m\in{\mathbb{N}}}\sqrt{\lambda_{\max}(C_{m})}\right)\,\|w\|^{2}_{[H^{1/2}_{,0}(\Sigma)]^{\prime}},$
(4.1)

where $\lambda_{\max}(C_{m})$ is the maximal eigenvalue of a symmetric matrix
$C_{m}\in\mathbb{R}^{(m+1)\times(m+1)}$. In the case of $T\leq L$, the matrix
$C_{m}$ is the zero matrix, i.e., $\lambda_{\max}(C_{m})=0$. However, in the
case $T>L$, the matrix $C_{m}$ is defined by the entries

$c_{\ell
i}=\sum\limits_{k=0}^{\infty}b_{k\ell}b_{ki}\quad\mbox{for}\;\ell,i=0,\ldots,m,$
$b_{kk}=2\,\left(1-\frac{L}{T}\right)\cos\left(\left(\frac{\pi}{2}+k\pi\right)\frac{L}{T}\right)\quad\mbox{for}\;k\in{\mathbb{N}}_{0},$
$b_{k\ell}=\frac{4}{\pi}\frac{\sqrt{2k+1}\sqrt{2\ell+1}}{(k+\ell+1)(k-\ell)}\cos\left((k+\ell+1)\frac{\pi}{2}\frac{L}{T}\right)\sin\left((\ell-k)\frac{\pi}{2}\frac{L}{T}\right)$

for $k,\ell\in{\mathbb{N}}_{0}$, $k-\ell=2j\neq 0$, $j\in{\mathbb{Z}}$, and
$b_{k\ell}=0$ else.

Proof. For $w=(w_{0},w_{L})\in L^{2}(\Sigma)$, we consider the Fourier series

$w_{0}(t)=\sum\limits_{k=0}^{\infty}\overline{w}_{0,k}\cos\left(\left(\frac{\pi}{2}+k\pi\right)\frac{t}{T}\right),\quad\overline{w}_{0,k}=\frac{2}{T}\int_{0}^{T}w_{0}(t)\,\cos\left(\left(\frac{\pi}{2}+k\pi\right)\frac{t}{T}\right)dt,$
$w_{L}(t)=\sum\limits_{k=0}^{\infty}\overline{w}_{L,k}\cos\left(\left(\frac{\pi}{2}+k\pi\right)\frac{t}{T}\right),\quad\overline{w}_{L,k}=\frac{2}{T}\int_{0}^{T}w_{L}(t)\,\cos\left(\left(\frac{\pi}{2}+k\pi\right)\frac{t}{T}\right)dt.$

In the case $T\leq L$, we explicitly compute

$\langle{\mathcal{H}}_{T}Vw,w\rangle_{L^{2}(\Sigma)}=\frac{T^{2}}{2}\sum\limits_{k=0}^{\infty}\frac{\overline{w}_{0,k}^{2}+\overline{w}_{L,k}^{2}}{(2k+1)\pi}=\frac{1}{2}\Big{(}\|w_{0}\|^{2}_{[H^{1/2}_{,0}(0,T)]^{\prime}}+\|w_{L}\|^{2}_{[H^{1/2}_{,0}(0,T)]^{\prime}}\Big{)},$

since there are no coupling terms.

In the case $T>L$, we have the representation

$\displaystyle\langle{\mathcal{H}}_{T}V$ $\displaystyle
w,w\rangle_{L^{2}(\Sigma)}$ $\displaystyle=$
$\displaystyle\,\frac{T^{2}}{2}\sum\limits_{k=0}^{\infty}\frac{\overline{w}_{0,k}^{2}+\overline{w}_{L,k}^{2}}{(2k+1)\pi}+\frac{T^{2}}{2}\sum\limits_{k=0}^{\infty}\overline{w}_{0,k}\overline{w}_{L,k}\frac{2}{(2k+1)\pi}\left(1-\frac{L}{T}\right)\cos\left(\left(\frac{\pi}{2}+k\pi\right)\frac{L}{T}\right)$
$\displaystyle+\frac{T^{2}}{2}\sum\limits_{k-\ell=2j\neq
0}\overline{w}_{0,\ell}\overline{w}_{L,k}\frac{4}{\pi^{2}}\frac{1}{(k+\ell+1)(k-\ell)}\cos\left((k+\ell+1)\frac{\pi}{2}\frac{L}{T}\right)\sin\left((\ell-k)\frac{\pi}{2}\frac{L}{T}\right)$
$\displaystyle=$
$\displaystyle\,\frac{T^{2}}{2}\sum\limits_{k=0}^{\infty}\Big{[}\widehat{w}_{0,k}^{2}+\widehat{w}_{L,k}^{2}\Big{]}+\frac{T^{2}}{2}\sum\limits_{k=0}^{\infty}2\widehat{w}_{0,k}\widehat{w}_{L,k}\left(1-\frac{L}{T}\right)\cos\left(\left(\frac{\pi}{2}+k\pi\right)\frac{L}{T}\right)$
$\displaystyle+\frac{T^{2}}{2}\sum\limits_{k-\ell=2j\neq
0}\widehat{w}_{0,\ell}\widehat{w}_{L,k}\frac{4}{\pi}\frac{\sqrt{2k+1}\sqrt{2\ell+1}}{(k+\ell+1)(k-\ell)}\cos\left((k+\ell+1)\frac{\pi}{2}\frac{L}{T}\right)\sin\left((\ell-k)\frac{\pi}{2}\frac{L}{T}\right),$

where

$\widehat{w}_{0,k}=\frac{\overline{w}_{0,k}}{\sqrt{(2k+1)\pi}},\quad\widehat{w}_{L,k}=\frac{\overline{w}_{L,k}}{\sqrt{(2k+1)\pi}}.$

When using the coefficients $b_{k\ell}$, we write the above result as

$\langle{\mathcal{H}}_{T}Vw,w\rangle_{L^{2}(\Sigma)}=\frac{T^{2}}{2}\left(\sum\limits_{k=0}^{\infty}\Big{[}\widehat{w}^{2}_{0,\ell}+\widehat{w}^{2}_{L,k}\Big{]}+\sum\limits_{k=0}^{\infty}\sum\limits_{\ell=0}^{\infty}b_{k\ell}\widehat{w}_{0,\ell}\widehat{w}_{L,k}\right)\,.$

Following [6, Chapter VIII], we consider the forms

$B(\widehat{w}_{0},\widehat{w}_{L}):=\sum\limits_{k=0}^{\infty}\sum\limits_{\ell=0}^{\infty}b_{k\ell}\widehat{w}_{0,\ell}\widehat{w}_{L,k},\quad
B_{m}(\widehat{w}_{0},\widehat{w}_{L}):=\sum\limits_{k=0}^{m}\sum\limits_{\ell=0}^{m}b_{k\ell}\widehat{w}_{0,\ell}\widehat{w}_{L,k}\,,$

and for the latter we estimate

$\displaystyle\Big{|}B_{m}(\widehat{w}_{0},\widehat{w}_{L})\Big{|}=\left|\sum\limits_{k=0}^{m}\sum\limits_{\ell=0}^{m}b_{k\ell}\widehat{w}_{0,\ell}\widehat{w}_{L,k}\right|$
$\displaystyle\leq\left[\sum\limits_{k=0}^{m}\widehat{w}_{L,k}^{2}\right]^{1/2}\left[\sum\limits_{k=0}^{m}\left(\sum\limits_{\ell=0}^{m}b_{k\ell}\widehat{w}_{0,\ell}\right)^{2}\right]^{1/2}$
$\displaystyle\leq\left[\sum\limits_{k=0}^{m}\widehat{w}_{L,k}^{2}\right]^{1/2}\left[\sum\limits_{k=0}^{\infty}\left(\sum\limits_{\ell=0}^{m}b_{k\ell}\widehat{w}_{0,\ell}\right)^{2}\right]^{1/2}\,.$

Hence, it remains to consider

$\displaystyle\sum\limits_{k=0}^{\infty}\left(\sum\limits_{\ell=0}^{m}b_{k\ell}\widehat{w}_{0,\ell}\right)^{2}$
$\displaystyle=\sum\limits_{\ell=0}^{m}\sum\limits_{j=0}^{m}\left(\sum\limits_{k=0}^{\infty}b_{k\ell}b_{kj}\right)\widehat{w}_{0,\ell}\widehat{w}_{0,j}$
$\displaystyle=\sum\limits_{\ell=0}^{m}\sum\limits_{j=0}^{m}c_{\ell
j}\widehat{w}_{0,\ell}\widehat{w}_{0,j}\,\leq\,\lambda_{\max}(C_{m})\sum\limits_{\ell=0}^{m}\widehat{w}_{0,\ell}^{2}\,.$

From this, we conclude

$\displaystyle\Big{|}B_{m}(\widehat{w}_{0},\widehat{w}_{L})\Big{|}$
$\displaystyle\leq\sqrt{\lambda_{\max}(C_{m})}\left[\sum\limits_{k=0}^{m}\widehat{w}_{L,k}^{2}\right]^{1/2}\left[\sum\limits_{\ell=0}^{m}\widehat{w}_{0,\ell}^{2}\right]^{1/2}$
$\displaystyle\leq\sup\limits_{m\in{\mathbb{N}}}\sqrt{\lambda_{\max}(C_{m})}\left[\sum\limits_{k=0}^{\infty}\widehat{w}_{L,k}^{2}\right]^{1/2}\left[\sum\limits_{\ell=0}^{\infty}\widehat{w}_{0,\ell}^{2}\right]^{1/2}$
$\displaystyle\leq\frac{1}{2}\,\sup\limits_{m\in{\mathbb{N}}}\sqrt{\lambda_{\max}(C_{m})}\left(\sum\limits_{k=0}^{\infty}\widehat{w}_{L,k}^{2}+\sum\limits_{\ell=0}^{\infty}\widehat{w}_{0,\ell}^{2}\right)$

for all $m\in{\mathbb{N}}$, and therefore

$\Big{|}B(\widehat{w}_{0},\widehat{w}_{L})\Big{|}\leq\frac{1}{2}\,\sup\limits_{m\in{\mathbb{N}}}\sqrt{\lambda_{\max}(C_{m})}\left(\sum\limits_{k=0}^{\infty}\widehat{w}_{L,k}^{2}+\sum\limits_{\ell=0}^{\infty}\widehat{w}_{0,\ell}^{2}\right)$

follows. With this, we finally obtain

$\displaystyle\langle{\mathcal{H}}_{T}Vw,w\rangle_{L^{2}(\Sigma)}$
$\displaystyle\geq\frac{T^{2}}{2}\left(1-\frac{1}{2}\,\sup\limits_{m\in{\mathbb{N}}}\sqrt{\lambda_{\max}(C_{m})}\right)\sum\limits_{k=0}^{\infty}\Big{[}\widehat{w}_{0,k}^{2}+\widehat{w}_{L,k}^{2}\Big{]}$
$\displaystyle=\frac{T^{2}}{2}\left(1-\frac{1}{2}\,\sup\limits_{m\in{\mathbb{N}}}\sqrt{\lambda_{\max}(C_{m})}\right)\sum\limits_{k=0}^{\infty}\frac{\overline{w}_{0,k}^{2}+\overline{w}_{L,k}^{2}}{(2k+1)\pi}$
$\displaystyle=\frac{T^{2}}{4}\left(1-\frac{1}{2}\,\sup\limits_{m\in{\mathbb{N}}}\sqrt{\lambda_{\max}(C_{m})}\right)\sum\limits_{k=0}^{\infty}\frac{\overline{w}_{0,k}^{2}+\overline{w}_{L,k}^{2}}{\frac{\pi}{2}+k\pi}$
$\displaystyle=\frac{1}{2}\left(1-\frac{1}{2}\,\sup\limits_{m\in{\mathbb{N}}}\sqrt{\lambda_{\max}(C_{m})}\right)\Big{(}\|w_{0}\|^{2}_{[H^{1/2}_{,0}(0,T)]^{\prime}}+\|w_{L}\|^{2}_{[H^{1/2}_{,0}(0,T)]^{\prime}}\Big{)},$

as stated. In both cases $T\leq L$ or $T>L$, the density of $L^{2}(\Sigma)$ in
$[H^{1/2}_{,0}(\Sigma)]^{\prime}$ yields the assertion.

###### Proposition 4.2

Numerical results indicate that

$\sup\limits_{m\in{\mathbb{N}}}\sqrt{\lambda_{\max}(C_{m})}=2-4\sin^{2}\left(\frac{\pi}{2(n+1)}\right),$

where $n$ is given in (2.6). Indeed, for $L=1$, $T\in[1,20]$ and $m=20000$,
the related results are given in Figure 1. Then, the ellipticity estimate
(4.1) becomes

$a_{{\mathcal{H}}_{T}}(w,w)=\langle{\mathcal{H}}_{T}Vw,w\rangle_{\Sigma}\geq\sin^{2}\left(\frac{\pi}{2(n+1)}\right)\,\|w\|^{2}_{[H^{1/2}_{,0}(\Sigma)]^{\prime}}\quad\mbox{for
all}\;w\in[H^{1/2}_{,0}(\Sigma)]^{\prime},$ (4.2)

where the ellipticity constant is the same as in (2.7), and in (2.9),
respectively. Hence, we can think of (4.2) being an interpolation of the
ellipticity estimates (2.7) and (2.9).

$1$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$0.5$$1$$1.5$$2$$\sqrt{\lambda_{\max}(C_{m})}$$2-4\sin^{2}\Big{(}\frac{\pi}{2(n+1)}\Big{)}$
Figure 1: Numerical evaluation of $\sqrt{\lambda_{\max}(C_{m})}$ for $L=1$,
$T\in[1,20]$, $m=20000$.

With the above results, we conclude unique solvability of the variational
formulation to find $w\in[H^{1/2}_{,0}(\Sigma)]^{\prime}$ such that

$\langle v,{\mathcal{H}}_{T}Vw\rangle_{\Sigma}=\langle
v,{\mathcal{H}}_{T}g\rangle_{\Sigma}\quad\mbox{for
all}\;v\in[H^{1/2}_{,0}(\Sigma)]^{\prime},$ (4.3)

where $g\in H^{1/2}_{0,}(\Sigma)$ is a given Dirichlet datum. Let
$W_{h}\subset[H^{1/2}_{,0}(\Sigma)]^{\prime}$ be some boundary element space,
e.g., of piecewise constant basis functions, which are defined with respect to
some decomposition of the lateral boundaries $\\{0\\}\times(0,T)$ and
$\\{L\\}\times(0,T)$, respectively. The space-time Galerkin boundary element
formulation of (4.3) is: Find $w_{h}\in W_{h}$ such that

$\langle v_{h},{\mathcal{H}}_{T}Vw_{h}\rangle_{\Sigma}=\langle
v_{h},{\mathcal{H}}_{T}g\rangle_{\Sigma}\quad\mbox{for all}\;v_{h}\in W_{h}.$

When assuming $w\in H^{s}(\Sigma)$ for some $s\in[0,1]$ and using standard
arguments, e.g., [12], we derive an a priori error estimate in the energy
norm,

$\|w-w_{h}\|_{[H^{1/2}_{,0}(\Sigma)]^{\prime}}\leq
c\,h^{s+\frac{1}{2}}\,\|w\|_{H^{s}(\Sigma)}.$

Moreover, using an inverse inequality, we also obtain an error estimate in
$L^{2}(\Sigma)$,

$\|w-w_{h}\|_{L^{2}(\Sigma)}\leq c\,h^{s}\,\|w\|_{H^{s}(\Sigma)}.$ (4.4)

##  5 Numerical results

Instead of the boundary integral equation (2.4) of the indirect approach, we
consider, as in [14], the boundary integral equation of the direct approach

$Vw=(\frac{1}{2}I+K)g\quad\text{ on }\Sigma,$ (5.1)

including the double layer boundary integral operator $K$ on the right hand
side. In this case, the unknown $w$ is the spatial normal derivative
$\partial_{n_{x}}u$ of the solution $u$ of (2.1).

For a boundary element approximation, consider a decomposition of the lateral
boundary

$\Sigma=\bigcup_{i=1}^{N_{0}+N_{L}}\overline{\tau}_{i}$

into $N_{0}+N_{L}$ boundary elements $\tau_{i}$ with maximal mesh size
$h=\max_{i}|\tau_{i}|$. Here, $N_{0}$ is the number of boundary elements for
the boundary $\\{0\\}\times(0,T)$ and $N_{L}$ is the number of boundary
elements for the boundary $\\{L\\}\times(0,T)$. The conforming ansatz space of
piecewise constant functions

$S_{h}^{0}(\Sigma):=S_{h_{0}}^{0}(0,T)\times
S_{h_{L}}^{0}(0,T)\subset[H^{1/2}_{,0}(\Sigma)]^{\prime}$

is used to define an approximate solution $w_{h}\in S_{h}^{0}(\Sigma).$ Then,
the Galerkin discretization of (5.1) to find $w_{h}\in S_{h}^{0}(\Sigma)$ such
that

$\langle v_{h},\mathcal{H}_{T}Vw_{h}\rangle_{L^{2}(\Sigma)}=\langle
v_{h},\mathcal{H}_{T}(\frac{1}{2}I+K)Q_{h}g\rangle_{L^{2}(\Sigma)}\quad\text{
for all }v_{h}\in S_{h}^{0}(\Sigma)$ (5.2)

is equivalent to the global linear system

$V_{h}\underline{w}=\underline{g}$ (5.3)

with the related system matrix
$V_{h}\in\mathbb{R}^{(N_{0}+N_{L})\times(N_{0}+N_{L})},$ the right-hand side
$\underline{g}\in\mathbb{R}^{N_{0}+N_{L}}$ and the vector of unknown
coefficients $\underline{w}\in\mathbb{R}^{N_{0}+N_{L}}$ of $w_{h}\in
S_{h}^{0}(\Sigma).$ Here, for an easier implementation, we approximate the
right-hand side $g\in H^{1/2}_{0,}(\Sigma)$ by $Q_{h}g$, where $Q_{h}$ is the
$L^{2}$ projection on the space of piecewise linear, continuous functions
fulfilling homogeneous initial conditions for $t=0.$ The assembling of the
matrix $V_{h}\in\mathbb{R}^{(N_{0}+N_{L})\times(N_{0}+N_{L})}$ and the right-
hand side $\underline{g}\in\mathbb{R}^{N_{0}+N_{L}}$, i.e., the realization of
$\mathcal{H}_{T}$, is done as proposed in [19, Subsection 2.2]. The integrals
for computing the projection $Q_{h}g$ are calculated by using high-order
quadrature rules. The global linear system (5.3) is solved by a direct solver.

In the numerical examples, we consider the spatial domain $\Omega=(0,3)$,
i.e., $L=3$, and the time interval $(0,6)$, i.e., $T=6$. The lateral
boundaries $\\{0\\}\times(0,T)$ and $\\{L\\}\times(0,T)$ are discretized
uniformly into $N_{0}=N_{L}=2^{\ell+1}$ boundary elements each,
$\ell=3,4,5,\ldots,12$.

In the first example, we consider the smooth solution

$u_{1}(x,t)=\left\\{\begin{array}[]{lcl}\frac{1}{2}\,(t-x-2)^{3}(x-t)^{3}&&\mbox{for}\;x\leq
t\leq 2+x,\\\\[2.84526pt] 0&&\mbox{otherwise}.\end{array}\right.$

Due to $w_{1}=\partial_{n_{x}}u_{1}\in H^{1}(\Sigma)$ and using the error
estimate (4.4), we expect a linear order of convergence, as confirmed by the
numerical results given in Table 1.

$\ell$ | $N_{0}+N_{L}$ | $\|w_{1}-w_{1,h}\|_{L^{2}(\Sigma)}$ | eoc
---|---|---|---
3 | 32 | 4.48 –1 |
4 | 64 | 2.11 –1 | 1.09
5 | 128 | 1.04 –1 | 1.02
6 | 256 | 5.18 –2 | 1.01
7 | 512 | 2.59 –2 | 1.00
8 | 1024 | 1.29 –2 | 1.00
9 | 2048 | 6.47 –3 | 1.00
10 | 4096 | 3.23 –3 | 1.00
11 | 8192 | 1.62 –3 | 1.00
12 | 16384 | 8.09 –4 | 1.00
Table 1: Numerical results for the boundary element method (5.2) in the case
$w_{1}\in H^{1}(\Sigma)$.

As a second example, we consider the singular solution

$u_{2}(x,t)=\left\\{\begin{array}[]{ccl}\displaystyle\frac{1}{2}\,|\sin(\pi(x-t))|&&\mbox{for}\;x\leq
t,\\\\[2.84526pt] 0&&\mbox{otherwise},\end{array}\right.$

where we have $w_{2}\in H^{s}(\Sigma)$ for $s<\frac{1}{2}$. Hence, using
(4.4), we expect the reduced order $\frac{1}{2}$ of convergence when
considering the error in $L^{2}(\Sigma)$. This is confirmed by the numerical
results as given in Table 2.

$\ell$ | $N_{0}+N_{L}$ | $\|w_{2}-w_{2,h}\|_{L^{2}(\Sigma)}$ | eoc
---|---|---|---
3 | 32 | 2.59 +0 | 0.34
4 | 64 | 1.75 +0 | 0.56
5 | 128 | 1.21 +0 | 0.53
6 | 256 | 8.45 –1 | 0.52
7 | 512 | 5.93 –1 | 0.51
8 | 1024 | 4.18 –1 | 0.51
9 | 2048 | 2.95 –1 | 0.50
10 | 4096 | 2.08 –1 | 0.50
11 | 8192 | 1.47 –1 | 0.50
12 | 16384 | 1.04 –1 | 0.50
Table 2: Numerical results for the boundary element method (5.2) in the case
$w_{2}\in H^{s}(\Sigma)$, $s<\frac{1}{2}$.

##  6 Conclusions

In this note, we have shown that the single layer boundary integral operator
of the wave equation in one space dimension is elliptic in the energy space
$[H^{1/2}_{,0}(\Sigma)]^{\prime}$, when composed with some modified Hilbert
transformation. This result corresponds to the well-known ellipticity results
for boundary integral operators related to second-order elliptic partial
differential equations. While this particular result is at this time
restricted to the spatially one-dimensional case, in the general case we were
already able to establish a related inf-sup stability condition [13] instead.
Although this is already sufficient to do a numerical analysis of related
boundary element methods, it remains open whether we can prove ellipticity
also in the multi-dimensional case. It is obvious that we can extend this
approach also to the hypersingular boundary integral operator, and to the
double layer boundary integral operator. Ellipticity of boundary integral
operators is an important ingredient in the a priori and a posteriori error
analysis of boundary element methods, in the construction of appropriate
preconditioners, and in the coupling with finite element methods. It goes
without saying that this proposed new approach requires more work in the
numerical analysis, and in the implementation of the proposed scheme,
including the composition of the single layer boundary integral operator and
the modified Hilbert transformation, which are both non-local. Nevertheless,
this work may give some more insight into the numerical analysis of existing
boundary element methods for the wave equation, and it presents an alternative
approach for a reliable and efficient numerical solution of the wave equation.

## Acknowledgment

The research of the second author was funded by the John Fell Oxford
University Press Research Fund.

## References

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