∎

††thanks: This work was supported by the Swedish Research Council under the
grant 2019–05396.11institutetext:  Emil Engström 22institutetext: Centre for
Mathematical Sciences, Lund University, P.O. Box 118, 221 00 Lund, Sweden
22email<EMAIL_ADDRESS>33institutetext: Eskil Hansen
44institutetext: Centre for Mathematical Sciences, Lund University, P.O. Box
118, 221 00 Lund, Sweden
44email<EMAIL_ADDRESS>
# Time-dependent Steklov–Poincaré operators and space-time Robin–Robin
decomposition for the heat equation ††thanks: This work was supported by the
Swedish Research Council under the grant 2019–05396.

Emil Engström  Eskil Hansen

###### Abstract

Domain decomposition methods are a set of widely used tools for
parallelization of partial differential equation solvers. Convergence is well
studied for elliptic equations, but in the case of parabolic equations there
are hardly any results for general Lipschitz domains in two or more
dimensions. The aim of this work is therefore to construct a new framework for
analyzing nonoverlapping domain decomposition methods for the heat equation in
a space-time Lipschitz cylinder. The framework is based on a variational
formulation, inspired by recent studies of space-time finite elements using
Sobolev spaces with fractional time regularity. In this framework, the time-
dependent Steklov–Poincaré operators are introduced and their essential
properties are proven. We then derive the interface interpretations of the
Dirichlet–Neumann, Neumann–Neumann and Robin–Robin methods and show that these
methods are well defined. Finally, we prove convergence of the Robin–Robin
method and introduce a modified method with stronger convergence properties.

###### Keywords:

Steklov–Poincaré operatorRobin–Robin method Nonoverlapping domain
decomposition Space-time Convergence

###### MSC:

65M5565J0835K20

##  1 Introduction

Domain decomposition methods enable the usage of parallel and distributed
hardware and are commonly employed when approximating the solutions to
elliptic equations. The basic idea is to first decompose the equation’s
spatial domain into subdomains. The numerical method then consists of
iteratively solving the elliptic equation on each subdomain and thereafter
communicating the results via the boundaries to the neighboring subdomains. An
in-depth survey of the topic can be found in the monographs quarteroni ;
widlund .

A recent development in the field is to apply this approach to parabolic
equations. The decomposition into spatial subdomains is then replaced by the
decomposition into space-time cylinders. In general, space-time decomposition
schemes enable additional parallelization and less storage requirements when
combined with a standard numerical method for parabolic problems. The methods
have especially gained attention in the contexts of parallel time integrators;
surveyed in gander15 , space-time finite elements; surveyed in steinbach19 ,
and parabolic problems with a spatial domain given by a union of domains with
very different material properties japhet20 ; japhet16 .

There have been several studies concerning the convergence and other
theoretical aspects of space-time decomposition schemes applied to parabolic
equations on one-dimensional or rectangular spatial domains; see gander07 ;
kwok16 ; kwok21 ; keller02 ; lemarie13a ; lemarie13b . However, there are
hardly any convergence results for more general settings, e.g., with spatial
Lipschitz domains in higher dimensions. The one exception is the convergence
analysis considered in japhet13  for an optimized Schwarz waveform relaxation,
where a method-specific variational approach is proposed. The latter is
similar to the analysis used by Lions for the Robin–Robin method applied to
elliptic equations lions3 .

This lack of convergence results is in stark contrast to the elliptic setting,
where the analysis of nonoverlapping domain decomposition schemes for
Lipschitz domains in higher dimensions is standard. The basic elliptic
framework is to reformulate the method as an iteration scheme posed on the
intersections of the subdomains in terms of Steklov–Poincaré operators. This
is the basis for the analysis of the Dirichlet–Neumann and Neumann–Neumann
methods quarteroni . The Steklov–Poincaré framework even yields the
convergence of the Robin–Robin method applied to nonlinear elliptic equations
EHEE22 . Note that the time-dependent Steklov–Poincaré operators were
introduced without any analysis in japhet13 .

Hence, the aim of this study is twofold. First, we strive to introduce a new
variational framework for time-dependent Steklov–Poincaré operators and to
derive the essential properties of these operators, e.g., coercivity and
bijectivity. The latter property implies that the space-time generalizations
of the Dirichlet–Neumann, Neumann–Neumann, and Robin–Robin methods are all
well defined in the parabolic setting. Second, we will employ the new
framework to give a first rigorous proof of the Robin–Robin method’s
convergence when applied to parabolic equations.

For sake of simplicity, we will restrict our attention to the heat equation
with homogeneous Dirichlet conditions. However, our analysis does not depend
on any symmetry of the underlying bilinear form, and a similar framework can
be used for more general parabolic problems and boundary conditions. To limit
the technicalities further, we consider the equation for all times
$t\in\mathbb{R}$, which allows a variational formulation with the same test
and trial spaces. Our model problem is therefore the heat equation on the
space-time cylinder $\Omega\times\mathbb{R}$, i.e.,

$\left\\{\begin{aligned} (\partial_{t}-\Delta)u&=f&&\text{in
}\Omega\times\mathbb{R},\\\ u&=0&&\text{on
}\partial\Omega\times\mathbb{R},\end{aligned}\right.$ (1)

where the spatial domain $\Omega\subset\mathbb{R}^{d}$, $d=2,3$, is bounded
with boundary $\partial\Omega$. Note that a homogeneous initial condition at
time $t_{0}$ can be incorporated by prescribing a source term $f$ that is zero
for times $t<t_{0}$.

Next, we decompose the spatial domain $\Omega$ into nonoverlapping subdomains
$\Omega_{i}$, $i=1,2$, with boundaries $\partial\Omega_{i}$, and denote the
interface separating the subdomains $\Omega_{i}$ by $\Gamma$. That is,

$\overline{\Omega}=\overline{\Omega}_{1}\cup\overline{\Omega}_{2},\quad\Omega_{1}\cap\Omega_{2}=\emptyset,\quad\text{and}\quad\Gamma=(\partial\Omega_{1}\cap\partial\Omega_{2})\setminus\partial\Omega.$
(2)

The space-time cylinder $\Omega\times\mathbb{R}$ is thereby decomposed into
$\Omega_{i}\times\mathbb{R}$, $i=1,2$, as illustrated in Figure 1. The current
setting is also valid for spatial subdomains $\Omega_{i}$ given as unions of
nonadjacent subdomains, i.e.,

$\Omega_{i}=\cup_{\ell=1}^{s}\Omega_{i\ell}\quad\text{and}\quad\overline{\Omega}_{i\ell}\cap\overline{\Omega}_{ij}=\emptyset\quad\text{for
}\ell\neq j.$

Figure 1: The nonoverlapping decomposition of the space-time cylinder.

With a fixed domain decomposition we can reformulate the heat equation as
equations on $\Omega_{i}\times\mathbb{R}$, $i=1,2$, connected via transmission
conditions on $\Gamma\times\mathbb{R}$. More precisely, we have the parabolic
transmission problem

$\left\\{\begin{aligned} (\partial_{t}-\Delta)u_{i}&=f&&\text{in
}\Omega_{i}\times\mathbb{R}&&\text{ for }i=1,2,\\\ u_{i}&=0&&\text{on
}(\partial\Omega_{i}\setminus\Gamma)\times\mathbb{R}&&\text{ for
}i=1,2,\\\\[5.0pt] u_{1}&=u_{2}&&\text{on }\Gamma\times\mathbb{R},&&\\\ \nabla
u_{1}\cdot\nu_{1}&=-\nabla u_{2}\cdot\nu_{2}&&\text{on
}\Gamma\times\mathbb{R},&&\end{aligned}\right.$ (3)

where $\nu_{i}$ denotes the unit outward normal vector of
$\partial\Omega_{i}$. Alternating between the decomposed space-time cylinders
and the transmission conditions generates the space-time generalizations of
the Dirichlet–Neumann and Neumann–Neumann methods. As $\nu_{1}=-\nu_{2}$ on
$\Gamma$, the transmission conditions are also equivalent to the Robin
conditions

$\nabla u_{1}\cdot\nu_{i}+su_{1}=\nabla u_{2}\cdot\nu_{i}+su_{2}\quad\text{on
}\Gamma\times\mathbb{R}\text{ for }i=1,2,$

where $s$ is a non-zero parameter. The latter reformulation gives rise to the
space-time Robin–Robin method, for which we will prove convergence. The method
is given by finding $(u^{n}_{1},u^{n}_{2})$ for $n=1,2,\ldots$ such that

$\left\\{\begin{aligned} (\partial_{t}-\Delta)u^{n+1}_{1}&=f&&\text{in
}\Omega_{1}\times\mathbb{R},\\\ u^{n+1}_{1}&=0&&\text{on
}(\partial\Omega_{1}\setminus\Gamma)\times\mathbb{R},\\\ \nabla
u^{n+1}_{1}\cdot\nu_{1}+su^{n+1}_{1}&=\nabla
u^{n}_{2}\cdot\nu_{1}+su^{n}_{2}&&\text{on }\Gamma\times\mathbb{R},\\\\[5.0pt]
(\partial_{t}-\Delta)u^{n+1}_{2}&=f&&\text{in }\Omega_{2}\times\mathbb{R},\\\
u^{n+1}_{2}&=0&&\text{on
}(\partial\Omega_{2}\setminus\Gamma)\times\mathbb{R},\\\ \nabla
u^{n+1}_{2}\cdot\nu_{2}+su^{n+1}_{2}&=\nabla
u^{n+1}_{1}\cdot\nu_{2}+su^{n+1}_{1}&&\text{on
}\Gamma\times\mathbb{R},\end{aligned}\right.$ (4)

where $u_{2}^{0}$ is a given initial guess, $s>0$ is a fixed method parameter,
and $u^{n}_{i}$ approximates $u|_{\Omega_{i}\times\mathbb{R}}$. Note that the
method per se is sequential, but the computation of each term $u^{n}_{i}$ can
be implemented in parallel if $\Omega_{i}$ is a union of nonadjacent
subdomains.

The standard variational framework for parabolic problems, based on solutions
in the space

$H^{1}\bigl{(}\mathbb{R},H^{-1}(\Omega_{i})\bigr{)}\cap
L^{2}\bigl{(}\mathbb{R},H^{1}(\Omega_{i})\bigr{)},$

is unfortunately not well suited for our domain decompositions. The issue is
that two functions $u_{i}$, $i=1,2$, in the above solution space, which
coincide on $\Gamma\times\mathbb{R}$ in the sense of trace, can not be
“pasted” together into a new function in
$H^{1}\bigl{(}\mathbb{R},H^{-1}(\Omega)\bigr{)}$; compare with (costabel90, ,
Example 2.14). Hence, the transmission problem Equation 3 does not necessarily
yield a solution to the heat equation Equation 1 in this context.

In order to remedy this, we consider a framework with solutions in

$H^{1/2}\bigl{(}\mathbb{R},L^{2}(\Omega_{i})\bigr{)}\cap
L^{2}\bigl{(}\mathbb{R},H^{1}(\Omega_{i})\bigr{)},$

which originates from lionsmagenes2  and resolves the above issue. This
$H^{1/2}$-setting also enables a trace theory valid for spatial Lipschitz
domains costabel90  and the related bilinear form becomes coercive-equivalent
fontesthesis . This will be the starting point for our analysis. Note that the
$H^{1/2}$-setting has been used by costabel90  in the context of boundary
element methods and was later employed for space-time finite elements
fontesthesis ; larsson15 ; steinbach20 .

The analysis is organized as follows: In Sections 2 and 3 we derive the
properties of the required function spaces and operators. We introduce the
variational formulations and prove equivalence between the heat equation and
the transmission problem in Section 4. The time-dependent Steklov–Poincaré
operators are analysed in Section 5 and three standard space-time domain
decompositions are proven to be well defined. In Sections 6 and 7 we prove
convergence of the space-time Robin–Robin method in
$L^{2}\bigl{(}\mathbb{R},H^{1}(\Omega_{1})\bigr{)}\times
L^{2}\bigl{(}\mathbb{R},H^{1}(\Omega_{2})\bigr{)}$ and the convergence of a
modified version of the method in

$H^{1/2}\bigl{(}\mathbb{R},L^{2}(\Omega_{1})\bigr{)}\cap
L^{2}\bigl{(}\mathbb{R},H^{1}(\Omega_{1})\bigr{)}\,\times\,H^{1/2}\bigl{(}\mathbb{R},L^{2}(\Omega_{2})\bigr{)}\cap
L^{2}\bigl{(}\mathbb{R},H^{1}(\Omega_{2})\bigr{)},$

respectively.

##  2 Preliminaries

The spatial domain $\Omega\subset\mathbb{R}^{d}$ and its decomposition
Equation 2 into $\Omega_{i}$, $i=1,2$, is assumed to satisfy the properties
below.

###### Assumption 1

The sets $\Omega,\,\Omega_{i},\,i=1,2$, are bounded and Lipschitz. The spatial
interface $\Gamma$ and the sets $\partial\Omega\setminus\partial\Omega_{i}$,
$i=1,2$, are all $(d-1)$-dimensional Lipschitz manifolds.

For a description of Lipschitz domains, see (kufner, , Chapter 6.2). The
assumptions are made in order ensure the existence of the spatial trace
operator, as well as, to allow the usage of Poincaré’s inequality. As a
notational convention, operators only depending on space or time are “hatted”
and their extensions to space-time are denoted without hats, e.g.,

$\hat{\Delta}:H_{0}^{1}(\Omega)\rightarrow
H^{-1}(\Omega)\quad\text{and}\quad\Delta:L^{2}\bigl{(}\mathbb{R},H_{0}^{1}(\Omega)\bigr{)}\rightarrow
L^{2}\bigl{(}\mathbb{R},H^{-1}(\Omega)\bigr{)}.$

Furthermore, $c$ and $C$ denote generic positive constants.

Consider the spatial function spaces

$V=H_{0}^{1}(\Omega),\quad
V_{i}^{0}=H_{0}^{1}(\Omega_{i}),\quad\text{and}\quad V_{i}=\\{v\in
H^{1}(\Omega_{i}):\left.(\hat{T}_{\partial\Omega_{i}}v)\right|_{\partial\Omega_{i}\setminus\Gamma}=0\\}.$

Here, $\hat{T}_{\partial\Omega_{i}}:H^{1}(\Omega_{i})\rightarrow
H^{1/2}(\partial\Omega_{i})$ denotes the spatial trace operator, see (kufner,
, Theorem 6.8.13). The norm on $V_{i}$ and $V_{i}^{0}$ is given by

$\|v\|_{V_{i}}=\bigl{(}\|\nabla
v\|^{2}_{L^{2}(\Omega_{i})^{d}}+\|v\|^{2}_{L^{2}(\Omega_{i})}\bigr{)}^{1/2}.$

By Poincaré’s inequality and Assumption 1 we have that $v\mapsto\|\nabla
v\|_{L^{2}(\Omega_{i})^{d}}$ is an equivalent norm on $V_{i}$ and $V_{i}^{0}$.
The Hilbert spaces $V$, $V_{i}$, and $V_{i}^{0}$ are all separable. The space
$H^{1/2}(\partial\Omega_{i})$ is defined as

$\displaystyle H^{1/2}(\partial\Omega_{i})=\\{v\in
L^{2}(\partial\Omega_{i}):\|v\|_{H^{1/2}(\partial\Omega_{i})}<\infty\\}\quad\text{with}$
(5)
$\displaystyle\|v\|_{H^{1/2}(\partial\Omega_{i})}=\Big{(}\int_{\partial\Omega_{i}}\int_{\partial\Omega_{i}}\frac{\left.|v(x)-v(y)\right|^{2}}{\left.|x-y\right|^{d}}\,\mathrm{d}x\,\mathrm{d}y+\|v\|^{2}_{L^{2}(\partial\Omega_{i})}\Big{)}^{1/2}.$

Denoting the extension by zero from $\Gamma$ to $\partial\Omega_{i}$ by
$\hat{E}_{i}$ we define the Lions–Magenes space as

$\Lambda=\\{\mu\in L^{2}(\Gamma):\hat{E}_{i}\mu\in
H^{1/2}(\partial\Omega_{i})\\}\quad\text{with}\quad\|\mu\|_{\Lambda}=\|\hat{E}_{i}\mu\|_{H^{1/2}(\partial\Omega_{i})}.$

Since $H^{1/2}(\partial\Omega_{i})$ is a separable Hilbert space, so is
$\Lambda$. On $V_{i}$ the trace operator takes the form

$\hat{T}_{i}:V_{i}\rightarrow\Lambda:v\mapsto\left.(\hat{T}_{\partial\Omega_{i}}v)\right|_{\Gamma},$

and is bounded; see (EHEE22, , Lemma 4.4).

###### Remark 1

By (widlund, , Lemma A.8) we can identify
$\Lambda=[H^{1/2}_{0}(\Gamma),L^{2}(\Gamma)]_{1/2}$, which shows that
$\Lambda$ is independent of $i=1,2$. This reference is without proof, but a
proof exists for the case of a smooth $\Omega$ in (lionsmagenes1, , Theorem
11.7). The independence of $\Lambda$ on $i=1,2$ is assumed to be true for the
sake of presentation, but it is not a necessary assumption for our convergence
analysis; see EHEE22 .

For the temporal function space $H^{s}(\mathbb{R})$, $s\in[0,1]$, we use the
Fourier definition

$\displaystyle H^{s}(\mathbb{R})=\\{v\in
L^{2}(\mathbb{R}):(1+(\cdot)^{2})^{s/2}\hat{\mathcal{F}}v\in
L_{\mathbb{C}}^{2}(\mathbb{R})\\}\quad\text{with}$ (6)
$\displaystyle\|v\|_{H^{s}(\mathbb{R})}=\|(1+(\cdot)^{2})^{s/2}\hat{\mathcal{F}}v\|_{L_{\mathbb{C}}^{2}(\mathbb{R})}.$

Here, $\hat{\mathcal{F}}$ is the Fourier transform and
$L_{\mathbb{C}}^{2}(\mathbb{R})$ is the complexification of the real Hilbert
space $L^{2}(\mathbb{R})$. Note that this is equivalent to the
Sobolev–Slobodetskii definition Equation 5 for $s=1/2$ and
$\partial\Omega_{i}=\mathbb{R}$; see (tartar, , Lemma 16.3). We then introduce
the temporal Hilbert transform by

$\hat{\mathcal{H}}v(t)=\lim_{\epsilon\rightarrow
0^{+}}\frac{1}{\pi}\int_{\left.|s\right|\geq\epsilon}\frac{1}{s}v(t-s)\,\mathrm{d}s.$

From (king, , Chapters 4, 5) we have that
$\hat{\mathcal{H}}:L^{2}(\mathbb{R})\rightarrow L^{2}(\mathbb{R})$ is an
isomorphism with inverse $\hat{\mathcal{H}}^{-1}=-\hat{\mathcal{H}}$ and

$\hat{\mathcal{H}}=\hat{\mathcal{F}}^{-1}\hat{M}_{\mathrm{sgn}}\hat{\mathcal{F}},$
(7)

where $\hat{M}_{\mathrm{sgn}}v(\xi)=-\mathrm{i}\,\mathrm{sgn}(\xi)v(\xi)$. The
formula Equation 7 combined with the definition Equation 6 shows that
$\hat{\mathcal{H}}:H^{s}(\mathbb{R})\rightarrow H^{s}(\mathbb{R})$ is an
isomorphism. We also introduce the temporal half-derivatives as

$\hat{\partial}^{1/2}_{\pm}=\hat{\mathcal{F}}^{-1}\hat{M}_{\pm}\hat{\mathcal{F}},$

where $\hat{M}_{+}v(\xi)=\sqrt{\mathrm{i}\xi}v(\xi)$ and
$\hat{M}_{-}v(\xi)=\overline{\sqrt{\mathrm{i}\xi}}v(\xi)$. It is clear from
the definition Equation 6 that
$\hat{\partial}^{1/2}_{\pm}:H^{1/2}(\mathbb{R})\rightarrow L^{2}(\mathbb{R})$
are bounded linear operators. The important relations between these operators
are given in the lemma below.

###### Lemma 1

For $v\in H^{1/2}(\mathbb{R})$ one has the equalities

$\hat{\partial}^{1/2}_{+}v=-\hat{\partial}^{1/2}_{-}\hat{\mathcal{H}}v\quad\text{and}\quad(\hat{\partial}^{1/2}_{+}v\,,\,\hat{\partial}^{1/2}_{-}v)_{L^{2}(\mathbb{R})}=0.$

Moreover, for $v\in H^{1}(\mathbb{R})$ and $w\in H^{1/2}(\mathbb{R})$ one has
the fractional integration by parts formula

$(\hat{\partial}_{t}v\,,\,w)_{L^{2}(\mathbb{R})}=(\hat{\partial}^{1/2}_{+}v\,,\,\hat{\partial}^{1/2}_{-}w)_{L^{2}(\mathbb{R})}.$

###### Proof

Let $v\in H^{1/2}(\mathbb{R})$ and observe that

$\sqrt{\mathrm{i}\xi}=-\overline{\sqrt{\mathrm{i}\xi}}(-\mathrm{i}\,\mathrm{sgn}(\xi)).$

The Fourier characterization of the operators
$\hat{\mathcal{H}},\hat{\partial}^{1/2}_{+},\hat{\partial}^{1/2}_{-}$ then
implies that

$\hat{\partial}^{1/2}_{+}v=-\hat{\partial}^{1/2}_{-}\hat{\mathcal{H}}v.$

A similar argument, together with the fact that $|\hat{\mathcal{F}}v|^{2}$ is
an even function, shows that

$(\hat{\partial}^{1/2}_{+}v,\hat{\partial}^{1/2}_{-}v)_{L^{2}(\mathbb{R})}=\int_{\mathbb{R}}i\xi|\hat{\mathcal{F}}v|^{2}\mathrm{d}\xi=0$

for $v\in H^{1/2}(\mathbb{R})$. Finally, the fractional integration by parts
formula follows by the Fourier characterization
$\hat{\partial}_{t}=\hat{\mathcal{F}}^{-1}\hat{M}\hat{\mathcal{F}}$, where
$\hat{M}v(\xi)=\mathrm{i}\xi v(\xi)$. ∎

The Fourier characterization of the operator $\hat{\partial}^{1/2}_{+}$ and
Equation 6 yield that

$v\mapsto\bigl{(}\|\hat{\partial}^{1/2}_{+}v\|^{2}_{L^{2}(\mathbb{R})}+\|v\|^{2}_{L^{2}(\mathbb{R})}\bigr{)}^{1/2}$

is an equivalent norm on $H^{1/2}(\mathbb{R})$.

##  3 Tensor spaces

Inspired by the finite element analysis in steinbach20 , we identify our
Bochner spaces in space-time as tensor spaces. A general introduction to
tensor spaces can be found in (weidmann, , Chapter 3.4).

We denote the algebraic tensor product of two (real) separable Hilbert spaces
$X,Y$ by $X\otimes Y$. For elements of the form $x\otimes y$ the inner product
is defined as

$(x_{1}\otimes y_{1},x_{2}\otimes
y_{2})_{X\,\tilde{\otimes}\,Y}=(x_{1},x_{2})_{X}(y_{1},y_{2})_{Y}$

and for arbitrary elements in $X\otimes Y$ the definition is extended by
linearity. The closure of $X\otimes Y$ with respect to this norm is denoted by
$X\,\tilde{\otimes}\,Y$. From these definitions it follows that
$X\,\tilde{\otimes}\,Y=Y\,\tilde{\otimes}\,X$.

If $\\{x_{k}\\}_{k\geq 0}$ and $\\{y_{\ell}\\}_{\ell\geq 0}$ are orthonormal
bases of $X$ and $Y$, respectively, then$\\{x_{k}\otimes
y_{\ell}\\}_{k,\ell\geq 0}$ is an orthonormal basis of $X\,\tilde{\otimes}\,Y$
and every $v\in X\,\tilde{\otimes}\,Y$ can be represented as

$v=\sum_{k,\ell=0}^{\infty}c_{k,\ell}(x_{k}\otimes
y_{\ell})=\sum_{k=0}^{\infty}x_{k}\otimes z_{k},\quad
z_{k}=\sum_{\ell=0}^{\infty}c_{k,\ell}y_{\ell}\in Y,$ (8)

where $\\{c_{k,\ell}\\}_{k,\ell\geq 0}$ are real coefficients. This follows
from (weidmann, , Theorem 3.12).

We recall the following result on the extensions of operators to tensor
spaces. The proof of Lemma 2 can be found in (aubin, , Section 12.4.1).

###### Lemma 2

Let $X_{k},Y_{k}$, $k=1,2$, be separable Hilbert spaces, and
$A:X_{1}\rightarrow X_{2}$, $B:Y_{1}\rightarrow Y_{2}$ be bounded linear
operators. Then there is a bounded linear operator

$A\,\tilde{\otimes}\,B:X_{1}\,\tilde{\otimes}\,Y_{1}\rightarrow
X_{2}\,\tilde{\otimes}\,Y_{2}$

such that $(A\,\tilde{\otimes}\,B)(x\otimes y)=Ax\otimes By$ for every $x\in
X_{1},y\in Y_{1}$.

From Lemma 2 it follows that the parabolic trace operators

$\displaystyle T_{\partial\Omega_{i}}$
$\displaystyle=I\,\tilde{\otimes}\,\hat{T}_{\partial\Omega_{i}}:L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega)\rightarrow
L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1/2}(\partial\Omega_{i}),$
$\displaystyle T_{i}$
$\displaystyle=I\,\tilde{\otimes}\,\hat{T}_{i}:L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}\rightarrow
L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda$

are bounded. Furthermore, we have the identity

$T_{i}v=\left.(T_{\partial\Omega_{i}}v)\right|_{\Gamma\times\mathbb{R}}\quad\text{for
}v\in L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}.$

This is easily proven by validating the identity on the dense subset
$L^{2}(\mathbb{R})\otimes V_{i}$ and using the continuity of the operators.
Note that the restrictions, as well as the extension by zero
$E_{i}=I\,\tilde{\otimes}\,\hat{E}_{i}$, are all well defined operations due
to Lemma 2.

For any separable Hilbert space $X$ one has the identification

$H^{s}(\mathbb{R})\,\tilde{\otimes}\,X\cong H^{s}(\mathbb{R},X),$

where $H^{s}(\mathbb{R},X)$, $s\in[0,1]$, is a Sobolev–Bochner space; see
(hytonen, , Chapter 2.5.d). This can be proven by noting that the norms
coincide on $H^{s}(\mathbb{R})\otimes X$, the identification
$H^{1}(\mathbb{R})\,\tilde{\otimes}\,X\cong H^{1}(\mathbb{R},X)$; see (aubin,
, Theorem 12.7.1), and the fact that $H^{1}(\mathbb{R})\otimes X$ and
$H^{1}(\mathbb{R},X)$ are dense in $H^{s}(\mathbb{R})\,\tilde{\otimes}\,X$ and
$H^{s}(\mathbb{R},X)$, respectively; see (weidmann, , Theorem 3.12) and
(lionsmagenes1, , Proposition 6.1).

The Sobolev–Bochner spaces below will make up the core of the analysis:

$\displaystyle W$
$\displaystyle=H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega)\,\cap\,L^{2}(\mathbb{R})\,\tilde{\otimes}\,V,$
$\displaystyle W_{i}^{0}$
$\displaystyle=H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})\,\cap\,L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}^{0},$
$\displaystyle W_{i}$
$\displaystyle=H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})\,\cap\,L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i},$
$\displaystyle Z$
$\displaystyle=H^{1/4}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Gamma)\,\cap\,L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda.$

###### Lemma 3

Suppose Assumption 1 holds. Then we have the identities

$\displaystyle L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}^{0}$
$\displaystyle=\\{v\in
L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i}):T_{\partial\Omega_{i}}v=0\\},$
(9) $\displaystyle L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}$
$\displaystyle=\\{v\in
L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i}):\left.(T_{\partial\Omega_{i}}v)\right|_{(\partial\Omega_{i}\setminus\Gamma)\times\mathbb{R}}=0\\},$
(10) $\displaystyle L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda$
$\displaystyle=\\{\mu\in
L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Gamma):E_{i}\mu\in
L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1/2}(\partial\Omega_{i})\\}.$ (11)

###### Proof

Throughout the proof let $\\{x_{k}\\}_{k\geq 0}$ be an orthonormal basis of
$L^{2}(\mathbb{R})$. Let $\\{y_{\ell}\\}_{\ell\geq 0}$ be an orthonormal basis
of a separable Hilbert space $Y$ given by the context, and the corresponding
$\\{z_{k}\\}_{k\geq 0}$ are defined as in Equation 8.

To prove the identity Equation 9 recall that $V_{i}^{0}=\\{y\in
H^{1}(\Omega_{i}):\hat{T}_{\partial\Omega_{i}}y=0\\}$; see (kufner, , Theorem
6.6.4). This observation together with the continuity of
$T_{\partial\Omega_{i}}$ then yields that every $v\in
L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}^{0}$ satisfies

$T_{\partial\Omega_{i}}v=\sum_{k,\ell=0}^{\infty}c_{k,\ell}(x_{k}\otimes\hat{T}_{\partial\Omega_{i}}y_{\ell})=0.$

Conversely, suppose that $v\in
L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})$ with
$T_{\partial\Omega_{i}}v=0$. By the continuity of $T_{\partial\Omega_{i}}$ we
have

$0=T_{\partial\Omega_{i}}v=\sum_{k=0}^{\infty}x_{k}\otimes\hat{T}_{\partial\Omega_{i}}z_{k}.$

It then follows by the orthonormality of $\\{x_{k}\\}_{k\geq 0}$ that

$0=(T_{\partial\Omega_{i}}v\,,\,x_{k}\otimes\hat{T}_{\partial\Omega_{i}}z_{k})_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\partial\Omega_{i})}=\|\hat{T}_{\partial\Omega_{i}}z_{k}\|^{2}_{L^{2}(\partial\Omega_{i})}.$
(12)

Hence, $z_{k}\in V_{i}^{0}$, $k=0,1,\dots$, and from Equation 8 we can
conclude that $v\in L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}^{0}$.

The proof of Equation 10 is similar after observing the definition
$V_{i}=\\{v\in
H^{1}(\Omega_{i}):\left.(\hat{T}_{\partial\Omega_{i}}v)\right|_{\partial\Omega_{i}\setminus\Gamma}=0\\}$
and employing the
$L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\partial\Omega_{i}\setminus\Gamma)$
inner product in Equation 12 instead of
$L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\partial\Omega_{i})$.

To prove Equation 11 first observe that $E_{i}$ can be interpreted as a map
from $L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda$ into
$L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1/2}(\partial\Omega_{i})$, by Lemma 2.
This immediately gives the inclusion from left to right. Conversely, assume
that $\mu\in L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Gamma)$ with
$E_{i}\mu\in L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1/2}(\partial\Omega_{i})$.
By Equation 8, we have the representation

$E_{i}\mu=\sum_{k=0}^{\infty}x_{k}\otimes z_{k},\quad z_{k}\in
H^{1/2}(\partial\Omega_{i}).$

As
$\left.(E_{i}\mu)\right|_{(\partial\Omega_{i}\setminus\Gamma)\times\mathbb{R}}=0$,
the orthonormality of $\\{x_{k}\\}_{k\geq 0}$ yields that

$0=(E_{i}\mu\,,\,x_{k}\otimes
z_{k})_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\partial\Omega_{i}\setminus\Gamma)}=\|z_{k}\|^{2}_{L^{2}(\partial\Omega_{i}\setminus\Gamma)},$

i.e., $\left.z_{k}\right|_{\partial\Omega_{i}\setminus\Gamma}=0$. As
$\hat{E}_{i}$ is an isometry from $\Lambda$ onto $\\{\mu\in
H^{1/2}(\partial\Omega_{i}):\left.\mu\right|_{\partial\Omega_{i}\setminus\Gamma}=0\\}$;
compare with (EHEE22, , Lemma 4.1), we obtain that
$\left.z_{k}\right|_{\Gamma}\in\Lambda$ for all $k=0,1,\dots$ Since

$\mu=\left.(E_{i}\mu)\right|_{\Gamma\times\mathbb{R}}=\sum_{k=0}^{\infty}x_{k}\otimes\left.z_{k}\right|_{\Gamma}\quad\text{in
}L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Gamma)$

and $\\{\sum_{k=0}^{n}x_{k}\otimes\left.z_{k}\right|_{\Gamma}\\}_{n\geq 0}$ is
a Cauchy sequence in $L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda$, we
conclude that$\mu\in L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda$. ∎

###### Lemma 4

If Assumption 1 holds, then $Z$ is dense in $L^{2}(\Gamma\times\mathbb{R})$.

###### Proof

The spaces $H^{1/4}(\mathbb{R})$ and $\Lambda$ are dense in
$L^{2}(\mathbb{R})$ and $L^{2}(\Gamma)$, respectively; see (tartar, , Lemma
15.10) and (EHEE22, , Lemma 4.2). Therefore, by (weidmann, , Theorem 3.12),
the corresponding algebraic tensor space $H^{1/4}(\mathbb{R})\otimes\Lambda$
is dense in $L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Gamma)\cong
L^{2}(\Gamma\times\mathbb{R})$. The density of $Z$ in
$L^{2}(\Gamma\times\mathbb{R})$ then follows as
$H^{1/4}(\mathbb{R})\otimes\Lambda\subset Z$. ∎

###### Lemma 5

If Assumption 1 holds, then $T_{i}:W_{i}\rightarrow Z$ is bounded.

###### Proof

It follows from the density of $C^{\infty}_{0}(\mathbb{R})\otimes
C^{\infty}(\bar{\Omega}_{i})$ in
$L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})$; see (weidmann, ,
Theorem 3.12), that our definition of $T_{\partial\Omega_{i}}$ coincide with
the definition given in (costabel90, , Lemma 2.4). Hence, the restricted trace
operator

$T_{\partial\Omega_{i}}:H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})\cap
L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})\rightarrow
H^{1/4}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})\cap
L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1/2}(\partial\Omega_{i})$

is well defined and bounded.

If $v\in W_{i}$ then
$T_{i}v=\left.(T_{\partial\Omega_{i}}v)\right|_{\Gamma\times\mathbb{R}}\in
H^{1/4}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Gamma)$. As $W_{i}\subset
L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}$, we have by definition that
$T_{i}v\in L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda$, i.e.,
$T_{i}:W_{i}\rightarrow Z$. The boundedness of $T_{i}:W_{i}\rightarrow Z$ then
follows as the operators
$T_{\partial\Omega_{i}}:H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})\cap
L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})\rightarrow
H^{1/4}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})$ and
$T_{i}:L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}\rightarrow
L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda$ are bounded. ∎

###### Lemma 6

Let $X$ be a separable Hilbert space and $s\in[0,1]$. The Hilbert transform

$\mathcal{H}=\hat{\mathcal{H}}\,\tilde{\otimes}\,I:H^{s}(\mathbb{R})\,\tilde{\otimes}\,X\rightarrow
H^{s}(\mathbb{R})\,\tilde{\otimes}\,X$

is then an isomorphism.

###### Proof

According to Lemma 2 the operator $\hat{\mathcal{H}}\otimes I$ extends to a
bounded linear operator
$\mathcal{H}:H^{s}(\mathbb{R})\,\tilde{\otimes}\,X\rightarrow
H^{s}(\mathbb{R})\,\tilde{\otimes}\,X$. From Equation 7 it follows that
$\mathcal{H}:H^{s}(\mathbb{R})\,\tilde{\otimes}\,X\rightarrow
H^{s}(\mathbb{R})\,\tilde{\otimes}\,X$ is an isomorphism. ∎

###### Lemma 7

If Assumption 1 holds then the restricted operators

$\mathcal{H}_{i}:W_{i}\rightarrow
W_{i}\quad\text{and}\quad\mathcal{H}_{\Gamma}:Z\rightarrow Z$

are isomorphisms and satisfy
$T_{i}\mathcal{H}_{i}v=\mathcal{H}_{\Gamma}T_{i}v$ for all $v\in W_{i}$.

###### Proof

By different choices of $s\in[0,1]$ and separable Hilbert spaces $X$ in Lemma
6 one obtains the isomorphisms

$\displaystyle\mathcal{H}_{i}$
$\displaystyle:L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})\rightarrow
L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})$
$\displaystyle\mathcal{H}_{i}$
$\displaystyle:H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})\rightarrow
H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})$
$\displaystyle\mathcal{H}_{i}$
$\displaystyle:L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}\rightarrow
L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}.$

Since the operators coincide on $W_{i}$ we have that
$\mathcal{H}_{i}:W_{i}\rightarrow W_{i}$ is an isomorphism. A similar argument
shows that $\mathcal{H}_{\Gamma}:Z\rightarrow Z$ is an isomorphism. For $v\in
L^{2}(\mathbb{R})\otimes V_{i}$ we have

$T_{i}\mathcal{H}_{i}v=(I\otimes\hat{T}_{i})(\hat{\mathcal{H}}\otimes
I_{L^{2}(\Omega_{i})})v=\hat{\mathcal{H}}\otimes\hat{T}_{i}v=(\hat{\mathcal{H}}\otimes
I_{L^{2}(\Gamma)})(I\otimes\hat{T}_{i})v=\mathcal{H}_{\Gamma}T_{i}v,$

and by density the identity holds for $v\in W_{i}\subset
L^{2}(\mathbb{R})\,\tilde{\otimes}\,V_{i}$. ∎

Finally, let $\varphi\in[0,\pi/2]$ be a parameter to be chosen later and
define

$\mathcal{H}_{i}^{\varphi}=\cos{(\varphi)}I-\sin{(\varphi)}\mathcal{H}_{i}\quad\text{and}\quad\mathcal{H}_{\Gamma}^{\varphi}=\cos{(\varphi)}I-\sin{(\varphi)}\mathcal{H}_{\Gamma}.$

It follows from Lemma 7 that $\mathcal{H}_{i}^{\varphi}:W_{i}\rightarrow
W_{i}$ and $\mathcal{H}_{\Gamma}^{\varphi}:Z\rightarrow Z$ are isomorphisms
and

$T_{i}\mathcal{H}_{i}^{\varphi}=\mathcal{H}_{\Gamma}^{\varphi}T_{i}.$ (13)

##  4 Variational space-time formulations

By Lemma 2 the extensions of the spatial gradient

$\nabla:L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})\rightarrow
L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})^{d}$

and the temporal half-derivatives

$\partial^{1/2}_{\pm}:H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})\rightarrow
L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})$

are all bounded linear operators. Note that we leave out the dependence on
$i=1,2$ on the above operators for the sake of readability.

The bilinear forms $a:W\times W\rightarrow\mathbb{R}$ and $a_{i}:W_{i}\times
W_{i}\rightarrow\mathbb{R}$, $i=1,2$, are defined by the formulas

$\displaystyle a(u,v)$
$\displaystyle=\int_{\mathbb{R}}\int_{\Omega}\partial^{1/2}_{+}u\partial^{1/2}_{-}v+\nabla
u\cdot\nabla v\,\mathrm{d}x\,\mathrm{d}t\quad\text{and}$ $\displaystyle
a_{i}(u_{i},v_{i})$
$\displaystyle=\int_{\mathbb{R}}\int_{\Omega_{i}}\partial^{1/2}_{+}u_{i}\partial^{1/2}_{-}v_{i}+\nabla
u_{i}\cdot\nabla v_{i}\,\mathrm{d}x\,\mathrm{d}t,$

respectively. For $f\in W^{*}$ the weak, or variational, formulation of the
heat equation Equation 1 is to find $u\in W$ such that

$a(u,v)=\langle f,v\rangle\quad\textrm{ for all }v\in W.$ (14)

Similarly, for $f\in(W_{i}^{0})^{*}$ the weak problems on
$\Omega_{i}\times\mathbb{R}$ is to find $u_{i}\in W_{i}^{0}$ such that

$a_{i}(u_{i},v_{i})=\langle f_{i},v_{i}\rangle\quad\textrm{ for all }v_{i}\in
W_{i}^{0}.$ (15)

Here, $X^{*}$ denotes the dual of the Hilbert space $X$ and
$\langle\cdot,\cdot\rangle$ denotes the dual pairing in $X^{*}\times X$. The
weak problem Equation 14 can be derived by multiplying Equation 1 by $v\in W$,
integrating and using the fractional integration by parts formula from Lemma 1
extended to the tensor setting. The weak formulations Equation 15 follow in
the same way.

A bilinear form $a:X\times X\rightarrow\mathbb{R}$ is referred to as coercive
in $Y$, where $X\subseteq Y$, if

$a(v,v)\geq c\|v\|_{Y}^{2}\quad\textrm{for all }v\in X,$

and coercive-equivalent in $X$ with the isomorphism $A:X\rightarrow X$ if

$a(v,Av)\geq c\|v\|_{X}^{2}\quad\textrm{for all }v\in X.$

The following result follows the argumentation in (larsson15, , Section 2.8),
which is a summary of the coercive-equivalency idea from fontesthesis .

###### Lemma 8

Let Assumption 1 be valid. The bilinear forms $a_{i}$ are then bounded in
$W_{i}$, coercive in $L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})$,
and coercive-equivalent in $W_{i}$ with the isomorphisms
$\mathcal{H}_{i}^{\varphi}$ for a sufficiently small $\varphi>0$. An analogous
result holds for the bilinear form $a$. In particular there exists unique
solutions to Equations 14 and 15, respectively.

###### Proof

We will only consider the case with the bilinear form $a_{i}$, as the same
proof holds for $a$. Here, the boundedness of $a_{i}$ on $W_{i}$ follows
directly by the Cauchy–Schwarz inequality together with the boundedness of the
operators $\nabla$ and $\partial^{1/2}_{\pm}$.

To prove the coercivity properties first observe that the identities in Lemma
1 can trivially be validated on $H^{1/2}(\mathbb{R})\otimes L^{2}(\Omega_{i})$
and extended to $H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})$ by
density and the boundedness of the related operators. That is,

$\partial^{1/2}_{+}v=-\partial^{1/2}_{-}\mathcal{H}_{i}v\quad\text{and}\quad(\partial^{1/2}_{+}v\,,\,\partial^{1/2}_{-}v)_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})}=0$

for every $v\in H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})$.
Second, as Assumption 1 is valid, a similar density argument together with the
observations regarding the equivalent norms in Section 2 yields the “extended”
Poincaré’s inequality

$\|\nabla v\|_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})^{d}}\geq
c\|v\|_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})}\quad\text{for
all }v\in L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i}),$

and the fact that

$v\mapsto\bigl{(}\|\partial^{1/2}_{+}v\|^{2}_{H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})}+\|\nabla
v\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})^{d}}\bigr{)}^{1/2}$

is an equivalent norm on $W_{i}$. Third, by Lemma 6, $\nabla\mathcal{H}_{i}$
is a bounded operator from
$L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})$ into
$L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})^{d}$. With these
observations we have the bound

$\displaystyle a_{i}(v,\mathcal{H}_{i}^{\varphi}v)$
$\displaystyle=\cos(\varphi)(\partial^{1/2}_{+}v\,,\,\partial^{1/2}_{-}v)_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})}$
$\displaystyle\quad+\sin(\varphi)(\partial^{1/2}_{+}v\,,\,-\partial^{1/2}_{-}\mathcal{H}_{i}v)_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})}$
$\displaystyle\quad+\cos(\varphi)\|\nabla
v\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})^{d}}-\sin(\varphi)(\nabla
v\,,\,\nabla\mathcal{H}_{i}v)_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})^{d}}$
$\displaystyle\geq\sin(\varphi)\|\partial^{1/2}_{+}v\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})}+\bigl{(}\cos(\varphi)-c\sin(\varphi)\bigr{)}\|\nabla
v\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})^{d}}$

for every $v\in W_{i}$. Thus, by choosing a sufficiently small $\varphi>0$ we
have the sought after coercive-equivalency

$a_{i}(v,\mathcal{H}_{i}^{\varphi}v)\geq c\|v\|_{W_{i}}^{2}\quad\text{for
every }v\in W_{i},$

and choosing $\varphi=0$ yields the coercivity bound

$a_{i}(v,v)\geq
c\|v\|_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\mathbb{R})}^{2}\quad\text{for
every }v\in W_{i}.$

The derived properties of $a_{i}$ yield the existence and uniqueness of a
solution for Equation 15, which follows by (larsson15, , Corollary 2.2). ∎

The next lemma is required in order to consider non-homogeneous boundary
values on the interface.

###### Lemma 9

Suppose that Assumption 1 holds. For any $\eta\in Z$ there exists $u_{i}\in
W_{i}$ such that $T_{i}u_{i}=\eta$ and

$a_{i}(u_{i},v_{i})=0\quad\text{for all }v_{i}\in W_{i}^{0}.$

Moreover, $\|u_{i}\|_{W_{i}}\leq C\|\eta\|_{Z}$.

###### Proof

From Equation 11 we have that $E_{i}\eta\in
H^{1/4}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\partial\Omega_{i})\cap
L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1/2}(\partial\Omega_{i})$. By
(costabel90, , Theorem 2.9 & Remark 2.10), there exists a weak solution

$u_{i}\in H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega_{i})\cap
L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})$

such that $T_{\partial\Omega_{i}}u_{i}=E_{i}\eta$. It follows from Equation 10
that $u_{i}\in W_{i}$. The bound $\|u_{i}\|_{W_{i}}\leq C\|\eta\|_{Z}$ follows
by (costabel90, , p. 515) together with the definitions of the related norms.
∎

According to Lemma 9 there exists a bounded linear operator
$F_{i}:Z\rightarrow W_{i}:\eta\mapsto u_{i}$ such that

$a(F_{i}\eta,v_{i})=0\quad\textrm{ for all }v_{i}\in W_{i}^{0},$ (16)

with left inverse $T_{i}$, i.e., $T_{i}F_{i}\eta=\eta$ for $\eta\in Z$. From
Lemma 8 there also exists a bounded linear operator
$G_{i}:(W_{i}^{0})^{*}\rightarrow W_{i}^{0}:f_{i}\mapsto u_{i}$ such that

$a_{i}(G_{i}f_{i},v_{i})=\langle f_{i},v_{i}\rangle\quad\textrm{for all
}v_{i}\in W_{i}^{0}.$ (17)

These two operators are the main ingredients for stating the transmission
problem and then reformulating the problem on the space-time interface
$\Gamma\times\mathbb{R}$. Before moving on to the transmission problem it is
necessary to prove that we can “paste” together functions in our
$H^{1/2}$-framework. This is the purpose of the following lemma.

###### Lemma 10

Suppose that Assumption 1 holds. Let $(v_{1},v_{2})\in W_{1}\times W_{2}$ and
define $v=\\{v_{i}\textrm{ on }\Omega_{i}\times\mathbb{R},\,i=1,2\\}$. If
$T_{1}v_{1}=T_{2}v_{2}$ then $v\in W$. Conversely, let $v\in W$ and define
$v_{i}=\left.v\right|_{\Omega_{i}\times\mathbb{R}}$. Then $T_{i}v_{i}\in Z$
and $T_{1}v_{1}=T_{2}v_{2}$.

###### Proof

Assume that $v_{i}\in W_{i}$, $i=1,2$, and $T_{1}v_{1}=T_{2}v_{2}$, then

$v=\\{v_{i}\textrm{ on }\Omega_{i}\times\mathbb{R},\,i=1,2\\}\in
L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega).$

In order to prove the $L^{2}(\mathbb{R})\,\tilde{\otimes}\,V$-regularity of
$v$, let $\\{x_{k}\\}_{k\geq 0}$ be an orthonormal basis of
$L^{2}(\mathbb{R})$ and $\\{(y_{i})_{\ell}\\}_{\ell\geq 0}$ be orthonormal
bases of $V_{i}$, $i=1,2$. The corresponding elements
$\\{(z_{i})_{k}\\}_{k\geq 0}\subset V_{i}$ are defined as in Equation 8. This
yields the representation

$v_{i}=\sum_{k=0}^{\infty}x_{k}\otimes(z_{i})_{k}$

and the equality

$0=(T_{1}v_{1}-T_{2}v_{2}\,,\,x_{k}\otimes(\hat{T}(z_{1})_{k}-\hat{T}_{2}(z_{2})_{k})_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Gamma)}=\|\hat{T}_{1}(z_{1})_{k}-\hat{T}_{2}(z_{2})_{k}\|^{2}_{L^{2}(\Gamma)}.$

That is, $\hat{T}_{1}(z_{1})_{k}=\hat{T}_{2}(z_{2})_{k}$, for all
$k=0,1,\ldots$, and from (EHEE22, , Lemma 4.6) it follows that
$z_{k}=\\{(z_{i})_{k}\textrm{ on }\Omega_{i}\times\mathbb{R},\,i=1,2\\}\in V$.
We also have the identification

$v=\sum_{k=0}^{\infty}x_{k}\otimes z_{k}\quad\text{in
}L^{2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega).$

As
$\|z_{k}\|^{2}_{V}=\|(z_{1})_{k}\|^{2}_{V_{1}}+\|(z_{2})_{k}\|^{2}_{V_{2}}$,
one has that $\\{\sum_{k=0}^{n}x_{k}\otimes z_{k}\\}_{n\geq 0}$ is a Cauchy
sequence in $L^{2}(\mathbb{R})\,\tilde{\otimes}\,V$. Hence, $v$ is also an
element in $L^{2}(\mathbb{R})\,\tilde{\otimes}\,V$.

The $H^{1/2}(\mathbb{R})\,\tilde{\otimes}\,L^{2}(\Omega)$-regularity of $v$
follows in a similar fashion, by expanding the $v_{i}$ elements in terms of an
orthonormal basis $\\{x_{k}\\}_{k\geq 0}$ of $H^{1/2}(\mathbb{R})$ and an
orthonormal basis $\\{(y_{i})_{k}\\}_{k\geq 0}$ of $L^{2}(\Omega_{i})$
together with the observation that
$\|z_{k}\|^{2}_{L^{2}(\Omega)}=\|(z_{1})_{k}\|^{2}_{L^{2}(\Omega_{1})}+\|(z_{2})_{k}\|^{2}_{L^{2}(\Omega_{2})}$.
In conclusion, $v\in W$.

Conversely, let $v\in W$. From Equation 10 we get that
$v_{i}=\left.v\right|_{\Omega_{i}\times\mathbb{R}}$ is an element in $W_{i}$
and, by Lemma 5, $T_{i}v_{i}\in Z$. Let $\\{x_{k}\\}_{k\geq 0}$ be an
orthonormal basis of $L^{2}(\mathbb{R})$ and $\\{y_{\ell}\\}_{\ell\geq 0}$ an
orthonormal basis of $V$. The related elements $\\{z_{k}\\}_{k\geq 0}\subset
V$ are given by Equation 8 and we obtain

$v=\sum_{k=0}^{\infty}x_{k}\otimes z_{k}\quad\text{and}\quad
v_{i}=\sum_{k=0}^{\infty}x_{k}\otimes(z_{i})_{k},$

where $(z_{i})_{k}=\left.z_{k}\right|_{\Omega_{i}\times\mathbb{R}}\in V_{i}$.
By (EHEE22, , Lemma 4.5) it follows that
$\hat{T}_{1}(z_{1})_{k}=\hat{T}_{2}(z_{2})_{k}$ for every $k=0,1,\ldots$
Therefore

$T_{1}v_{1}=\sum_{k,\ell=0}^{\infty}x_{k}\otimes\hat{T}_{1}(z_{1})_{k}=\sum_{k,\ell=0}^{\infty}x_{k}\otimes\hat{T}_{2}(z_{2})_{k}=T_{2}v_{2},$

and the sought after equality is obtained. ∎

The existence and uniqueness theory holds for any $f\in W^{*}$, but for the
convergence analysis we need a stricter requirement. For simplicity we assume
the following:

###### Assumption 2

The source term $f$ is an element in $L^{2}(\Omega\times\mathbb{R})$.

With Assumption 2 we can define
$f_{i}=\left.f\right|_{\Omega_{i}\times\mathbb{R}}$, which we will assume to
hold for the remainder of the analysis. The weak transmission problem is then
to find $(u_{1},u_{2})\in W_{1}\times W_{2}$ such that

$\left\\{\begin{aligned}
a_{i}(u_{i},v_{i})&=(f_{i},v_{i})_{L^{2}(\Omega_{i})}&&\text{for all }v_{i}\in
W_{i}^{0},\,i=1,2,\\\ T_{1}u_{1}&=T_{2}u_{2},&&\\\
\textstyle\sum_{i=1}^{2}a_{i}(&u_{i},F_{i}\mu)-(f_{i},F_{i}\mu)_{L^{2}(\Omega_{i})}=0&&\text{for
all }\mu\in Z.\end{aligned}\right.$ (18)

###### Lemma 11

Suppose that Assumptions 1 and 2 hold. Then the weak heat equation is
equivalent to the weak transmission problem in the following way: If $u$
solves Equation 14 then
$(u_{1},u_{2})=(\left.u\right|_{\Omega_{1}\times\mathbb{R}},\left.u\right|_{\Omega_{2}\times\mathbb{R}})$
solves Equation 18. Conversely, if $(u_{1},u_{2})$ solves Equation 18 then
$u=\\{u_{i}\textrm{ on }\Omega_{i}\times\mathbb{R}\,,i=1,2\\}$ solves Equation
14. In particular, there exists a unique solution to Equation 18.

###### Remark 2

The proof of Lemma 11 follows by the same argument as (quarteroni, , Lemma
1.2.1) and requires that Lemma 10 holds. This is one of the reason that the
analysis is performed in the $H^{1/2}$-setting. As already stated in the
introduction, analogous results to Lemma 10 are not always true; see for
instance (costabel90, , Example 2.14) for a counterexample in
$H^{1}(\mathbb{R})\,\tilde{\otimes}\,H^{-1}(\Omega)\cap
L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega)$.

The weak form of the space-time Robin–Robin method is the following: Let
$u_{2}^{0}\in W_{2}$ be an initial guess and $s>0$ be the fixed method
parameter. For each $n=1,2,\ldots$ find functions $(u_{1}^{n},u_{2}^{n})\in
W_{1}\times W_{2}$ such that

$\left\\{\begin{aligned}
&a_{1}(u_{1}^{n+1},v_{1})=(f_{1},v_{1})_{L^{2}(\Omega_{1}\times\mathbb{R})}&&\text{for
all }v_{1}\in W_{1}^{0},\\\
&a_{1}(u_{1}^{n+1},F_{1}\mu)-(f_{1},F_{1}\mu)_{L^{2}(\Omega_{1}\times\mathbb{R})}+a_{2}(u_{2}^{n},F_{2}\mu)&&\\\
&\quad-(f_{2},F_{2}\mu)_{L^{2}(\Omega_{2}\times\mathbb{R})}=s(T_{2}u_{2}^{n}-T_{1}u_{1}^{n+1},\mu)_{L^{2}(\Gamma\times\mathbb{R})}&&\text{for
all }\mu\in Z,\\\\[5.0pt]
&a_{2}(u_{2}^{n+1},v_{2})=(f_{2},v_{2})_{L^{2}(\Omega_{2}\times\mathbb{R})}&&\text{for
all }v_{2}\in W_{2}^{0},\\\
&a_{2}(u_{2}^{n+1},F_{2}\mu)-(f_{2},F_{2}\mu)_{L^{2}(\Omega_{2}\times\mathbb{R})}+a_{1}(u_{1}^{n+1},F_{1}\mu)&&\\\
&\quad-(f_{1},F_{1}\mu)_{L^{2}(\Omega_{1}\times\mathbb{R})}=s(T_{1}u_{1}^{n+1}-T_{2}u_{2}^{n+1},\mu)_{L^{2}(\Gamma\times\mathbb{R})}&&\text{for
all }\mu\in Z.\end{aligned}\right.$ (19)

The weak forms of the Dirichlet–Neumann and Neumann–Neumann methods can be
given in a similar form, but are not stated here since they are the same as
their elliptical counterparts; see (quarteroni, , Chapter 1.3).

##  5 Time-dependent Steklov–Poincaré operators

The goal is now to reformulate the transmission problem and the space-time
domain decomposition methods to problems and methods given on the interface
$\Gamma\times\mathbb{R}$. In order to do so, we first introduce the time-
dependent Steklov–Poincaré operators $S_{i}:Z\rightarrow Z^{*}$ and the
functionals $\chi_{i}\in Z^{*}$ by

$\langle
S_{i}\eta,\mu\rangle=a_{i}(F_{i}\eta,F_{i}\mu)\quad\text{and}\quad\langle\chi_{i},\mu\rangle=(f_{i},F_{i}\mu)_{L^{2}(\Omega_{i}\times\mathbb{R})}-a_{i}(G_{i}f_{i},F_{i}\mu),$

respectively. We also write $S=S_{1}+S_{2}$ and $\chi=\chi_{1}+\chi_{2}$. We
can then introduce the weak Steklov–Poincaré equation $S\eta=\chi$ in $Z^{*}$
or, equivalently,

$\sum_{i=1}^{2}\langle
S_{i}\eta,\mu\rangle=\sum_{i=1}^{2}\langle\chi_{i},\mu\rangle\quad\textrm{for
all }\mu\in Z.$ (20)

###### Remark 3

The Steklov–Poincaré operators do not depend on the choice of extension
$F_{i}$. For an arbitrary extension $R_{i}:Z\rightarrow W_{i}$ such that
$T_{i}R_{i}\mu=\mu$ we have, by Equation 9, that $F_{i}\mu-R_{i}\mu\in
W_{i}^{0}$. Combining this with Equation 16 implies that

$\langle
S_{i}\eta,\mu\rangle=a_{i}(F_{i}\eta,F_{i}\mu)=a_{i}(F_{i}\eta,F_{i}\mu-
R_{i}\mu)+a_{i}(F_{i}\eta,R_{i}\mu)=a_{i}(F_{i}\eta,R_{i}\mu).$

The Steklov–Poincaré operators have the corresponding properties as the
bilinear forms $a_{i}$ in Lemma 8.

###### Lemma 12

Suppose that Assumptions 1 and 2 hold. Then the time-dependent
Steklov–Poincaré operator $S_{i}$ is bounded and satisfies the coercivity
bound

$\langle S_{i}\eta,\eta\rangle\geq
c\|F_{i}\eta\|_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})}^{2}\quad\text{for
all }\eta\in Z.$ (21)

Moreover, $S_{i}$ is coercive-equivalent in $Z$ with the isomorphism
$\mathcal{H}_{\Gamma}^{\varphi}$ for $\varphi>0$ small enough. That is,

$\langle S_{i}\eta,\mathcal{H}_{\Gamma}^{\varphi}\eta\rangle\geq
c\|\eta\|_{Z}^{2}\quad\text{for all }\eta\in Z.$

Analogous results hold for $S$.

###### Proof

Throughout the proof, let $\eta,\mu\in Z$ be arbitrary elements. The
boundedness of $S_{i}$ is proved by Lemmas 8 and 9, since

$|\langle S_{i}\eta,\mu\rangle|=|a_{i}(F_{i}\eta,F_{i}\mu)|\leq
C\|F_{i}\eta\|_{W_{i}}\|F_{i}\mu\|_{W_{i}}\leq C\|\eta\|_{Z}\|\mu\|_{Z}.$

We also obtain the coercivity bound

$\langle S_{i}\eta,\eta\rangle=a_{i}(F_{i}\eta,F_{i}\eta)\geq
c\|F_{i}\eta\|_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})}^{2},$

using Lemma 8. From Equation 13 we have

$T_{i}(F_{i}\mathcal{H}_{\Gamma}^{\varphi}-\mathcal{H}_{i}^{\varphi}F_{i})\eta=T_{i}F_{i}\mathcal{H}_{\Gamma}^{\varphi}\eta-
T_{i}\mathcal{H}_{i}^{\varphi}F_{i}\eta=\mathcal{H}_{\Gamma}^{\varphi}\eta-\mathcal{H}_{\Gamma}^{\varphi}T_{i}F_{i}\eta=0$

and therefore
$(F_{i}\mathcal{H}_{\Gamma}^{\varphi}-\mathcal{H}_{i}^{\varphi}F_{i})\eta\in
W_{i}^{0}$ by Equation 9. This together with Equation 16 implies that

$\displaystyle a_{i}(F_{i}\eta,F_{i}\mathcal{H}_{\Gamma}^{\varphi}\eta)$
$\displaystyle=a_{i}\bigl{(}F_{i}\eta,(F_{i}\mathcal{H}_{\Gamma}^{\varphi}-\mathcal{H}_{i}^{\varphi}F_{i})\eta\bigr{)}+a_{i}(F_{i}\eta,\mathcal{H}_{i}^{\varphi}F_{i}\eta)$
$\displaystyle=a_{i}(F_{i}\eta,\mathcal{H}_{i}^{\varphi}F_{i}\eta).$

Then Lemmas 5 and 8 give that

$\langle
S_{i}\eta,\mathcal{H}_{\Gamma}^{\varphi}\eta\rangle=a_{i}(F_{i}\eta,\mathcal{H}_{i}^{\varphi}F_{i}\eta)\geq
c\|F_{i}\eta\|_{W_{i}}^{2}\geq c\|T_{i}F_{i}\eta\|_{Z}^{2}=c\|\eta\|_{Z}^{2}.$

The bounds for $S$ follow by summing the bounds for $S_{1}$ and $S_{2}$. ∎

From Lemma 4 we have the Gelfand triple

$Z\hookrightarrow L^{2}(\Gamma\times\mathbb{R})\cong
L^{2}(\Gamma\times\mathbb{R})^{*}\hookrightarrow Z^{*}.$

We introduce the Riesz isomorphism $J:L^{2}(\Gamma\times\mathbb{R})\rightarrow
L^{2}(\Gamma\times\mathbb{R})^{*}$ and note that the Gelfand triple implies
the identity

$\langle
J\eta,\mu\rangle=(\eta,\mu)_{L^{2}(\Gamma\times\mathbb{R})}\quad\text{ for all
}\eta\in L^{2}(\Gamma\times\mathbb{R}),\,\mu\in Z.$ (22)

We trivially also have the bounds

$\langle
J\eta,\mu\rangle\leq\|\eta\|_{L^{2}(\Gamma\times\mathbb{R})}\|\mu\|_{L^{2}(\Gamma\times\mathbb{R})}\quad\text{
for all }\eta\in L^{2}(\Gamma\times\mathbb{R}),\,\mu\in Z$ (23)

and

$\langle
J\eta,\eta\rangle\geq\|\eta\|_{L^{2}(\Gamma\times\mathbb{R})}^{2}\quad\text{
for all }\eta\in Z.$ (24)

###### Theorem 5.1

Suppose that Assumptions 1 and 2 hold and let $s\geq 0$. Then the operators
$sJ+S_{i},sJ+S:Z\rightarrow Z^{*}$ are isomorphisms. In particular, there
exists a unique solution to the weak Steklov–Poincaré equation Equation 20.

###### Proof

For simplicity we assume that $s=0$. Then $S_{i}$ is bijective if and only if
for any $z\in Z^{*}$ there exists a unique solution $\eta\in Z$ to

$\langle S_{i}\eta,\mu\rangle=\langle z,\mu\rangle\quad\textrm{ for all
}\mu\in Z.$

Since $\mathcal{H}_{\Gamma}^{\varphi}:Z\rightarrow Z$ is an isomorphism this
is equivalent to

$\langle S_{i}\eta,\mathcal{H}_{\Gamma}^{\varphi}\mu\rangle=\langle
z,\mathcal{H}_{\Gamma}^{\varphi}\mu\rangle\quad\textrm{ for all }\mu\in Z.$
(25)

Note that $\mu\mapsto\langle z,\mathcal{H}_{\Gamma}^{\varphi}\mu\rangle$ is
also an element in $Z^{*}$. Moreover, by Lemma 12 the bilinear form
$(\eta,\mu)\mapsto\langle S_{i}\eta,\mathcal{H}_{\Gamma}^{\varphi}\mu\rangle$
is bounded and coercive in $Z$. Therefore, by the Lax–Milgram lemma (lax, ,
Chapter 6, Theorem 6) there exists a unique solution $\eta$ to Equation 25. It
follows from Lemma 12 that $S_{i}$ and $S_{i}^{-1}$ are bounded and therefore
$S_{i}$ is an isomorphism. Since we have Equations 23 and 24 the proof can be
generalized to any $s\geq 0$. The statement for $S$ is proven similarly to
$S_{i}$. ∎

###### Lemma 13

Suppose that Assumptions 1 and 2 hold. The weak transmission problem and the
weak Steklov–Poincaré equation are equivalent in the following way: If
$(u_{1},u_{2})$ solves Equation 18 then $\eta=T_{i}u_{i}$ solves Equation 20.
Conversely, if $\eta$ solves Equation 20 then
$(u_{1},u_{2})=(F_{1}\eta+G_{1}f_{1},F_{2}\eta+G_{2}f_{2})$ solves Equation
18.

The proof of Lemma 13 is immediate after writing out the definitions of the
Steklov–Poincaré operators $S_{i}$, $i=1,2$.

We will now introduce the iterations on the interface $\Gamma\times\mathbb{R}$
and show equivalence to the corresponding domain decomposition methods. The
first one is the weak Peaceman–Rachford iteration, which is given by finding
$(\eta^{n}_{1},\eta^{n}_{2})\in Z\times Z$, for $n=1,2,\ldots,$ such that

$\left\\{\begin{aligned}
\langle(sJ+S_{1})\eta_{1}^{n+1}-\chi_{1},\mu\rangle&=\langle(sJ-
S_{2})\eta_{2}^{n}+\chi_{2},\mu\rangle\quad&\text{for all }\mu\in Z,\\\
\langle(sJ+S_{2})\eta_{2}^{n+1}-\chi_{2},\mu\rangle&=\langle(sJ-
S_{1})\eta_{1}^{n+1}+\chi_{1},\mu\rangle\quad&\text{for all }\mu\in
Z.\end{aligned}\right.$ (26)

Here, $\eta_{2}^{0}\in Z$ is an initial guess.

###### Lemma 14

Suppose that Assumptions 1 and 2 hold. Then the weak forms of the Robin–Robin
method and the Peaceman–Rachford iteration are equivalent in the following
way: If $(u_{1}^{n},u_{2}^{n})_{n\geq 1}$ solves Equation 19 then
$(\eta_{1}^{n},\eta_{2}^{n})_{n\geq 1}$, defined as
$\eta_{i}^{n}=T_{i}u^{n}_{i}$, solves Equation 26 with
$\eta_{2}^{0}=T_{2}u^{0}_{2}$. Conversely, if
$(\eta_{1}^{n},\eta_{2}^{n})_{n\geq 1}$ solves Equation 26 then
$(u_{1}^{n},u_{2}^{n})_{n\geq 1}$, defined as
$u_{i}^{n}=F_{i}\eta_{i}^{n}+G_{i}f_{i}$, solves Equation 19 with
$u_{2}^{0}=F_{2}\eta_{2}^{0}+G_{2}f_{2}$.

###### Proof

Let $(u_{1}^{n},u_{2}^{n})_{n\geq 1}\subset W_{1}\times W_{2}$ be a weak
Robin–Robin approximation solving Equation 19 and define
$\eta_{i}^{n}=T_{i}u^{n}_{i}\in Z$. This together with the first and third
equation in Equation 19 yields the identification
$u_{i}^{n}=F_{i}\eta_{i}^{n}+G_{i}f_{i}$. Entering these observations into the
second and fourth equation in Equation 19 gives the equalities

$\displaystyle
s(\eta_{1}^{n+1},\mu)_{L^{2}(\Gamma\times\mathbb{R})}+a_{1}(F_{1}\eta_{1}^{n+1}+G_{1}f_{1},F_{1}\mu)-(f_{1},F_{1}\mu)_{L^{2}(\Omega_{1}\times\mathbb{R})}$
(27)
$\displaystyle=s(\eta_{2}^{n},\mu)_{L^{2}(\Gamma\times\mathbb{R})}-a_{2}(F_{2}\eta_{2}^{n}+G_{2}f_{2},F_{2}\mu)+(f_{2},F_{2}\mu)_{L^{2}(\Omega_{2}\times\mathbb{R})},\quad\text{and}$
$\displaystyle
s(\eta_{2}^{n+1},\mu)_{L^{2}(\Gamma\times\mathbb{R})}+a_{2}(F_{2}\eta_{2}^{n+1}+G_{2}f_{2},F_{2}\mu)-(f_{2},F_{2}\mu)_{L^{2}(\Omega_{2}\times\mathbb{R})}$
$\displaystyle=s(\eta_{1}^{n+1},\mu)_{L^{2}(\Gamma\times\mathbb{R})}-a_{1}(F_{1}\eta_{1}^{n+1}+G_{1}f_{1},F_{1}\mu)+(f_{1},F_{1}\mu)_{L^{2}(\Omega_{1}\times\mathbb{R})}$

for all $\mu\in Z$. By the definitions of $S_{i}$ and $\chi_{i}$, the
equalities Equation 27 are equivalent to Equation 26, i.e.,
$(\eta_{1}^{n},\eta_{2}^{n})_{n\geq 1}$ solves Equation 26 with
$\eta_{2}^{0}=T_{2}u^{0}_{2}$.

Conversely, let $(\eta_{1}^{n},\eta_{2}^{n})_{n\geq 1}\subset Z\times Z$ be a
weak Peaceman–Rachford approximation solving Equation 26 and define
$u_{i}^{n}=F_{i}\eta_{i}^{n}+G_{i}f_{i}$. This definition ensures that the
first and third equation in Equation 19 are fulfilled, as well as
$T_{i}u_{i}^{n}=\eta_{i}^{n}$. As already observed, one Peaceman–Rachford
iterate Equation 26 is equivalent to Equation 27. Inserting the definition of
$u_{i}^{n}$ into Equation 27 then yields that the second and fourth equation
in Equation 19, i.e., $(u_{1}^{n},u_{2}^{n})_{n\geq 1}$ solves Equation 19
with $u_{2}^{0}=F_{2}\eta_{2}^{0}+G_{2}f_{2}$. ∎

###### Corollary 1

Suppose that Assumptions 1 and 2 hold. Then the weak forms of the Robin–Robin
method and the Peaceman–Rachford iteration are well defined, that is,
Equations 19 and 26 have unique solutions at each iteration.

###### Proof

It follows from Theorem 5.1 that Equation 26 has a unique solution at each
iteration and, by Lemma 14, so does Equation 19. ∎

The theory above applies to the weak forms of the Dirichlet–Neumann and
Neumann–Neumann methods as well. To this end, we introduce two preconditioned
fixed point iterations on the interface $\Gamma\times\mathbb{R}$. The first
one is given by finding $\eta^{n}\in Z$, for $n=1,2,\ldots,$ such that

$\langle S_{2}\eta^{n+1},\mu\rangle=\langle
S_{2}\eta^{n},\mu\rangle+s_{0}\langle-S\eta^{n}+\chi,\mu\rangle\quad\text{ for
all }\mu\in Z.$ (28)

Here, $\eta^{0}$ is an initial guess and $s_{0}\in(0,1)$ is a method
parameter. The second one is given by finding
$(\lambda_{1}^{n},\lambda_{2}^{n},\eta^{n})\in Z\times Z\times Z$, for
$n=1,2,\ldots,$ such that

$\left\\{\begin{aligned} \langle
S_{i}\lambda_{i}^{n+1},\mu\rangle&=\langle-S\eta^{n}+\chi,\mu\rangle&&\text{
for all }\mu\in Z,\,i=1,2,\\\
\eta^{n+1}&=\eta^{n}+s_{1}\lambda_{1}^{n+1}+s_{2}\lambda_{2}^{n+1}.&&\end{aligned}\right.$
(29)

Again, $\eta^{0}$ is an initial guess and $s_{1},s_{2}>0$ are method
parameters.

###### Lemma 15

Suppose that Assumptions 1 and 2 hold. The weak Dirichlet–Neumann method is
equivalent to the interface iteration Equation 28 and the weak Neumann–Neumann
method is equivalent to the interface iteration Equation 29.

Here, the definition of the methods being equivalent and the proofs thereof
follow in the same fashion as for Lemma 14. By Theorems 5.1 and 15, we now
trivially have the following existence and uniqueness results.

###### Corollary 2

Suppose that Assumptions 1 and 2 hold. The weak forms of the Dirichlet–Neumann
and Neumann–Neumann methods, and their corresponding interface iterations
Equation 28 and Equation 29, respectively, are all well defined.

The standard convergence theories of the Dirichlet–Neumann and Neumann–Neumann
methods (quarteroni, , Chapter 4) do not apply for the heat equation since the
time-dependent Steklov–Poincaré operators are not symmetric. In the following
sections we instead prove convergence results for Robin–Robin-type methods.

##  6 Convergence of the Robin–Robin method

The variational framework appears to be too general for analyzing the
convergence of the Peaceman–Rachford iteration. Therefore, we introduce the
Steklov–Poincaré operators as affine unbounded operators on
$L^{2}(\Gamma\times\mathbb{R})$ before we prove that the iteration converges.
To this end, let

$\displaystyle D(\mathcal{S}_{i})$ $\displaystyle=\\{\eta\in
Z:S_{i}\eta-\chi_{i}\in L^{2}(\Gamma\times\mathbb{R})^{*}\\},$ $\displaystyle
D(\mathcal{S})$ $\displaystyle=\\{\eta\in Z:S\eta-\chi\in
L^{2}(\Gamma\times\mathbb{R})^{*}\\}$

and define the unbounded affine operators

$\displaystyle\mathcal{S}_{i}$ $\displaystyle:D(\mathcal{S}_{i})\subseteq
L^{2}(\Gamma\times\mathbb{R})\rightarrow
L^{2}(\Gamma\times\mathbb{R}):\eta\mapsto J^{-1}(S_{i}\eta-\chi_{i}),$
$\displaystyle\mathcal{S}$ $\displaystyle:D(\mathcal{S})\subseteq
L^{2}(\Gamma\times\mathbb{R})\rightarrow
L^{2}(\Gamma\times\mathbb{R}):\eta\mapsto J^{-1}(S\eta-\chi).$

In $L^{2}(\Gamma\times\mathbb{R})$ the Steklov–Poincaré equation is to find
$\eta\in D(\mathcal{S})$ such that

$\mathcal{S}\eta=0,$ (30)

and the Peaceman–Rachford iteration takes the following form: For each
$n=1,2,\dots$ find $(\eta_{1}^{n},\eta_{2}^{n})\in D(\mathcal{S}_{1})\times
D(\mathcal{S}_{2})$ such that

$\left\\{\begin{aligned}
(sI+\mathcal{S}_{1})\eta_{1}^{n+1}&=(sI-\mathcal{S}_{2})\eta_{2}^{n},\\\
(sI+\mathcal{S}_{2})\eta_{2}^{n+1}&=(sI-\mathcal{S}_{1})\eta_{1}^{n+1}.\end{aligned}\right.$
(31)

Here, $\eta_{2}^{0}\in D(\mathcal{S}_{2})$ is a given initial guess. We now
verify that Equation 30 is indeed a restriction of the weak Steklov–Poincaré
equation Equation 20.

###### Lemma 16

Suppose that Assumptions 1 and 2 hold. If $\eta\in D(\mathcal{S})$ solves the
$L^{2}$-Steklov–Poincaré equation Equation 30, then $\eta$ also solves the
weak Steklov–Poincaré equation Equation 20.

###### Proof

From Equation 30 we have

$(J^{-1}(S\eta-\chi),\mu)_{L^{2}(\Gamma\times\mathbb{R})}=0\quad\text{ for all
}\mu\in L^{2}(\Gamma\times\mathbb{R}).$

Therefore, we get by Equation 22 that

$\langle
S\eta-\chi,\mu\rangle=(J^{-1}(S\eta-\chi),\mu)_{L^{2}(\Gamma\times\mathbb{R})}=0$

for all $\mu\in Z$. ∎

A similar result holds for the Peaceman–Rachford iteration. The proof is left
out since it is similar to the proof of Lemma 16.

###### Lemma 17

Suppose that Assumptions 1 and 2 hold. If $(\eta_{1}^{n},\eta_{2}^{n})_{n\geq
1}$ solves the $L^{2}$-Peaceman–Rachford iteration Equation 31 with
$\eta_{2}^{0}\in D(\mathcal{S}_{2})$, then $(\eta_{1}^{n},\eta_{2}^{n})_{n\geq
1}$ also solves the weak Peaceman–Rachford iteration Equation 26 with the same
initial guess.

###### Lemma 18

Suppose that Assumptions 1 and 2 hold. Then $\mathcal{S}_{i},\,i=1,2$, satisfy
the monotonicity property

$(\mathcal{S}_{i}\eta-\mathcal{S}_{i}\mu,\eta-\mu)_{L^{2}(\Gamma\times\mathbb{R})}\geq
c\|F_{i}(\eta-\mu)\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})}\quad\text{
for all }\eta,\mu\in D(\mathcal{S}_{i}).$ (32)

Moreover, for any $s\geq 0$ the operators
$sI+\mathcal{S}_{i}:D(\mathcal{S}_{i})\rightarrow
L^{2}(\Gamma\times\mathbb{R}),\,i=1,2$, are bijective. Similar results hold
for $\mathcal{S}$. In particular, there exists a unique solution to Equation
30 and the iteration Equation 31 is well defined.

###### Proof

The monotonicity follows from Equations 22 and 21, since

$\displaystyle(\mathcal{S}_{i}\eta-\mathcal{S}_{i}\mu,\eta-\mu)_{L^{2}(\Gamma\times\mathbb{R})}$
$\displaystyle=\langle(S_{i}\eta-\chi_{i})-(S_{i}\mu-\chi_{i}),\eta-\mu\rangle$
$\displaystyle=\langle S_{i}(\eta-\mu),\eta-\mu\rangle\geq
c\|F_{i}(\eta-\mu)\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{i})}$

for all $\eta,\mu\in D(\mathcal{S}_{i})$. Let $\mu\in
L^{2}(\Gamma\times\mathbb{R})$ be arbitrary. Then $\chi_{i}+J\mu\in Z^{*}$
and, by Theorem 5.1, there exists a unique $\eta\in Z$ such that
$(sJ+S_{i})\eta=\chi_{i}+J\mu$ in $Z^{*}$. Rearranging yields that
$S_{i}\eta-\chi_{i}=J(\mu-s\eta)\in L^{2}(\Gamma\times\mathbb{R})^{*}$, i.e.,
$\eta\in D(\mathcal{S}_{i})$ with

$(sI+\mathcal{S}_{i})\eta=s\eta+J^{-1}(S_{i}\eta-\chi_{i})=\mu.$

Thus, we have shown that $(sI+\mathcal{S}_{i})$ is bijective. The proof for
$\mathcal{S}$ is similar and is therefore left out. ∎

In order to proceed with the convergence analysis, we require the following
regularity of the solution to the weak heat equation Equation 14.

###### Assumption 3

The functionals

$\mu\mapsto
a_{i}(\left.u\right|_{\Omega_{i}\times\mathbb{R}},F_{i}\mu)-(f_{i},F_{i}\mu)_{L^{2}(\Omega_{i}\times\mathbb{R})},\quad{i=1,2},$

are elements in $L^{2}(\Gamma\times\mathbb{R})^{*}$, where $u\in W$ is the
solution to Equation 14.

###### Remark 4

The assumption is somewhat implicit, but can be interpreted as the solution
having a normal derivative $\nabla u\cdot\nu_{i}$ on the space-time interface
belonging to $L^{2}(\Gamma\times\mathbb{R})$. This holds, for instance, if the
solution satisfies the additional regularity

$u\in W\cap L^{2}\bigl{(}\mathbb{R},H^{2}(\Omega)\bigr{)}\quad\text{or}\quad
u\in W\cap L^{2}\bigl{(}\mathbb{R},C^{1}(\bar{\Omega})\bigr{)}.$

To see this, first observe that a Lipschitz manifold $\partial\Omega_{i}$ has
a normal vector $\nu_{i}$ in $L^{\infty}(\partial\Omega_{i})^{d}$. The
additional regularity of $u$ then yields that each term
$\partial_{j}u\,(\nu_{i})_{j}$ of the normal derivative becomes an element in
$L^{2}(\Gamma\times\mathbb{R})$.

###### Lemma 19

Suppose that Assumptions 1, 2 and 3 hold. If $\eta$ is the solution to the
$L^{2}$-Steklov–Poincaré equation Equation 30 then $\eta\in
D(\mathcal{S}_{1})\cap D(\mathcal{S}_{2})$.

###### Proof

Let $\eta\in D(\mathcal{S})$ be the solution to Equation 30. By Lemmas 13 and
11, we have the identification
$\left.u\right|_{\Omega_{i}\times\mathbb{R}}=F_{i}\eta+G_{i}f_{i}$, where
$u\in W$ is the solution to the weak heat equation Equation 14. Assumption 3
then yields that

$\displaystyle\mu\mapsto\langle S_{i}\eta-\chi_{i},\mu\rangle$
$\displaystyle=a_{i}(F_{i}\eta,F_{i}\mu)+a_{i}(G_{i}f_{i},F_{i}\mu)-(f_{i},F_{i}\mu)_{L^{2}(\Omega_{i}\times\mathbb{R})}$
$\displaystyle=a_{i}(\left.u\right|_{\Omega_{i}\times\mathbb{R}},F_{i}\mu)-(f_{i},F_{i}\mu)_{L^{2}(\Omega_{i}\times\mathbb{R})}$

is an element in $L^{2}(\Gamma\times\mathbb{R})^{*}$, i.e., $\eta\in
D(\mathcal{S}_{i})$ for $i=1,2$. ∎

###### Lemma 20

Suppose that Assumptions 1, 2 and 3 hold. Let $\eta$ be the solution to the
$L^{2}$-Steklov–Poincaré equation Equation 30 and
$(\eta_{1}^{n},\eta_{2}^{n})_{n\geq 1}$ be the $L^{2}$-Peaceman–Rachford
iterates Equation 31 with $\eta_{2}^{0}\in D(\mathcal{S}_{2})$. Then we have
the limit

$(\mathcal{S}_{i}\eta^{n+1}_{i}-\mathcal{S}_{i}\eta,\eta^{n+1}_{i}-\eta)_{L^{2}(\Gamma\times\mathbb{R})}\rightarrow
0,$ (33)

as $n$ tends to infinity.

Lemma 20 is a consequence of the abstract result (lionsmercier, , Proposition
1). The latter requires the monotonicity Equation 32 and the fact that the
solution to Equation 30 satisfies $\eta\in D(\mathcal{S}_{1})\cap
D(\mathcal{S}_{2})$, which follows from Lemma 19. A simpler proof of Lemma 20
can be found in (EHEE22, , Lemma 8.8).

We are now in a position to prove that the Robin–Robin method converges.

###### Theorem 6.1

Suppose that Assumptions 1, 2 and 3 hold. Let $u$ be the solution to the weak
heat equation Equation 14 and $\eta$ be the solution to the
$L^{2}$-Steklov–Poincaré equation Equation 30. The iterates
$(\eta_{1}^{n},\eta_{2}^{n})_{n\geq 1}$ of the $L^{2}$-Peaceman–Rachford
iteration Equation 31 converges to $\eta$, i.e.,

$\|\eta^{n}_{1}-\eta\|_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda}+\|\eta^{n}_{2}-\eta\|_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda}\rightarrow
0,$

as $n$ tends to infinity. Moreover the weak Robin–Robin approximation
$(u_{1}^{n},u_{2}^{n})_{n\geq 1}$ converges to
$(u_{1},u_{2})=(\left.u\right|_{\Omega_{1}\times\mathbb{R}},\left.u\right|_{\Omega_{2}\times\mathbb{R}})$,
i.e.,

$\|u^{n}_{1}-u_{1}\|_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{1})}+\|u^{n}_{2}-u_{2}\|_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{2})}\rightarrow
0,$

as $n$ tends to infinity.

###### Proof

From Equations 33 and 32 we obtain

$\displaystyle\|\eta^{n}_{1}-\eta\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda}+\|\eta^{n}_{2}-\eta\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda}$
$\displaystyle\quad=\|T_{1}F_{1}(\eta^{n}_{1}-\eta)\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda}+\|T_{2}F_{2}(\eta^{n}_{2}-\eta)\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,\Lambda}$
$\displaystyle\quad\leq
C\bigl{(}\|F_{1}(\eta^{n}_{1}-\eta)\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{1})}+\|F_{2}(\eta^{n}_{2}-\eta)\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{2})}\bigr{)}$
$\displaystyle\quad\leq
C\bigl{(}(\mathcal{S}_{1}\eta^{n+1}_{1}-\mathcal{S}_{1}\eta,\eta^{n+1}_{1}-\eta)_{L^{2}(\Gamma\times\mathbb{R})}+(\mathcal{S}_{2}\eta^{n}_{2}-\mathcal{S}_{2}\eta,\eta^{n}_{2}-\eta)_{L^{2}(\Gamma\times\mathbb{R})}\bigr{)}\rightarrow
0,$

as $n$ tends to infinity. By Lemmas 13, 11 and 14, one has the identities

$(u_{1},u_{2})=(F_{1}\eta+G_{1}f_{1},F_{2}\eta+G_{2}f_{2})\quad\text{and}\quad(u_{1}^{n},u_{2}^{n})=(F_{1}\eta_{1}^{n}+G_{1}f_{1},F_{2}\eta_{2}^{n}+G_{2}f_{2}).$

This together with the last bound in the limit above yields that

$\displaystyle\|u^{n}_{1}-u_{1}\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{1})}+\|u^{n}_{2}-u_{2}\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{2})}$
$\displaystyle\quad=\|F_{1}(\eta^{n}_{1}-\eta)\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{1})}+\|F_{2}(\eta^{n}_{2}-\eta)\|^{2}_{L^{2}(\mathbb{R})\,\tilde{\otimes}\,H^{1}(\Omega_{2})}\rightarrow
0,$

as $n$ tends to infinity. ∎

###### Remark 5

Note that the Robin–Robin method is unlikely to have a geometric convergence
rate in the present continuous framework. This is indicated already in the
elliptic case, by observing that the spatially discrete method has a
convergence rate of the form $1-\mathcal{O}(\sqrt{h})$; see gander06 . That
is, a rate deteriorating to one as the spatial discretization parameter $h$
tends to zero.

##  7 Modified Robin–Robin method

One can achieve a slightly stronger convergence result if one allows the
evaluation of the (non-local) Hilbert transform in the method. Instead of the
standard Robin–Robin method, we therefore also consider the method given by
finding, for each $n=1,2,\dots$, functions $(u_{1}^{n},u_{2}^{n})\in
W_{1}\times W_{2}$ such that

$\left\\{\begin{aligned}
&a_{1}(u_{1}^{n+1},v_{1})=(f_{1},v_{1})_{L^{2}(\Omega_{1}\times\mathbb{R})}&&\text{for
all }v_{1}\in W_{1}^{0},\\\
&a_{1}(u_{1}^{n+1},F_{1}\mathcal{H}^{\varphi}_{\Gamma}\mu)-(f_{1},F_{1}\mathcal{H}^{\varphi}_{\Gamma}\mu)_{L^{2}(\Omega_{1}\times\mathbb{R})}&&\\\
&\quad+a_{2}(u_{2}^{n},F_{2}\mathcal{H}^{\varphi}_{\Gamma}\mu)-(f_{2},F_{2}\mathcal{H}^{\varphi}_{\Gamma}\mu)_{L^{2}(\Omega_{2}\times\mathbb{R})}&&\\\
&\quad=s(T_{2}u_{2}^{n}-T_{1}u_{1}^{n+1},\mu)_{L^{2}(\Gamma\times\mathbb{R})}&&\text{for
all }\mu\in Z,\\\\[5.0pt]
&a_{2}(u_{2}^{n+1},v_{2})=(f_{2},v_{2})_{L^{2}(\Omega_{2}\times\mathbb{R})}&&\text{for
all }v_{2}\in W_{2}^{0},\\\
&a_{2}(u_{2}^{n+1},F_{2}\mathcal{H}^{\varphi}_{\Gamma}\mu)-(f_{2},F_{2}\mathcal{H}^{\varphi}_{\Gamma}\mu)_{L^{2}(\Omega_{2}\times\mathbb{R})}&&\\\
&\quad+a_{1}(u_{1}^{n+1},F_{1}\mathcal{H}^{\varphi}_{\Gamma}\mu)-(f_{1},F_{1}\mathcal{H}^{\varphi}_{\Gamma}\mu)_{L^{2}(\Omega_{1}\times\mathbb{R})}&&\\\
&\quad=s(T_{1}u_{1}^{n+1}-T_{2}u_{2}^{n+1},\mu)_{L^{2}(\Gamma\times\mathbb{R})}&&\text{for
all }\mu\in Z.\end{aligned}\right.$ (34)

Here, $u_{2}^{0}\in W_{2}$ is an initial guess and $s,\varphi>0$ are suitably
chosen method parameters. We refer to Equation 34 as the weak formulation of
the modified Robin–Robin method. Moreover, we consider the weak modified
Peaceman–Rachford iteration, as finding functions
$(\eta_{1}^{n},\eta_{2}^{n})\in Z\times Z$, for each $n=1,2,\dots$, such that

$\left\\{\begin{aligned}
&\langle(sJ+\mathcal{H}_{\Gamma}^{\varphi*}S_{1})\eta_{1}^{n+1}-\mathcal{H}_{\Gamma}^{\varphi*}\chi_{1},\mu\rangle&&\\\
&\quad=\langle(sJ-\mathcal{H}_{\Gamma}^{\varphi*}S_{2})\eta_{2}^{n}+\mathcal{H}_{\Gamma}^{\varphi*}\chi_{2},\mu\rangle&&\text{for
all }\mu\in Z,\\\
&\langle(sJ+\mathcal{H}_{\Gamma}^{\varphi*}S_{2})\eta_{2}^{n+1}-\mathcal{H}_{\Gamma}^{\varphi*}\chi_{2},\mu\rangle&&\\\
&\quad=\langle(sJ-\mathcal{H}_{\Gamma}^{\varphi*}S_{1})\eta_{1}^{n+1}+\mathcal{H}_{\Gamma}^{\varphi*}\chi_{1},\mu\rangle&&\text{for
all }\mu\in Z.\end{aligned}\right.$ (35)

Here, $\eta_{2}^{0}\in Z$ is an initial guess and
$\mathcal{H}_{\Gamma}^{\varphi*}$ denotes the adjoint of
$\mathcal{H}_{\Gamma}^{\varphi}$ in $Z$, i.e.

$\langle\mathcal{H}_{\Gamma}^{\varphi*}z,\mu\rangle=\langle
z,\mathcal{H}_{\Gamma}^{\varphi}\mu\rangle.$

for all $z\in Z^{*}$ and $\mu\in Z$.

###### Lemma 21

Suppose that Assumptions 1 and 2 hold. Then the weak forms of the modified
Robin–Robin method Equation 34 and the modified Peaceman–Rachford iteration
Equation 35 are equivalent.

The proof of Lemma 21 is similar to the proof of Lemma 14 and is therefore
left out. As in the case of the Robin–Robin method we restrict the
Steklov–Poincaré operators to $L^{2}(\Gamma\times\mathbb{R})$. Let

$\displaystyle D(\mathcal{S}_{i}^{\varphi})$ $\displaystyle=\\{\eta\in
Z:\mathcal{H}_{\Gamma}^{\varphi*}(S_{i}\eta-\chi_{i})\in
L^{2}(\Gamma\times\mathbb{R})^{*}\\},$ $\displaystyle
D(\mathcal{S}^{\varphi})$ $\displaystyle=\\{\eta\in
Z:\mathcal{H}_{\Gamma}^{\varphi*}(S\eta-\chi)\in
L^{2}(\Gamma\times\mathbb{R})^{*}\\}$

and define the unbounded affine operators

$\displaystyle\mathcal{S}_{i}^{\varphi}$
$\displaystyle:D(\mathcal{S}_{i}^{\varphi})\subseteq
L^{2}(\Gamma\times\mathbb{R})\rightarrow
L^{2}(\Gamma\times\mathbb{R}):\eta\mapsto
J^{-1}\mathcal{H}_{\Gamma}^{\varphi*}(S_{i}\eta-\chi_{i}),$
$\displaystyle\mathcal{S}^{\varphi}$
$\displaystyle:D(\mathcal{S}^{\varphi})\subseteq
L^{2}(\Gamma\times\mathbb{R})\rightarrow
L^{2}(\Gamma\times\mathbb{R}):\eta\mapsto
J^{-1}\mathcal{H}_{\Gamma}^{\varphi*}(S\eta-\chi).$

The $L^{2}$-formulation of the modified Peaceman–Rachford iteration is then
the following: For each $n=1,2,\dots$ find $(\eta_{1}^{n+1},\eta_{2}^{n+1})\in
D(\mathcal{S}_{1}^{\varphi})\times D(\mathcal{S}_{2}^{\varphi})$ such that

$\left\\{\begin{aligned}
(sI+\mathcal{S}_{1}^{\varphi})\eta_{1}^{n+1}&=(sI-\mathcal{S}_{2}^{\varphi})\eta_{2}^{n},\\\
(sI+\mathcal{S}_{2}^{\varphi})\eta_{2}^{n+1}&=(sI-\mathcal{S}_{1}^{\varphi})\eta_{1}^{n+1}.\end{aligned}\right.$
(36)

Here $\eta_{2}^{0}\in D(\mathcal{S}_{2}^{\varphi})$ is an initial guess. We
have an analogous result to Lemma 17 so that Equation 36 and Equation 35 are
equivalent for $\eta_{0}^{n}\in D(\mathcal{S}_{2}^{\varphi})$. Moreover, we
have the following result on the domain $D(\mathcal{S}_{i}^{\varphi})$:

###### Lemma 22

Suppose that Assumptions 1 and 2 hold. Then

$D(\mathcal{S}_{i}^{\varphi})=D(\mathcal{S}_{i}),\quad i=1,2.$

If Assumption 3 also holds and $\eta$ is the solution to Equation 30 then
$\eta\in D(\mathcal{S}_{1}^{\varphi})\cap D(\mathcal{S}_{2}^{\varphi})$.

###### Proof

Since $\mathcal{H}_{\Gamma}^{\varphi}:L^{2}(\Gamma\times\mathbb{R})\rightarrow
L^{2}(\Gamma\times\mathbb{R})$ is an isomorphism we have that
$\mathcal{H}_{\Gamma}^{\varphi*}:L^{2}(\Gamma\times\mathbb{R})^{*}\rightarrow
L^{2}(\Gamma\times\mathbb{R})^{*}$ also is an isomorphism. Therefore
$D(\mathcal{S}_{i}^{\varphi})=D(\mathcal{S}_{i})$ and the remaining statement
follows from Lemma 19. ∎

The theorem below follows by the same argument as Theorem 6.1, the difference
being that the operator $\mathcal{H}_{\Gamma}^{\varphi*}S_{i}$ is coercive in
the stronger norm of $Z$; see Lemma 12. This in turn gives the stronger
monotonicity property

$(\mathcal{S}_{i}^{\varphi}\eta-\mathcal{S}_{i}^{\varphi}\mu,\eta-\mu)_{L^{2}(\Gamma\times\mathbb{R})}\geq
c\|F_{i}(\eta-\mu)\|^{2}_{Z}\quad\text{ for all }\eta,\mu\in
D(\mathcal{S}_{i}^{\varphi}).$ (37)

###### Theorem 7.1

Suppose that Assumptions 1, 2 and 3 hold. Let $u$ be the solution to the weak
heat equation Equation 14 and $\eta$ be the solution to the
$L^{2}$-Steklov–Poincaré equation Equation 30. The iterates
$(\eta_{1}^{n},\eta_{2}^{n})_{n\geq 1}$ of the $L^{2}$-modified
Peaceman–Rachford iteration Equation 36 converges to $\eta$, i.e.,

$\|\eta^{n}_{1}-\eta\|_{Z}+\|\eta^{n}_{2}-\eta\|_{Z}\rightarrow 0,$

as $n$ tends to infinity. Moreover, the weak modified Robin–Robin
approximation$(u_{1}^{n},u_{2}^{n})_{n\geq 1}$ converges to
$(u_{1},u_{2})=(\left.u\right|_{\Omega_{1}\times\mathbb{R}},\left.u\right|_{\Omega_{2}\times\mathbb{R}})$,
i.e.,

$\|u^{n}_{1}-u_{1}\|_{W_{1}}+\|u^{n}_{2}-u_{2}\|_{W_{2}}\rightarrow 0,$

as $n$ tends to infinity.

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