assumptionAssumption remarkRemark exampleExample Convergence Analysis of the
Nonoverlapping Robin–Robin MethodE. Engström, E. Hansen

# Convergence Analysis of the Nonoverlapping Robin–Robin Method for Nonlinear
Elliptic Equations††thanks: Version of . This work was supported by the
Swedish Research Council under the grant 2019– 05396.

Emil Engström Centre for Mathematical Sciences, Lund University, P.O. Box 118,
SE-22100 Lund, Sweden<EMAIL_ADDRESS>Eskil Hansen Centre for
Mathematical Sciences, Lund University, P.O. Box 118, SE-22100 Lund, Sweden,
<EMAIL_ADDRESS>
###### Abstract

We prove convergence for the nonoverlapping Robin–Robin method applied to
nonlinear elliptic equations with a $p$-structure, including degenerate
diffusion equations governed by the $p$-Laplacian. This nonoverlapping domain
decomposition is commonly encountered when discretizing elliptic equations, as
it enables the usage of parallel and distributed hardware. Convergence has
been derived in various linear contexts, but little has been proven for
nonlinear equations. Hence, we develop a new theory for nonlinear
Steklov–Poincaré operators based on the $p$-structure and the
$L^{p}$-generalization of the Lions–Magenes spaces. This framework allows the
reformulation of the Robin–Robin method into a Peaceman–Rachford splitting on
the interfaces of the subdomains, and the convergence analysis then follows by
employing elements of the abstract theory for monotone operators. The analysis
is performed on Lipschitz domains and without restrictive regularity
assumptions on the solutions.

###### keywords:

Robin–Robin method, Nonoverlapping domain decomposition, Nonlinear elliptic
equation, Convergence, Steklov–Poincaré operator

65N55, 65J15, 35J70, 47N20

##  1 Introduction

Approximating the solution of an elliptic partial differential equation (PDE)
typically demands large-scale computations requiring the usage of parallel and
distributed hardware. In this context, a nonoverlapping domain decomposition
method is a suitable choice, as it can be implemented in parallel with local
communication. After decomposing the equation’s spatial domain into
nonoverlapping subdomains, the method consists of an iterative procedure that
solves the equation on each subdomain and thereafter communicates the results
via the boundaries to the adjacent subdomains. For a general introduction we
refer to [18, 23].

There is a vast amount of methods in the literature, employing different
transmission conditions between the subdomains. The standard examples are
based on the alternate use of Dirichlet and Neumann boundary conditions, but a
competitive alternative is the Robin–Robin method, where the same type of
Robin boundary condition is used for all subdomains. The Robin–Robin method
was introduced in [13] together with a convergence proof when applied to
linear elliptic equations. After applying a finite element discretization,
convergence rates of the form $1-\mathcal{O}(\sqrt{h})$, with $h$ denoting the
mesh width, have been derived in various linear contexts [9, 15, 24]; also see
[7]. For generalizations and further applications of the Robin–Robin method
applied to linear PDEs we refer to [3, 9] and references therein.

When considering nonlinear elliptic PDEs the literature is more limited.
Convergence studies relating to overlapping Schwarz methods are given in [6,
21, 22]. However, there are hardly any results dealing with nonoverlapping
domain decomposition schemes. One exception is [2], where the convergence of
the Dirichlet–Neumann and Robin–Robin methods are analyzed for a family of
one-dimensional elliptic equations. A related study is [20], where the
equivalence between a class of nonlinear elliptic equations and the
corresponding transmission problems are proven for nonoverlapping
decompositions with cross points, but no numerical scheme is considered. Apart
from [22], all these nonlinear studies rely on frameworks similar to the
linear case, e.g., assuming that the diffusion is uniformly positive.

Hence, the aim of this paper is to derive a genuinely nonlinear extension of
the linear convergence result given in [13] for the nonoverlapping Robin–Robin
method. We will focus on nonlinear elliptic equations of the form

(1) $\left\\{\begin{aligned} -\nabla\cdot\alpha(\nabla u)+g(u)&=f&&\text{in
}\Omega,\\\ u&=0&&\text{on }\partial\Omega,\end{aligned}\right.$

where $\Omega$ is a bounded domain in $\mathbb{R}^{d}$, $d=1,2,\ldots,$ with
boundary $\partial\Omega$. The functions $\alpha$ and $g$ are assumed to have
a $p$-structure; defined in Section 2. This structure enables a clear-cut
convergence analysis for a broad family of degenerate elliptic equations,
i.e., $\alpha(\nabla u)$ may vanish for nonzero values of $u$. The latter
typically prevents the existence of a strong solution in $W^{2,p}(\Omega)$.

The archetypical examples of nonlinear elliptic equations with a $p$-structure
are those governed by the $p$-Laplacian, where $\alpha(z)=|z|^{p-2}z$.
Examples include the computation of the nonlinear resolvent

(2) $-\nabla\cdot(|\nabla u|^{p-2}\nabla u)+\lambda u=f,$

arising in the context of an implicit Euler discretization of the parabolic
$p$-Laplace equation, and the nonlinear reaction-diffusion problem

(3) $-\nabla\cdot(|\nabla u|^{p-2}\nabla u)+\lambda|u|^{p-2}u=f.$

For sake of simplicity, we decompose the original domain $\Omega$ into two
nonoverlapping subdomains $\\{\Omega_{i},\text{ for }i=1,2\\}$, with
boundaries denoted by $\partial\Omega_{i}$, and separated by the interface
$\Gamma$, i.e.,

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

Two examples of such decompositions are illustrated in Figs. 1(a) and 1(b),
respectively. The analysis presented here can also, in a trivial fashion, be
extended to the case when $\Omega_{i}$ is a family of nonadjacent subdomains,
e.g., the stripewise domain decomposition illustrated in Fig. 1(c).

(a)

(b)

(c)

(d)

Figure 1: Examples of different domain decompositions: (a) illustrates a
domain decomposition with two intersection points; (b) a decomposition without
intersection points; (c) a stripewise decomposition without cross points; (d)
a full decomposition with cross points.

The strong form of the Robin–Robin method applied to Eq. 1 is then to find
$(u^{n}_{1},u^{n}_{2})$, for $n=1,2,\ldots,$ such that

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

where $\nu_{i}$ denotes the unit outward normal vector of
$\partial\Omega_{i}$, $u_{2}^{0}$ is a given initial guess and $s>0$ is a
method parameter. Here, $u^{n}_{i}$ and $u^{n}_{i}|_{\Gamma}$ approximate the
solution $u$ restricted to $\Omega_{i}$ and $\Gamma$, respectively.

The convergence analysis is organized as follows. For linear elliptic
equations, i.e., equations with a $2$-structure, the analysis relies on the
trace operator from $H^{1}(\Omega_{i})$ onto $H^{1/2}(\partial\Omega_{i})$,
and the Lions–Magenes spaces $H^{1/2}_{00}(\Gamma)$. We therefore start by
introducing the generalized $p$-version of the trace operator, now given from
$W^{1,p}(\Omega_{i})$ onto $W^{1-1/p,p}(\partial\Omega_{i})$, and the
corresponding Lions–Magenes spaces $\Lambda_{i}$; see Sections 3 and 4. There
is a surprising lack of proofs in the literature dealing with this generalized
$p$-setting. We therefore make an effort to give precise definitions and proof
references.

With the correct function spaces in place, we prove that the nonlinear
transmission problem is equivalent to the weak form of Eq. 1 in Theorem 5.2,
and introduce the new nonlinear Steklov–Poincaré operators, as maps from
$\Lambda_{i}$ to $\Lambda_{i}^{*}$, in Section 6. The latter yields that the
transmission problem can be stated as a problem on $\Gamma$ and the
Robin–Robin method reduces to the Peaceman–Rachford splitting. The main
challenge is then to derive the fundamental properties of the nonlinear
Steklov–Poincaré operators from the $p$-structure, which is achieved in
Section 7. By interpreting the nonlinear Steklov–Poincaré operators as
unbounded, monotone maps on $L^{2}(\Gamma)$ we finally prove that the
Robin–Robin method is well defined on $W^{1,p}(\Omega_{1})\times
W^{1,p}(\Omega_{2})$; see Corollary 8.8, and convergent in the same space; see
Theorem 8.13. The latter relies on the abstract theory of the
Peaceman–Rachford splittings [14].

The continuous convergence analysis presented here also holds in the finite
dimensional case obtained after a suitable spatial discretization, e.g., by
employing finite elements. However, we will limit ourselves to the continuous
case in this paper. Hence, important issues including convergence rates for
the finite-dimensional case and the influence of the mesh width on the optimal
choice of the method parameter $s$ will be explored elsewhere.

Finally, $c_{i}$ and $C_{i}$ will denote generic positive constants that
assume different values at different occurrences.

##  2 Nonlinear elliptic equations with $p$-structure

Throughout the paper, we will consider the nonlinear elliptic equation Eq. 1
with $f\in L^{2}(\Omega)$ and $\Omega$ being a bounded Lipschitz domain. The
equation is assumed to have a $p$-structure of the form:

The parameters $(p,r)$ and the functions
$\alpha:\mathbb{R}^{d}\to\mathbb{R}^{d},g:\mathbb{R}\to\mathbb{R}$ satisfy the
properties below.

  * •

Let $p\in[2,\infty)$ and $r\in(1,\infty)$. If $p<d$ then $r\leq
dp/\bigl{(}2(d-p)\bigr{)}+1$.

  * •

The functions $\alpha$ and $g$ are continuous and satisfy the growth
conditions

$|\alpha(z)|\leq C_{1}|z|^{p-1}\quad\text{and}\quad|g(x)|\leq
C_{2}|x|^{r-1},\quad\text{for all }z\in\mathbb{R}^{d},x\in\mathbb{R}.$

  * •

The function $\alpha$ is strictly monotone with the bound

$\bigl{(}\alpha(z)-\alpha(\tilde{z})\bigr{)}\cdot(z-\tilde{z})\geq
c_{1}|z-\tilde{z}|^{p},\quad\text{for all }z,\tilde{z}\in\mathbb{R}^{d}.$

  * •

The function $\alpha$ is coercive with the bound

$\alpha(z)\cdot z\geq c_{2}|z|^{p},\quad\text{for all }z\in\mathbb{R}^{d}.$

  * •

The function $g$ is strictly monotone and coercive with the bounds

$\bigl{(}g(x)-g(\tilde{x})\bigr{)}(x-\tilde{x})\geq
c_{3}|x-\tilde{x}|^{r}\quad\text{and}\quad g(x)x\geq
c_{4}|x|^{r},\quad\text{for all }x,\tilde{x}\in\mathbb{R}.$

######  Example 2.1.

The equation Eq. 2 satisfies Section 2 with $\alpha(z)=|z|^{p-2}z$,
$\lambda>0$, $g(x)=\lambda x$ and r=2. The same holds for equation Eq. 3 with
$g(x)=\lambda|x|^{p-2}x$ and $r=p$.

######  Remark 2.2.

The last assertion of Section 2 is made in order to ensure that the
convergence analysis of the domain decomposition is valid without employing
the Poincaré inequality, which allows decompositions where
$\partial\Omega\setminus\partial\Omega_{i}=\emptyset$; see Fig. 1(b). If the
latter setting is excluded, then the analysis is valid for a broader class of
functions $g$, especially $g=0$.

######  Remark 2.3.

Possible generalizations of Section 2, which we omit for sake of notational
simplicity, include the cases: dependence on the spatial variable and first-
order terms, e.g., $\alpha(\nabla u)=\alpha(x,u,\nabla u)$; the parameter
choice $p\in(2d/(d+1),2)$, which requires an additional set of embedding
results for trace spaces; other variations similar to the $p$-Laplacian, e.g.,

$\alpha(z)=(|z_{1}|^{p-2}z_{1},\ldots,|z_{d}|^{p-2}z_{d}),$

which only require slight reformulations of the monotonicity and coercivity
conditions.

Let $V=W^{1,p}_{0}(\Omega)$ and define the form $a:V\times
V\rightarrow\mathbb{R}$ by

$a(u,v)=\int_{\Omega}\alpha(\nabla u)\cdot\nabla v+g(u)v\,\mathrm{d}x.$

The weak form of Eq. 1 is to find $u\in V$ such that

(5) $a(u,v)=(f,v)_{L^{2}(\Omega)},\quad\text{for all }v\in V.$

The $p$-structure implies that there exist a unique weak solution of Eq. 5;
see, e.g., [19, Theorem 2.36]. A central part of the existence proof, and our
convergence analysis as well, is to observe that the $p$-structure directly
implies that the form $a$ is bounded, strictly monotone and coercive.

######  Lemma 2.4.

If Section 2 holds, then $a:V\times V\rightarrow\mathbb{R}$ is well defined
and satisfies the upper bound

$|a(u,v)|\leq C_{1}\bigl{(}\|\nabla u\|^{p-1}_{L^{p}(\Omega)^{d}}\|\nabla
v\|^{\phantom{p-1}}_{L^{p}(\Omega)^{d}}+\|u\|_{L^{r}(\Omega)}^{r-1}\|v\|^{\phantom{r-1}}_{L^{r}(\Omega)}\bigr{)},$

the strict monotonicity bound

$a(u,u-v)-a(v,u-v)\geq
c_{1}\bigl{(}\|\nabla(u-v)\|^{p}_{L^{p}(\Omega)^{d}}+\|u-v\|^{r}_{L^{r}(\Omega)}\bigr{)}$

and the coercivity bound

$a(u,u)\geq c_{2}\bigl{(}\|\nabla
u\|^{p}_{L^{p}(\Omega)^{d}}+\|u\|^{r}_{L^{r}(\Omega)}\bigr{)},$

for all $u,v\in V$.

In order to conduct the convergence analysis, we also make the following
additional regularity assumption on the weak solution. The weak solution $u\in
V$ of Eq. 5 satisfies $\alpha(\nabla u)\in C(\overline{\Omega})^{d}$. Note
that the above regularity assumption does _not_ imply that $u$ is a strong
solution in $W^{2,p}(\Omega)$. A possible generalization of Section 2 is
discussed in Remark 8.4.

######  Example 2.5.

Consider the equations given by the $p$-Laplacian in Example 2.1. If $p\geq d$
then the weak solution $u\in V$ is also in $C(\overline{\Omega})$. If in
addition $f\in L^{\infty}(\Omega)$ and the boundary $\partial\Omega$ is
$C^{1,\beta}$, then [12, Theorem 1] yields that $u\in
C^{1,\beta}(\overline{\Omega})$. The latter implies that Section 2 is valid in
this context.

Finally, we will make frequent use of the fact that, under Section 2, the
standard $W^{1,p}(\Omega)$-norm is equivalent to the norm

(6) $u\mapsto\|\nabla u\|_{L^{p}(\Omega)^{d}}+\|u\|_{L^{r}(\Omega)}.$

For $r\geq p$ this follows directly by the Sobolev embedding theorem together
with the assumed restrictions on $(p,r)$. For $r<p$ the equivalence follows by
an additional bootstrap argument.

##  3 Function spaces and trace operators on $\Omega_{i}$

We start by considering a manifold $\mathcal{M}$ in $\mathbb{R}^{d}$, which
will play the role of $\partial\Omega_{i}$ or $\Gamma$. The manifold
$\mathcal{M}$ is said to be Lipschitz if there exist open, overlapping sets
$\Theta_{r}$ such that

$\mathcal{M}=\bigcup_{r=1}^{m}\Theta_{r},$

where each $\Theta_{r}$ can be described as the graph of a Lipschitz
continuous function $b_{r}$. More precisely, there exists $(d-1)$-dimensional
cubes $\theta_{r}$ and local charts $\psi_{r}:\Theta_{r}\rightarrow\theta_{r}$
that are bijective and Lipschitz continuous. The charts have the structure
$\psi_{r}^{-1}=A^{-1}_{r}\circ Q_{r}$, where
$A_{r}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is a coordinate transformation
and

$Q_{r}:\theta_{r}\rightarrow\mathbb{R}^{d}:x_{r}\mapsto\bigl{(}x_{r},b_{r}(x_{r})\bigr{)}$

for the Lipschitz continuous map $b_{r}:\theta_{r}\rightarrow\mathbb{R}$. A
function $\mu:\mathcal{M}\rightarrow\mathbb{R}$ now has the local components
$\mu\circ\psi^{-1}_{r}$. We refer to [10, Section 6.2] for further details.

On a Lipschitz manifold we may introduce a measure [16, Chapter 3] and thus
define the integral and the space $L^{p}(\mathcal{M})$; see, e.g., [4]. From
[4, Chapters 3.4–3.5] it follows that $L^{p}(\mathcal{M})$ is a Banach space
and that $L^{2}(\mathcal{M})$ is a Hilbert space with the inner product

$(\eta,\mu)_{L^{2}(\mathcal{M})}=\int_{\mathcal{M}}\eta\mu\,\mathrm{d}S.$

Let $\\{\varphi_{r}\\}$ be a partition of unity of $\mathcal{M}$. The integral
then satisfies

(7)
$\int_{\mathcal{M}}\mu\,\mathrm{d}S=\sum_{r=1}^{m}\int_{\theta_{r}}(\mu\varphi_{r})\circ\psi^{-1}_{r}|n_{r}|\,\mathrm{d}x,$

where
$n_{r}=(\partial_{1}b_{r},\partial_{2}b_{r},\dots,\partial_{d-1}b_{r},-1)$;
see [16, Theorem 3.9]. Seemingly obvious properties of the integral, including

$\int_{\mathcal{M}}\mu\,\mathrm{d}S=\int_{\mathcal{M}_{0}}\mu\,\mathrm{d}S+\int_{\mathcal{M}\setminus\mathcal{M}_{0}}\mu\,\mathrm{d}S,$

relies heavily on the observation that the integral is independent of the
representation $(\Theta_{r},A_{r},b_{r})$ and the choice of partition of unity
$\\{\varphi_{r}\\}$; see [16, Theorems 3.5 and 3.7] and the comments
thereafter.

The equality Eq. 7 also shows that our integral and $L^{p}$-spaces are
equivalent to the ones used in [10]. Moreover, by [10, Lemma 6.3.5], the
$L^{p}$-norm used here is equivalent to the norm

$\mu\mapsto\Big{(}\sum_{r=1}^{m}\|\mu\circ\psi^{-1}_{r}\|_{L^{p}(\theta_{r})}^{p}\Big{)}^{1/p}.$

Finally, recall that for a Lipschitz manifold $\mathcal{M}$ the unit outward
normal vector $\nu=(\nu^{1},\ldots,\nu^{d})$ is defined almost everywhere; see
[10, Section 6.10.1]. The normal vector is given locally by
$\nu\circ\psi^{-1}_{r}=n_{r}/|n_{r}|$ and the Lipschitz continuity of $b_{r}$
yields that $\nu^{\ell}\in L^{\infty}(\mathcal{M})$.

The boundaries $\partial\Omega_{i}$ and the interface $\Gamma$ are all
$(d-1)$-dimensional Lipschitz manifolds.

We use the notation
$(\Theta_{r}^{i},\theta_{r}^{i},m_{i},\psi_{r}^{i},b_{r}^{i},\phi_{r}^{i},\nu_{i})$
for the quantities related to the local representations of
$\partial\Omega_{i}$. For later use, we note that

$\nu_{1}=-\nu_{2}\quad\text{ on }\Gamma.$

Next, we define the fractional Sobolev spaces on the $(d-1)$-dimensional cubes
$\theta_{r}$. Let $0<s<1$, then $W^{s,p}(\theta_{r})$ is defined as all $u\in
L^{p}(\theta_{r})$ such that

$|u|_{s,\theta_{r}}=\Big{(}\int_{\theta_{r}}\int_{\theta_{r}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{d-1+sp}}\,\mathrm{d}x\,\mathrm{d}y\Big{)}^{1/p}<\infty.$

The corresponding norm is given by

$\|u\|_{W^{s,p}(\theta_{r})}=\|u\|_{L^{p}(\theta_{r})}+|u|_{s,\theta_{r}}.$

Having defined the fractional Sobolev space on $\theta^{i}_{r}$ we can also
define them on $\partial\Omega_{i}$. For $0<s<1$, introduce

$W^{s,p}(\partial\Omega_{i})=\\{\mu\in
L^{p}(\partial\Omega_{i}):\mu\circ(\psi_{r}^{i})^{-1}\in
W^{s,p}(\theta_{r}^{i}),\text{ for }r=1,\dots,m_{i}\\},$

equipped with the norm

$\|\mu\|_{W^{s,p}(\partial\Omega_{i})}=\Big{(}\sum_{r=1}^{m_{i}}\|\mu\circ(\psi_{r}^{i})^{-1}\|^{p}_{W^{s,p}(\theta_{r}^{i})}\Big{)}^{1/p}.$

By the definitions of the norms, it follows directly that

$\|\mu\|_{L^{p}(\partial\Omega_{i})}\leq
C\|\mu\|_{W^{s,p}(\partial\Omega_{i})}.$

Furthermore, the space $W^{s,p}(\partial\Omega_{i})$ is complete and
reflexive; see [10, Definition 6.8.6] and the comment thereafter. Next, we
recapitulate the trace theorem for $W^{1,p}$-functions on Lipschitz domains;
see, e.g., [10, Theorems 6.8.13 and 6.9.2].

######  Lemma 3.1.

If the Sections 2 and 3 are valid, then there exists a surjective bounded
linear operator $T_{\partial\Omega_{i}}:W^{1,p}(\Omega_{i})\rightarrow
W^{1-1/p,p}(\partial\Omega_{i})$ such that
$T_{\partial\Omega_{i}}u=u|_{\partial\Omega_{i}}$ when $u\in
C^{\infty}(\overline{\Omega}_{i})$. The operator $T_{\partial\Omega_{i}}$ has
a bounded linear right inverse
$R_{\partial\Omega_{i}}:W^{1-1/p,p}(\partial\Omega_{i})\rightarrow
W^{1,p}(\Omega_{i})$.

We can then define the Sobolev spaces on $\Omega_{i}$ required for the domain
decomposition, namely

$V_{i}^{0}=W^{1,p}_{0}(\Omega_{i})\quad\text{and}\quad V_{i}=\\{v\in
W^{1,p}(\Omega_{i}):(T_{\partial\Omega_{i}}v)|_{\partial\Omega_{i}\setminus\Gamma}=0\\}.$

The spaces are equipped with the norm

$\|v\|_{V_{i}}=\|\nabla
v\|_{L^{p}(\Omega_{i})^{d}}+\|v\|_{L^{r}(\Omega_{i})}.$

As for Eq. 6, this norm is equivalent to the standard
$W^{1,p}(\Omega_{i})$-norm under Section 2. Furthermore, the spaces
$V_{i}^{0}$ and $V_{i}$ are reflexive Banach spaces.

##  4 Function spaces and trace operators on $\Gamma$

The $L^{p}$-form of the Lions–Magenes spaces can be defined as

$\Lambda_{i}=\\{\mu\in L^{p}(\Gamma):E_{i}\mu\in
W^{1-1/p,p}(\partial\Omega_{i})\\},\quad\text{with}\quad\|\mu\|_{\Lambda_{i}}=\|E_{i}\mu\|_{W^{1-1/p,p}(\partial\Omega_{i})}.$

Here, $E_{i}\mu$ denotes the extension by zero of $\mu$ to
$\partial\Omega_{i}$. We also define the trace space

$\Lambda=\\{\mu\in L^{p}(\Gamma):\mu\in\Lambda_{i},\text{ for
}i=1,2\\},\quad\text{with}\quad\|\mu\|_{\Lambda}=\|\mu\|_{\Lambda_{1}}+\|\mu\|_{\Lambda_{2}}.$

######  Lemma 4.1.

If the Sections 2 and 3 hold, then $\Lambda_{i}$ and $\Lambda$ are reflexive
Banach spaces.

######  Proof 4.2.

Observe that $E_{i}$ is a linear isometry from $\Lambda_{i}$ onto

(8)
$R(E_{i})=\\{\mu\in{W^{1-1/p,p}(\partial\Omega_{i})}:\mu|_{\partial\Omega\setminus\Gamma}=0\\}.$

Next, consider a sequence $\\{\mu^{k}\\}\subset R(E_{i})$ such that
$\mu^{k}\to\mu$ in $W^{1-1/p,p}(\partial\Omega_{i})$. Then
$\mu^{k}|_{\Omega_{i}\setminus\Gamma}=0$ and $\mu^{k}\to\mu$ in
$L^{p}(\partial\Omega_{i})$, which implies that

(9)
$\int_{\partial\Omega_{i}\setminus\Gamma}|\mu|^{p}\,\mathrm{d}S=\int_{\partial\Omega_{i}\setminus\Gamma}|\mu^{k}-\mu|^{p}\,\mathrm{d}S\leq\int_{\partial\Omega_{i}}|\mu^{k}-\mu|^{p}\,\mathrm{d}S\to
0,\quad\text{as }k\to\infty.$

Hence, $\mu\in R(E_{i})$ and consequently $R(E_{i})$ is a closed subset of
$W^{1-1/p,p}(\partial\Omega_{i})$. The space $\Lambda_{i}$ is therefore
isomorphic to a closed subset of the reflexive Banach space
$W^{1-1/p,p}(\partial\Omega_{i})$, i.e., $\Lambda_{i}$ is complete and
reflexive [11, Chapter 8, Theorem 15].

To prove that the same holds true for $\Lambda$ introduce the reflexive Banach
space $X=W^{1-1/p,p}(\partial\Omega_{1})\times
W^{1-1/p,p}(\partial\Omega_{2})$, with the norm

$\|(\mu_{1},\mu_{2})\|_{X}=\|\mu_{1}\|_{W^{1-1/p,p}(\partial\Omega_{1})}+\|\mu_{2}\|_{W^{1-1/p,p}(\partial\Omega_{2})},$

and the operator $E:\Lambda\rightarrow X$ defined by
$E\mu=(E_{1}\mu,E_{2}\mu)$. As $E$ is a linear isometry from $\Lambda$ onto

$R(E)=\\{(\mu_{1},\mu_{2})\in
X:{\mu_{1}}|_{\partial\Omega_{1}\setminus\Gamma}=0,\,{\mu_{2}}|_{\partial\Omega_{2}\setminus\Gamma}=0,\,{\mu_{1}}|_{\Gamma}={\mu_{2}}|_{\Gamma}\\},$

it is again sufficient to prove that $R(E)$ is a closed subset of $X$. Let
$\\{(\mu_{1}^{k},\mu_{2}^{k})\\}\subset R(E)$ be a convergent sequence in $X$
with the limit $(\mu_{1},\mu_{2})$. By the same argument as Eq. 9, we obtain
that $\mu_{i}|_{\Omega_{i}\setminus\Gamma}=0$. As
${\mu^{k}_{1}}|_{\Gamma}={\mu^{k}_{2}}|_{\Gamma}$, we also have the limit

$\int_{\Gamma}|\mu_{1}-\mu_{2}|^{p}\,\mathrm{d}S\leq
2^{p-1}\big{(}\int_{\Gamma}|\mu_{1}^{k}-\mu_{1}|^{p}\,\mathrm{d}S+\int_{\Gamma}|\mu_{2}^{k}-\mu_{2}|^{p}\,\mathrm{d}S\big{)}\to
0,\quad\text{as }k\to\infty,$

i.e., ${\mu_{1}}|_{\Gamma}={\mu_{2}}|_{\Gamma}$ in $L^{p}(\Gamma)$ and we
obtain that $(\mu_{1},\mu_{2})\in R(E)$. Thus, $R(E)$ is closed and $\Lambda$
is therefore a reflexive Banach space.

######  Lemma 4.3.

If the Sections 2 and 3 hold, then $\Lambda_{i}$ and $\Lambda$ are dense in
$L^{2}(\Gamma)$.

The proof of the lemma is almost identical to the proof of [10, Theorem 6.6.3]
and is therefore left out.

######  Remark 4.4.

We conjecture that $\Lambda_{1}=\Lambda_{2}$. However, we will move on to a
$L^{2}(\Gamma)$-framework for which it is not necessary to make this
identification.

Collecting these results yield the Gelfand triplets

$\Lambda_{i}\overset{d}{\hookrightarrow}L^{2}(\Gamma)\cong
L^{2}(\Gamma)^{*}\overset{d}{\hookrightarrow}\Lambda_{i}^{*}\quad\text{and}\quad\Lambda\overset{d}{\hookrightarrow}L^{2}(\Gamma)\cong
L^{2}(\Gamma)^{*}\overset{d}{\hookrightarrow}\Lambda^{*}.$

For future use, we introduce the Riesz isomorphism on $L^{2}(\Gamma)$ given by

$J:L^{2}(\Gamma)\to L^{2}(\Gamma)^{*}:\mu\mapsto(\mu,\cdot)_{L^{2}(\Gamma)},$

which satisfies the relations

$\langle
J\eta,\mu_{i}\rangle_{\Lambda_{i}^{*}\times\Lambda_{i}}=(\eta,\mu_{i})_{L^{2}(\Gamma)}\quad\text{and}\quad\langle
J\eta,\mu\rangle_{\Lambda^{*}\times\Lambda}=(\eta,\mu)_{L^{2}(\Gamma)},$

for all $\eta\in L^{2}(\Gamma)$, $\mu_{i}\in\Lambda_{i}$ and $\mu\in\Lambda$.
Here, $\langle\cdot,\cdot\rangle_{X^{*}\times X}$ denotes the dual pairing
between a Banach space $X$ and its dual $X^{*}$. In the following we will drop
the subscripts on the dual parings.

In order to relate the spaces $V_{i}$ and $\Lambda_{i}$, we observe that for
$v\in V_{i}$ one has $T_{\partial\Omega_{i}}v\in R(E_{i})$; see Eq. 8. Hence,
the trace operator

$T_{i}:V_{i}\to\Lambda_{i}:v\mapsto\bigl{(}T_{\partial\Omega_{i}}v\bigr{)}|_{\Gamma}$

is well defined. We also introduce the linear operator

$R_{i}:\Lambda_{i}\to V_{i}:\mu\mapsto R_{\partial\Omega_{i}}E_{i}\mu.$

######  Lemma 4.5.

If the Sections 2 and 3 hold, then $T_{i}$ and $R_{i}$ are bounded, and
$R_{i}$ is a right inverse of $T_{i}$.

######  Proof 4.6.

For $v\in V_{i}$ and $\mu\in\Lambda_{i}$ we have, by Lemma 3.1, that

$\displaystyle\|T_{i}v\|_{\Lambda_{i}}=\|E_{i}\bigl{(}(T_{\partial\Omega_{i}}v)|_{\Gamma}\bigr{)}\|_{W^{1-1/p,p}(\partial\Omega_{i})}=\|T_{\partial\Omega_{i}}v\|_{W^{1-1/p,p}(\partial\Omega_{i})}\leq
C_{i}\|v\|_{V_{i}}$
$\displaystyle\text{and}\quad\|R_{i}\mu\|_{V_{i}}=\|R_{\partial\Omega_{i}}E_{i}\mu\|_{V_{i}}\leq
C_{i}\|E_{i}\mu\|_{W^{1-1/p,p}(\partial\Omega_{i})}=C_{i}\|\mu\|_{\Lambda_{i}}.$

Hence, the linear operators $T_{i}$ and $R_{i}$ are bounded. Furthermore, for
every $\mu\in\Lambda_{i}$ we have

$T_{i}R_{i}\mu=\bigl{(}T_{\partial\Omega_{i}}R_{\partial\Omega_{i}}E_{i}\mu\bigr{)}|_{\Gamma}=(E_{i}\mu)|_{\Gamma}=\mu,$

i.e., $R_{i}$ is a right inverse of $T_{i}$.

We continue by deriving a few useful properties related to the operator
$T_{i}$.

######  Lemma 4.7.

If the Sections 2 and 3 hold and $v\in V$, then
$\mu=T_{1}v|_{\Omega_{1}}=T_{2}v|_{\Omega_{2}}$ is an element in $\Lambda$.

######  Proof 4.8.

Let $v\in V$. As $C^{\infty}_{0}(\Omega)$ is dense in $V$, there exists a
sequence $\\{v^{k}\\}\subset C^{\infty}_{0}(\Omega)$ such that $v^{k}\to v$ in
$V$. Set $v_{i}=v|_{\Omega_{i}}$ and $v_{i}^{k}=v^{k}|_{\Omega_{i}}$. Clearly,
$T_{1}v^{k}_{1}=T_{2}v^{k}_{2}$. Since $v^{k}\to v$ in $V$, we also have that
$v_{i}^{k}\to v_{i}$ in $V_{i}$. The continuity of $T_{i}$ then implies that
$T_{i}v_{i}^{k}\to T_{i}v_{i}$ in $\Lambda_{i}$. Putting this together gives
us

$\Lambda_{1}\ni
T_{1}v_{1}=\lim_{k\to\infty}T_{1}v_{1}^{k}=\lim_{k\to\infty}T_{2}v_{2}^{k}=T_{2}v_{2}\in\Lambda_{2}\quad\text{in
}L^{p}(\Gamma).$

If we now define $\mu=T_{1}v_{1}=T_{2}v_{2}$, then $\mu$ is an element in
$\Lambda=\Lambda_{1}\cap\Lambda_{2}$.

######  Lemma 4.9.

Let the Sections 2 and 3 hold. If two elements $v_{1}\in V_{1}$ and $v_{2}\in
V_{2}$ satisfies $T_{1}v_{1}=T_{2}v_{2}$, then $v=\\{v_{1}\textrm{ on
}\Omega_{1};\,v_{2}\textrm{ on }\Omega_{2}\\}$ is an element in $V$.

######  Proof 4.10.

It is clear that $v\in L^{p}(\Omega)$. For each component $1\leq\ell\leq d$,
there exists a weak derivative $\partial_{\ell}v_{i}\in L^{p}(\Omega_{i})$ of
$v_{i}\in V_{i}\subset W^{1,p}(\Omega_{i})$. If we define

$z_{\ell}=\\{\partial_{\ell}v_{1}\textrm{ on
}\Omega_{1};\,\partial_{\ell}v_{2}\textrm{ on }\Omega_{2}\\},$

then $z_{\ell}\in L^{p}(\Omega)$. Let $w\in C^{\infty}_{0}(\Omega)$ and set
$w_{i}=w|_{\Omega_{i}}\in C^{\infty}(\Omega_{i})$. The
$W^{1,p}(\Omega_{i})$-version of Green’s formula [17, Section 3.1.2] yields
that

$\displaystyle\int_{\Omega}z_{\ell}w\,\mathrm{d}x$
$\displaystyle=\sum_{i=1}^{2}\int_{\Omega_{i}}\partial_{\ell}v_{i}\,w_{i}\,\mathrm{d}x=\sum_{i=1}^{2}-\int_{\Omega_{i}}v_{i}\,\partial_{\ell}w_{i}\,\mathrm{d}x+\int_{\partial\Omega_{i}}(T_{\partial\Omega_{i}}v_{i})w_{i}\nu_{i}^{\ell}\,\mathrm{d}S$
$\displaystyle=-\int_{\Omega}v\partial_{\ell}w\,\mathrm{d}x+\sum_{i=1}^{2}\int_{\Gamma}(T_{i}v_{i})w\nu_{i}^{\ell}\,\mathrm{d}S=-\int_{\Omega}v\partial_{\ell}w\,\mathrm{d}x,$

i.e., $z_{\ell}$ is the $\ell$th weak partial derivative of $v$. By
construction $T_{\partial\Omega}v=0$, and $v$ is therefore an element in $V$.

##  5 Transmission problem and the Robin–Robin method

In order to state the Robin–Robin method we reformulate the nonlinear elliptic
equation Eq. 5 on $\Omega$ into two equations on $\Omega_{i}$ connected via
$\Gamma$, i.e., we consider a nonlinear transmission problem. To this end, on
each $V_{i}$ we define $a_{i}:V_{i}\times V_{i}\rightarrow\mathbb{R}$ by

$a_{i}(u_{i},v_{i})=\int_{\Omega_{i}}\alpha(\nabla u_{i})\cdot\nabla
v_{i}+g(u_{i})v_{i}\,\mathrm{d}x.$

We also define $f_{i}=f|_{\Omega_{i}}\in L^{2}(\Omega_{i})$.

######  Lemma 5.1.

If the Sections 2 and 3 hold, then $a_{i}:V_{i}\times
V_{i}\rightarrow\mathbb{R}$ is well defined and satisfies the growth, strict
monotonicity and coercivity bounds stated in Lemma 2.4, with the terms
$(a,V,\Omega)$ replaced by $(a_{i},V_{i},\Omega_{i})$.

The weak form of the nonlinear transmission problem is then to find
$(u_{1},u_{2})\in V_{1}\times V_{2}$ such that

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

The framework given in Section 4 enables us to prove equivalence between the
nonlinear elliptic equation and the nonlinear transmission problem along the
same lines as done for linear equations [18, Lemma 1.2.1].

######  Theorem 5.2.

Let Sections 2 and 3 hold. If $u\in V$ solves Eq. 5, then
$(u_{1},u_{2})=(u|_{\Omega_{1}},u|_{\Omega_{2}})$ solves Eq. 10. Conversely,
if $(u_{1},u_{2})$ solves Eq. 10, then $u=\\{u_{1}\textrm{ on
}\Omega_{1};\,u_{2}\textrm{ on }\Omega_{2}\\}$ solves Eq. 5.

######  Proof 5.3.

Assume that $u\in V$ solves Eq. 5 and define
$(u_{1},u_{2})=(u|_{\Omega_{1}},u|_{\Omega_{2}})\in V_{1}\times V_{2}$. For
$v_{i}\in V_{i}^{0}$ we can extend by zero to $w_{i}\in V$, by using Lemma
4.9. Then

$a_{i}(u_{i},v_{i})=a(u,w_{i})=(f,w_{i})_{L^{2}(\Omega)}=(f_{i},v_{i})_{L^{2}(\Omega_{i})}.$

Moreover, $T_{1}u_{1}=T_{2}u_{2}$ follows immediately from Lemma 4.7. For an
arbitrary $\mu\in\Lambda$ let $v_{i}=R_{i}\mu$. Then, by Lemma 4.9,
$v=\\{v_{1}\textrm{ on }\Omega_{1};\,v_{2}\textrm{ on }\Omega_{2}\\}$ is an
element in $V$ and

$a_{1}(u_{1},R_{1}\mu)+a_{2}(u_{2},R_{2}\mu)=a(u,v)=(f,v)_{L^{2}(\Omega)}=(f_{1},R_{1}\mu)_{L^{2}(\Omega_{1})}+(f_{2},R_{2}\mu)_{L^{2}(\Omega_{2})}.$

This proves that $(u_{1},u_{2})$ solves Eq. 10. Conversely, let
$(u_{1},u_{2})\in V_{1}\times V_{2}$ be a solution to Eq. 10 and define
$u=\\{u_{1}\textrm{ on }\Omega_{1};\,u_{2}\textrm{ on }\Omega_{2}\\}$. By
Lemma 4.9, we have that $u\in V$. Next, consider $v\in V$ and let
$v_{i}=v|_{\Omega_{i}}\in V_{i}$. From Lemma 4.7 we have that $\mu=T_{i}v_{i}$
is well defined and $\mu\in\Lambda$. The observation that $v_{i}-R_{i}\mu\in
V_{i}^{0}$, for $i=1,2$, then implies the equality

$\displaystyle a(u,v)$
$\displaystyle=\sum_{i=1}^{2}a_{i}(u_{i},v_{i}-R_{i}\mu)+a_{i}(u_{i},R_{i}\mu)=\sum_{i=1}^{2}(f_{i},v_{i}-R_{i}\mu)_{L^{2}(\Omega_{i})}+a_{i}(u_{i},R_{i}\mu)$
$\displaystyle=\sum_{i=1}^{2}(f_{i},v_{i})_{L^{2}(\Omega_{i})}+a_{i}(u_{i},R_{i}\mu)-(f_{i},R_{i}\mu)_{L^{2}(\Omega_{i})}=(f,v)_{L^{2}(\Omega)}.$

As $v$ can be chosen arbitrarily, $u$ solves Eq. 5.

######  Remark 5.4.

As the nonlinear elliptic equation Eq. 5 has a unique weak solution, Theorem
5.2 implies that the same holds true for the nonlinear transmission problem
Eq. 10.

######  Remark 5.5.

The equivalence can easily be generalized to more than two subdomains without
cross points as in Fig. 1(c). However, for domain decompositions with cross
points such as in Fig. 1(d), the situation is more delicate. A proof of the
equivalence for quasilinear equations in $H^{1}(\Omega)$ can be found in [20].
This result can most likely be generalized to our $W^{1,p}(\Omega)$-setting,
but we will study this aspects elsewhere.

In order to approximate the weak solution $(u^{n}_{1},u^{n}_{2})\in
V_{1}\times V_{2}$ of the nonlinear transmission problem in a parallell
fashion, we consider the Robin–Robin method. Multiplying by test functions and
formally applying Green’s formula to the equations Eq. 4 yield that the weak
form of the method is given by finding $(u^{n}_{1},u^{n}_{2})\in V_{1}\times
V_{2}$, for $n=1,2,\ldots,$ such that

(11) $\left\\{\begin{aligned}
&a_{1}(u_{1}^{n+1},v_{1})=(f_{1},v_{1})_{L^{2}(\Omega_{1})},&&\text{for all
}v_{1}\in V_{1}^{0}\\\
&a_{1}(u_{1}^{n+1},R_{1}\mu)-(f_{1},R_{1}\mu)_{L^{2}(\Omega_{1})}+a_{2}(u_{2}^{n},R_{2}\mu)&&\\\
&\qquad-(f_{2},R_{2}\mu)_{L^{2}(\Omega_{2})}=s(T_{2}u_{2}^{n}-T_{1}u_{1}^{n+1},\mu)_{L^{2}(\Gamma)},&&\text{for
all }\mu\in\Lambda,\\\\[5.0pt]
&a_{2}(u_{2}^{n+1},v_{2})=(f_{2},v_{2})_{L^{2}(\Omega_{2})},&&\text{for all
}v_{2}\in V_{2}^{0}\\\
&a_{2}(u_{2}^{n+1},R_{2}\mu)-(f_{2},R_{2}\mu)_{L^{2}(\Omega_{2})}+a_{1}(u_{1}^{n+1},R_{1}\mu)&&\\\
&\qquad-(f_{1},R_{1}\mu)_{L^{2}(\Omega_{1})}=s(T_{1}u_{1}^{n+1}-T_{2}u_{2}^{n+1},\mu)_{L^{2}(\Gamma)},&&\text{for
all }\mu\in\Lambda,\end{aligned}\right.$

where $u^{0}_{2}\in V_{2}$ is an initial guess and $s>0$ is the given method
parameter.

##  6 Interface formulations

The ambition is now to reformulate the nonlinear transmission problem and the
Robin–Robin method, which are all given on the domains $\Omega_{i}$, into
problems and methods only stated on the interface $\Gamma$. As a preparation,
we observe that nonlinear elliptic equations on $\Omega_{i}$ with
inhomogeneous Dirichlet conditions have unique weak solutions.

######  Lemma 6.1.

If the Sections 2 and 3 hold, then for each $\eta\in\Lambda_{i}$ there exists
a unique $u_{i}\in W^{1,p}(\Omega_{i})$ such that

(12) $a_{i}(u_{i},v_{i})=(f_{i},v_{i})_{L^{2}(\Omega_{i})},\quad\text{for all
}v_{i}\in V^{0}_{i},$

and $T_{\partial\Omega_{i}}u_{i}=E_{i}\eta$ in
$W^{1-1/p,p}(\partial\Omega_{i})$.

The proof can, e.g., be found in [19, Theorem 2.36]. With the notation of
Lemma 6.1, consider the operator

$F_{i}:\eta\mapsto u_{i},$

i.e., the map from a given boundary value on $\Gamma$ to the corresponding
weak solution of the nonlinear elliptic problem Eq. 12 on $\Omega_{i}$. From
the statement of Lemma 6.1 we see that

$F_{i}:\Lambda_{i}\to V_{i}\quad\text{and}\quad T_{i}F_{i}\eta=\eta\text{ for
}\eta\in\Lambda_{i}.$

In other words, the operator $F_{i}$ is a nonlinear right inverse to $T_{i}$.
This property will be frequently used as it, together with the boundedness and
linearity of $T_{i}$, gives rise to bounds of the forms

$\|\eta\|_{\Lambda_{i}}\leq
C_{i}\|F_{i}\eta\|_{V_{i}}\quad\text{and}\quad\|\eta-\mu\|_{\Lambda_{i}}\leq
C_{i}\|F_{i}\eta-F_{i}\mu\|_{V_{i}}.$

We can now define the nonlinear Steklov–Poincaré operators as

$\displaystyle\langle
S_{i}\eta,\mu\rangle=a_{i}(F_{i}\eta,R_{i}\mu)-(f_{i},R_{i}\mu)_{L^{2}(\Omega_{i})},\quad\text{for
all }\eta,\mu\in\Lambda_{i},\quad\text{and}$ $\displaystyle\langle
S\eta,\mu\rangle=\sum_{i=1}^{2}\langle
S_{i}\eta,\mu\rangle=\sum_{i=1}^{2}a_{i}(F_{i}\eta,R_{i}\mu)-(f_{i},R_{i}\mu)_{L^{2}(\Omega_{i})},\quad\text{for
all }\eta,\mu\in\Lambda.$

Thus, we may restate the nonlinear transmission problem Eq. 10 as the
Steklov–Poincaré equation, i.e., finding $\eta\in\Lambda$ such that

(13) $\langle S\eta,\mu\rangle=0,\quad\text{for all }\mu\in\Lambda.$

That the reformulation is possible follows directly from the definitions of
the operators $F_{i}$ and $S$, but we state this as a lemma for future
reference.

######  Lemma 6.2.

Let the Sections 2 and 3 hold. If $(u_{1},u_{2})$ solves Eq. 10, then
$\eta=T_{1}u_{1}=T_{2}u_{2}$ solves Eq. 13. Conversely, if $\eta$ solves Eq.
13, then $(u_{1},u_{2})=(F_{1}\eta,F_{2}\eta)$ solves Eq. 10.

We next turn to the Robin–Robin method, which is equivalent to the
Peaceman–Rachford splitting on the interface $\Gamma$. The weak form of the
splitting is given by finding
$(\eta^{n}_{1},\eta^{n}_{2})\in\Lambda_{1}\times\Lambda_{2}$, for
$n=1,2,\ldots,$ such that

(14) $\left\\{\begin{aligned}
\langle(sJ+S_{1})\eta_{1}^{n+1},\mu\rangle&=\langle(sJ-
S_{2})\eta_{2}^{n},\mu\rangle,\\\
\langle(sJ+S_{2})\eta_{2}^{n+1},\mu\rangle&=\langle(sJ-
S_{1})\eta_{1}^{n+1},\mu\rangle,\end{aligned}\right.$

for all $\mu\in\Lambda$, where $\eta_{2}^{0}\in\Lambda_{2}$ is an initial
guess.

The observation that the Robin–Robin method and the Peaceman–Rachford
splitting are equivalent for linear elliptic equations was made in [1]. The
equivalence was also utilized in [5, Section 4.4.1] for the linear setting of
the Stokes–Darcy coupling.

######  Lemma 6.3.

Let the Sections 2 and 3 be valid. If $(u^{n}_{1},u^{n}_{2})_{n\geq 1}$ is a
weak Robin–Robin approximation Eq. 11, then
$(\eta^{n}_{1},\eta^{n}_{2})_{n\geq 1}=(T_{1}u^{n}_{1},T_{2}u^{n}_{2})_{n\geq
1}$ is a weak Peaceman–Rachford approximation Eq. 14, with
$\eta^{0}_{2}=T_{2}u^{0}_{2}$. Conversely, if
$(\eta^{n}_{1},\eta^{n}_{2})_{n\geq 1}$ is given by Eq. 14, then
$(u^{n}_{1},u^{n}_{2})_{n\geq 1}=(F_{1}\eta^{n}_{1},F_{2}\eta^{n}_{2})_{n\geq
1}$ fulfils Eq. 11, with $u^{0}_{2}=F_{2}\eta^{0}_{2}$.

######  Proof 6.4.

First assume that $(u^{n}_{1},u^{n}_{2})_{n\geq 1}\subset V_{1}\times V_{2}$
is a weak Robin–Robin approximation and define
$\eta^{n}_{i}=T_{i}u^{n}_{i}\in\Lambda_{i}$. This definition, the existence of
a unique solution of Eq. 12 and the first and third assertions of Eq. 11 yield
the identification $u_{i}^{n}=F_{i}\eta_{i}^{n}$. Inserting this into the
second and fourth assertion of Eq. 11 gives us

$\displaystyle
s(T_{1}F_{1}\eta_{1}^{n+1},\mu)_{L^{2}(\Gamma)}+a_{1}(F_{1}\eta_{1}^{n+1},R_{1}\mu)-(f,R_{1}\mu)_{L^{2}(\Omega_{1})}$
$\displaystyle\quad=s(T_{2}F_{2}\eta_{2}^{n},\mu)_{L^{2}(\Gamma)}-a_{2}(F_{2}\eta_{2}^{n},R_{2}\mu)+(f,R_{2}\mu)_{L^{2}(\Omega_{2})},\quad\text{and}$
$\displaystyle
s(T_{2}F_{2}\eta_{2}^{n+1},\mu)_{L^{2}(\Gamma)}+a_{2}(F_{2}\eta_{2}^{n+1},R_{2}\mu)-(f,R_{2}\mu)_{L^{2}(\Omega_{2})}$
$\displaystyle\quad=s(T_{1}F_{1}\eta_{1}^{n+1},\mu)_{L^{2}(\Gamma)}-a_{1}(F_{1}\eta_{1}^{n+1},R_{1}\mu)+(f,R_{1}\mu)_{L^{2}(\Omega_{1})},$

for all $\mu\in\Lambda$, which is precisely the weak form of the
Peaceman–Rachford splitting Eq. 14, with $\eta^{0}_{2}=T_{2}u^{0}_{2}$.
Conversely, suppose that $(\eta^{n}_{1},\eta^{n}_{2})_{n\geq
1}\subset\Lambda_{1}\times\Lambda_{2}$ is a weak Peaceman–Rachford
approximation and define $u_{i}^{n}=F_{i}\eta^{n}_{i}\in V_{i}$. Inserting
this into Eq. 14 directly gives that $(u^{n}_{1},u^{n}_{2})_{n\geq 1}$, with
$u_{2}^{0}=F_{2}\eta^{0}_{2}$, is a weak Robin–Robin approximation Eq. 11.

######  Remark 6.5.

For now we do not know if the weak Robin–Robin and Peaceman–Rachford
approximations actually exist, but we will return to this issue shortly.

##  7 Properties of the nonlinear Steklov–Poincaré operators

We proceed by deriving the central properties of the Steklov–Poincaré
operators $S_{i},S$ when interpreted as maps from $\Lambda_{i},\Lambda$ into
the corresponding dual spaces.

######  Lemma 7.1.

If the Sections 2 and 3 hold, then
$S_{i}:\Lambda_{i}\rightarrow\Lambda_{i}^{*}$ and
$S:\Lambda\rightarrow\Lambda^{*}$ are well defined.

######  Proof 7.2.

The linearity of the functionals $S_{i}\eta$ and $S\eta$ follow by definition.
As $F_{i}\eta\in V_{i}$ we have, by Lemma 5.1, that

$\displaystyle|\langle S_{i}\eta,\mu\rangle|$
$\displaystyle\leq|a_{i}(F_{i}\eta,R_{i}\mu)|+|(f_{i},R_{i}\mu)_{L^{2}(\Omega_{i})}|$
$\displaystyle\leq c_{i}(\|\nabla
F_{i}\eta\|^{p-1}_{L^{p}(\Omega_{i})^{d}}\|\nabla
R_{i}\mu\|^{\phantom{p-1}}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\eta\|_{L^{r}(\Omega_{i})}^{r-1}\|R_{i}\mu\|^{\phantom{r-1}}_{L^{r}(\Omega_{i})})$
$\displaystyle\quad+\|f_{i}\|_{L^{2}(\Omega_{i})}\|R_{i}\mu\|_{L^{2}(\Omega_{i})}$
$\displaystyle\leq C_{i}(\|\nabla
F_{i}\eta\|_{V_{i}},\|f_{i}\|_{L^{2}(\Omega_{i})})\|R_{i}\mu\|_{V_{i}}\leq
C_{i}\|\mu\|_{\Lambda_{i}},$

for all $\mu\in\Lambda_{i}$. Thus, $S_{i}\eta$ is a bounded functional on
$\Lambda_{i}$. The boundedness of $S\eta$ follows directly by summing up the
bounds for $S_{i}$ .

######  Lemma 7.3.

If the Sections 2 and 3 hold, then the operators
$S_{i}:\Lambda_{i}\rightarrow\Lambda_{i}^{*}$ and
$S:\Lambda\rightarrow\Lambda^{*}$ are strictly monotone with

$\langle S_{i}\eta-S_{i}\mu,\eta-\mu\rangle\geq
c_{i}\bigl{(}\|\nabla(F_{i}\eta-
F_{i}\mu)\|_{L^{p}(\Omega_{i})^{d}}^{p}+\|F_{i}\eta-
F_{i}\mu\|_{L^{r}(\Omega_{i})}^{r}\bigr{)},$

for all $\eta,\mu\in\Lambda_{i}$, and

$\langle S\eta-S\mu,\eta-\mu\rangle\geq
c\sum_{i=1}^{2}\bigl{(}\|\nabla(F_{i}\eta-
F_{i}\mu)\|_{L^{p}(\Omega_{i})^{d}}^{p}+\|F_{i}\eta-
F_{i}\mu\|_{L^{r}(\Omega_{i})}^{r}\bigr{)},$

for all $\eta,\mu\in\Lambda$, respectively.

######  Proof 7.4.

Since, $w_{i}=R_{i}(\eta-\mu)-(F_{i}\eta-F_{i}\mu)\in V_{i}^{0}$, for all
$\eta,\mu\in\Lambda_{i}$, we have according to the definition of $F_{i}$ that

(15) $a_{i}(F_{i}\eta,w_{i})-a_{i}(F_{i}\mu,w_{i})=0.$

By this equality and Lemma 5.1, it follows that

$\displaystyle\langle S_{i}\eta-S_{i}\mu$
$\displaystyle,\eta-\mu\rangle=a_{i}\bigl{(}F_{i}\eta,R_{i}(\eta-\mu)\bigr{)}-a_{i}\bigl{(}F_{i}\mu,R_{i}(\eta-\mu)\bigr{)}$
$\displaystyle=a_{i}(F_{i}\eta,w_{i})+a_{i}(F_{i}\eta,F_{i}\eta-
F_{i}\mu)-a_{i}(F_{i}\mu,w_{i})-a_{i}(F_{i}\mu,F_{i}\eta-F_{i}\mu)$
$\displaystyle\geq c_{i}\bigl{(}\|\nabla(F_{i}\eta-
F_{i}\mu)\|_{L^{p}(\Omega_{i})^{d}}^{p}+\|F_{i}\eta-
F_{i}\mu\|_{L^{r}(\Omega_{i})}^{r}\bigr{)},$

for all $\eta,\mu\in\Lambda_{i}$, which proves the monotonicity bound for
$S_{i}$. The bound for $S$ follows directly by summing up the bounds for
$S_{i}$.

######  Lemma 7.5.

If the Sections 2 and 3 hold, then the operators
$S_{i}:\Lambda_{i}\rightarrow\Lambda_{i}^{*}$ and
$S:\Lambda\rightarrow\Lambda^{*}$ are coercive, i.e,

$\lim_{\|\eta\|_{\Lambda_{i}}\to\infty}\frac{\langle
S_{i}\eta,\eta\rangle}{\|\eta\|_{\Lambda_{i}}}=\infty\quad\text{and}\quad\lim_{\|\eta\|_{\Lambda}\to\infty}\frac{\langle
S\eta,\eta\rangle}{\|\eta\|_{\Lambda}}=\infty.$

######  Proof 7.6.

As $\|F_{i}\eta\|_{V_{i}}=\|\nabla
F_{i}\eta\|_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\eta\|_{L^{r}(\Omega_{i})}\geq
c_{i}\|\eta\|_{\Lambda_{i}}$, we have that

$P(\|\nabla
F_{i}\eta\|_{L^{p}(\Omega_{i})^{d}},\|F_{i}\eta\|_{L^{r}(\Omega_{i})})\to\infty,\quad\text{as
}\|\eta\|_{\Lambda_{i}}\to\infty,$

where $P(x,y)=(x^{p}+y^{r})/(x+y)$. In particular, we assume from now on that

$P(\|\nabla
F_{i}\eta\|_{L^{p}(\Omega_{i})^{d}},\|F_{i}\eta\|_{L^{r}(\Omega_{i})})\geq\|f_{i}\|_{L^{2}(\Omega_{i})}.$

By observing that $R_{i}\eta-F_{i}\eta\in V_{i}^{0}$, Lemma 5.1 yields the
lower bound

$\displaystyle\langle S_{i}\eta,\eta\rangle$
$\displaystyle=a_{i}(F_{i}\eta,F_{i}\eta)+a_{i}(F_{i}\eta,R_{i}\eta-
F_{i}\eta)-(f_{i},R_{i}\eta)_{L^{2}(\Omega_{i})}$
$\displaystyle=a_{i}(F_{i}\eta,F_{i}\eta)+(f_{i},R_{i}\eta-
F_{i}\eta)_{L^{2}(\Omega_{i})}-(f_{i},R_{i}\eta)_{L^{2}(\Omega_{i})}$
$\displaystyle\geq c_{i}(\|\nabla
F_{i}\eta\|^{p}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\eta\|^{r}_{L^{r}(\Omega_{i})})-(f_{i},F_{i}\eta)_{L^{2}(\Omega_{i})}$
$\displaystyle\geq c_{i}P(\|\nabla
F_{i}\eta\|_{L^{p}(\Omega_{i})^{d}},\|F_{i}\eta\|_{L^{r}(\Omega_{i})})\|F_{i}\eta\|_{V_{i}}-\|f_{i}\|_{L^{2}(\Omega_{i})}\|F_{i}\eta\|_{V_{i}}$
$\displaystyle\geq c_{i}\bigl{(}P(\|\nabla
F_{i}\eta\|_{L^{p}(\Omega_{i})^{d}},\|F_{i}\eta\|_{L^{r}(\Omega_{i})})-\|f_{i}\|_{L^{2}(\Omega_{i})}\bigr{)}\|\eta\|_{\Lambda_{i}},$

which implies that $S_{i}$ is coercive. For $S$ we obtain that

$\frac{\langle
S\eta,\eta\rangle}{\|\eta\|_{\Lambda}}\geq\frac{\sum_{i=1}^{2}\bigl{(}c_{i}\,P(\|\nabla
F_{i}\eta\|_{L^{p}(\Omega_{i})^{d}},\|F_{i}\eta\|_{L^{r}(\Omega_{i})})-\|f_{i}\|_{L^{2}}\bigr{)}\|\eta\|_{\Lambda_{i}}}{\|\eta\|_{\Lambda_{1}}+\|\eta\|_{\Lambda_{2}}},$

which tends to infinity as $\|\eta\|_{\Lambda}$ tends to infinity. Thus, $S$
is also coercive.

In order to prove that the operators $S_{i},S$ are demicontinuous, we first
consider the continuity of the operators $F_{i}$.

######  Lemma 7.7.

If the Sections 2 and 3 hold, then the nonlinear operators
$F_{i}:\Lambda_{i}\to V_{i}$ are continuous.

######  Proof 7.8.

Let $\eta,\mu$ be elements in $\Lambda_{i}$. Using the equality Eq. 15
together with Lemma 5.1 gives us the bound

(16) $\displaystyle c_{i}$ $\displaystyle(\|\nabla(F_{i}\eta-
F_{i}\mu)\|^{p}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\eta-
F_{i}\mu\|^{r}_{L^{r}(\Omega_{i})})$ $\displaystyle\leq
a_{i}(F_{i}\eta,F_{i}\eta-F_{i}\mu)-a_{i}(F_{i}\mu,F_{i}\eta-F_{i}\mu)$
$\displaystyle=a_{i}\bigl{(}F_{i}\eta,R_{i}(\eta-\mu)\bigr{)}-a_{i}(F_{i}\eta,w_{i})-a_{i}\bigl{(}F_{i}\mu,R_{i}(\eta-\mu)\bigr{)}+a_{i}(F_{i}\mu,w_{i})$
$\displaystyle\leq C_{i}\bigl{(}\|\nabla
F_{i}\eta\|^{p-1}_{L^{p}(\Omega_{i})^{d}}\|\nabla
R_{i}(\eta-\mu)\|^{\phantom{p-1}}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\eta\|_{L^{r}(\Omega_{i})}^{r-1}\|R_{i}(\eta-\mu)\|^{\phantom{r-1}}_{L^{r}(\Omega_{i})}$
$\displaystyle\quad+\|\nabla F_{i}\mu\|^{p-1}_{L^{p}(\Omega_{i})^{d}}\|\nabla
R_{i}(\eta-\mu)\|^{\phantom{p-1}}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\mu\|_{L^{r}(\Omega_{i})}^{r-1}\|R_{i}(\eta-\mu)\|^{\phantom{r-1}}_{L^{r}(\Omega_{i})}\bigr{)}$
$\displaystyle\leq C_{i}\big{(}\|\nabla
F_{i}\eta\|^{p-1}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\eta\|_{L^{r}(\Omega_{i})}^{r-1}$
$\displaystyle\quad+\|\nabla
F_{i}\mu\|^{p-1}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\mu\|_{L^{r}(\Omega_{i})}^{r-1}\big{)}\|\eta-\mu\|_{\Lambda_{i}}.$

Letting $\mu=0$ in Eq. 16 and employing the inequality
$|x|^{p}-2^{p-1}|y|^{p}\leq 2^{p-1}|x-y|^{p}$, twice, yield that

$\displaystyle c_{i}$ $\displaystyle(\|\nabla
F_{i}\eta\|^{p}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\eta\|^{r}_{L^{r}(\Omega_{i})}-2^{p-1}\|\nabla
F_{i}0\|^{p}_{L^{p}(\Omega_{i})^{d}}-2^{r-1}\|F_{i}0\|^{r}_{L^{r}(\Omega_{i})})$
$\displaystyle\leq c_{i}(2^{p-1}\|\nabla(F_{i}\eta-
F_{i}0)\|^{p}_{L^{p}(\Omega_{i})^{d}}+2^{r-1}\|F_{i}\eta-
F_{i}0\|^{r}_{L^{r}(\Omega_{i})})$ $\displaystyle\leq C_{i}\big{(}\|\nabla
F_{i}\eta\|^{p-1}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\eta\|_{L^{r}(\Omega_{i})}^{r-1}+\|\nabla
F_{i}0\|^{p-1}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}0\|_{L^{r}(\Omega_{i})}^{r-1}\big{)}\|\eta\|_{\Lambda_{i}}.$

Thus, we have a bound of the form

(17) $\frac{\|\nabla
F_{i}\eta\|^{p}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\eta\|^{r}_{L^{r}(\Omega_{i})}-c_{1}}{\|\nabla
F_{i}\eta\|^{p-1}_{L^{p}(\Omega_{i})^{d}}+\|F_{i}\eta\|_{L^{r}(\Omega_{i})}^{r-1}+c_{2}}\leq
C_{i}\|\eta\|_{\Lambda_{i}},$

for every $\eta\in\Lambda_{i}$, where $c_{\ell}=c_{\ell}(\|\nabla
F_{i}0\|_{L^{p}(\Omega_{i})^{d}},\|F_{i}0\|_{L^{r}(\Omega_{i})})\geq 0$.

Assume that $\eta^{k}\to\eta$ in $\Lambda_{i}$. As $\eta^{k}$ is bounded in
$\Lambda_{i}$, the bound Eq. 17 implies that $\nabla F_{i}\eta^{k}$ and
$F_{i}\eta^{k}$ are bounded in $L^{p}(\Omega_{i})^{d}$ and
$L^{r}(\Omega_{i})$, respectively. By setting $\mu=\eta^{k}$ in Eq. 16, we
finally obtain that $\nabla F_{i}\eta^{k}\to\nabla F_{i}\eta$ in in
$L^{p}(\Omega_{i})^{d}$ and $F_{i}\eta^{k}\to F_{i}\eta$ in
$L^{r}(\Omega_{i})$, i.e., $F_{i}\eta^{k}\to F_{i}\eta$ in $V_{i}$.

######  Lemma 7.9.

If the Sections 2 and 3 hold, then the operators
$S_{i}:\Lambda_{i}\rightarrow\Lambda_{i}^{*}$ and
$S:\Lambda\rightarrow\Lambda^{*}$ are demicontinuous, i.e., if
$\eta_{i}^{k}\to\eta_{i}$ in $\Lambda_{i}$ and $\eta^{k}\to\eta$ in $\Lambda$
then

$\langle S_{i}\eta^{k}_{i}-S_{i}\eta_{i},\mu_{i}\rangle\to
0\quad\text{and}\quad\langle S\eta^{k}-S\eta,\mu\rangle\to 0,$

for all $\mu_{i}\in\Lambda_{i}$ and $\mu\in\Lambda$.

######  Proof 7.10.

Assume that $\eta_{i}^{k}\to\eta_{i}$ in $\Lambda_{i}$. Lemma 7.7 then implies
that $\nabla F_{i}\eta^{k}\to\nabla F_{i}\eta$ in $L^{p}(\Omega_{i})^{d}$ and
$F_{i}\eta^{k}\to F_{i}\eta$ in $L^{r}(\Omega_{i})$. By the assumed continuity
and boundedness of the functions
$\alpha:z\mapsto\bigl{(}\alpha_{1}(z),\ldots,\alpha_{d}(z)\bigr{)}$ and $g$,
we also have that the corresponding Nemyckii operators
$\alpha_{\ell}:L^{p}(\Omega_{i})^{d}\to
L^{\nicefrac{{p}}{{(p-1)}}}(\Omega_{i})$ and $g:L^{r}(\Omega_{i})\to
L^{\nicefrac{{r}}{{(r-1)}}}(\Omega_{i})$ are continuous [25, Proposition
26.6]. The demicontinuity of $S_{i}$ then holds, as

$\displaystyle|\langle S_{i}\eta^{k}_{i}-S_{i}\eta_{i},\mu_{i}\rangle|$
$\displaystyle\leq\bigl{(}\,\sum_{\ell=1}^{d}\|\alpha_{\ell}(\nabla
F_{i}\eta_{i}^{k})-\alpha_{\ell}(\nabla
F_{i}\eta_{i})\|_{L^{\nicefrac{{p}}{{(p-1)}}}(\Omega_{i})}$
$\displaystyle\quad+\|g(F_{i}\eta_{i}^{k})-g(F_{i}\eta_{i})\|_{L^{\nicefrac{{r}}{{(r-1)}}}(\Omega_{i})}\bigr{)}\|R_{i}\mu_{i}\|_{V_{i}},$

for every $\mu_{i}\in\Lambda_{i}$. The demicontinuity of $S_{i}$ directly
implies the same property for $S$, as a convergent sequence $\\{\eta^{k}\\}$
in $\Lambda$ is also convergent in $\Lambda_{1}$ and $\Lambda_{2}.$

######  Theorem 7.11.

If the Sections 2 and 3 hold, then the nonlinear Steklov–Poincaré operators
$S_{i}:\Lambda_{i}\rightarrow\Lambda_{i}^{*}$ and
$S:\Lambda\rightarrow\Lambda^{*}$ are bijective.

######  Proof 7.12.

The spaces $\Lambda_{i}$ and $\Lambda$ are real, reflexive Banach spaces and,
by Lemmas 7.3, 7.5 and 7.9, the operators
$S_{i}:\Lambda_{i}\rightarrow\Lambda_{i}^{*}$ and
$S:\Lambda\rightarrow\Lambda^{*}$ are all strictly monotone, coercive and
demicontinuous. With these properties, the Browder–Minty theorem; see, e.g.,
[25, Theorem 26.A(a,c,f)], implies that the operators are bijective.

The next corollary follows by the same argumentation as for the bijectivity of
$S_{i}$.

######  Corollary 7.13.

If the Sections 2 and 3 hold, then the operators
$sJ+S_{i}:\Lambda_{i}\to\Lambda_{i}^{*}$ are bijective, for every $s>0$.

##  8 Existence and convergence of the Robin–Robin method

The weak form of the Peaceman–Rachford splitting Eq. 14 seems to be too
general for a convergence analysis. To remedy this, we will restrict the
domains of the operators $S_{i},S$ such that the Steklov–Poincaré equation Eq.
13 and the Peaceman–Rachford splitting can be interpreted on $L^{2}(\Gamma)$
instead of on the dual spaces $\Lambda_{i}^{*},\Lambda^{*}$. This comes at the
cost of requiring more regularity of the weak solution and of the initial
guess $\eta^{0}_{2}$.

More precisely, we define the operators
$\mathcal{S}_{i}:D(\mathcal{S}_{i})\subseteq L^{2}(\Gamma)\to L^{2}(\Gamma)$
as

$D(\mathcal{S}_{i})=\\{\mu\in\Lambda_{i}:S_{i}\mu\in
L^{2}(\Gamma)^{*}\\}\quad\text{and}\quad\mathcal{S}_{i}\mu=J^{-1}S_{i}\mu\text{
for }\mu\in D(\mathcal{S}_{i}).$

Analogously, we introduce $\mathcal{S}:D(\mathcal{S})\subseteq
L^{2}(\Gamma)\to L^{2}(\Gamma)$ given by

$D(\mathcal{S})=\\{\mu\in\Lambda:S\mu\in
L^{2}(\Gamma)^{*}\\}\quad\text{and}\quad\mathcal{S}\mu=J^{-1}S\mu\text{ for
}\mu\in D(\mathcal{S}).$

As the zero functional obviously is an element in $L^{2}(\Gamma)^{*}$, the
unique solution $\eta\in\Lambda$ of the Steklov–Poincaré equation is in
$D(\mathcal{S})$ and

$\mathcal{S}\eta=0.$

######  Remark 8.1.

By the above construction, one obtains that $D(\mathcal{S}_{1})\cap
D(\mathcal{S}_{2})\subseteq D(\mathcal{S})$ and

$\mathcal{S}\mu=\mathcal{S}_{1}\mu+\mathcal{S}_{2}\mu,\quad\text{ for all
}\mu\in D(\mathcal{S}_{1})\cap D(\mathcal{S}_{2}).$

However, the definition of the domains do not ensure that $D(\mathcal{S})$ is
equal to $D(\mathcal{S}_{1})\cap D(\mathcal{S}_{2})$, as $(S_{1}+S_{2})\mu\in
L^{2}(\Gamma)^{*}$ does not necessarily imply that $S_{i}\mu\in
L^{2}(\Gamma)^{*}$.

If the weak solution of the nonlinear elliptic equation Eq. 5 satisfies the
additional regularity property stated in Section 2, then the corresponding
solution of $\mathcal{S}\eta=0$ is in fact an element in
$D(\mathcal{S}_{1})\cap D(\mathcal{S}_{2})$. This propagation of regularity
will be crucial when proving convergence of the Peaceman–Rachford splitting.

######  Lemma 8.2.

If the Sections 2, 2 and 3 hold and $\mathcal{S}\eta=0$, then $\eta\in
D(\mathcal{S}_{1})\cap D(\mathcal{S}_{2})$.

######  Proof 8.3.

As $u=\\{F_{1}\eta\text{ on }\Omega_{1};F_{2}\eta\text{ on }\Omega_{2}\\}$ is
the weak solution of Eq. 5, we have that

$\int_{\Omega}\alpha(\nabla u)\cdot\nabla
v\,\mathrm{d}x=-\int_{\Omega}\bigl{(}g(u)-f\bigr{)}v\,\mathrm{d}x,\quad\text{for
all }v\in C^{\infty}_{0}(\Omega).$

The restrictions on $r$ in Section 2 yield that $u\in
W^{1,p}(\Omega)\hookrightarrow L^{2(r-1)}(\Omega)$. This together with the
observation $|g(u)|^{2}\leq C|u|^{2(r-1)}$ implies that $g(u)-f\in
L^{2}(\Omega)$, i.e., the distributional divergence of $\alpha(\nabla u)$ is
in $L^{2}(\Omega)^{d}$. By Section 2 and restricting to $\Omega_{i}$, we
arrive at

$\displaystyle\alpha(\nabla F_{i}\eta)\in H(\mathrm{div},\Omega_{i})\cap
C(\overline{\Omega}_{i})^{d},\quad\alpha(\nabla F_{i}\eta)\cdot\nu_{i}\in
L^{\infty}(\partial\Omega_{i})\quad\text{and}$
$\displaystyle\quad\nabla\cdot\alpha(\nabla F_{i}\eta)=g(F_{i}\eta)-f_{i}\in
L^{2}(\Omega_{i}).$

The $H(\mathrm{div},\Omega_{i})$-version of Green’s formula [8, Chapter 1,
Corollary 2.1] then gives us

$\int_{\Omega_{i}}\alpha(\nabla F_{i}\eta)\cdot\nabla
v\,\mathrm{d}x=-\int_{\Omega_{i}}\nabla\cdot\alpha(\nabla
F_{i}\eta)v\,\mathrm{d}x+\int_{\partial\Omega_{i}}\alpha(\nabla
F_{i}\eta)\cdot\nu_{i}\,T_{\partial\Omega_{i}}v\,\mathrm{d}S,$

for all $v\in H^{1}(\Omega_{i})$. Hence,

$\displaystyle\langle S_{i}\eta,\mu\rangle$
$\displaystyle=\int_{\Omega_{i}}\alpha(\nabla F_{i}\eta)\cdot\nabla
R_{i}\mu\,\mathrm{d}x+\int_{\Omega_{i}}\bigl{(}g(F_{i}\eta)-f_{i}\bigr{)}R_{i}\mu\,\mathrm{d}x$
$\displaystyle=\int_{\partial\Omega_{i}}\alpha(\nabla
F_{i}\eta)\cdot\nu_{i}\,T_{\partial\Omega_{i}}R_{i}\mu\,\mathrm{d}S=\bigl{(}\alpha(\nabla
F_{i}\eta)\cdot\nu_{i},\mu\bigr{)}_{L^{2}(\Gamma)},\quad\text{for all
}\mu\in\Lambda_{i},$

which implies that $\eta\in D(\mathcal{S}_{i})$, for $i=1,2$.

######  Remark 8.4.

From the proof it is clear that the regularity assumption $\alpha(\nabla u)\in
C(\overline{\Omega})^{d}$ is stricter than necessary, and could be replaced by
assuming that the normal component of $\alpha(\nabla u)$ on $\Gamma$ can be
interpreted as an element in $L^{2}(\Gamma)^{*}$. However, characterizing the
spatial regularity of $u$ required to satisfy this weaker assumption demands a
more elaborate trace theory than the one considered in Section 3.

######  Lemma 8.5.

If the Sections 2 and 3 hold, then the operators $\mathcal{S}_{i}$ are
monotone, i.e.,

$(\mathcal{S}_{i}\eta-\mathcal{S}_{i}\mu,\eta-\mu)_{L^{2}(\Gamma)}\geq
0,\quad\text{for all }\eta,\mu\in D(\mathcal{S}_{i}),$

and the operators $sI+\mathcal{S}_{i}:D(\mathcal{S}_{i})\to L^{2}(\Gamma)$ are
bijective for any $s>0$.

######  Proof 8.6.

The monotonicity follows by Lemma 7.3, as

$\displaystyle(\mathcal{S}_{i}\eta-\mathcal{S}_{i}\mu,\eta-\mu)_{L^{2}(\Gamma)}$
$\displaystyle=(J^{-1}S_{i}\eta-J^{-1}S_{i}\mu,\eta-\mu)_{L^{2}(\Gamma)}$
$\displaystyle=\langle S_{i}\eta-S_{i}\mu,\eta-\mu\rangle\geq 0,\quad\text{for
all }\eta,\mu\in D(\mathcal{S}_{i})\subseteq\Lambda_{i}.$

For a fixed $s>0$ and an arbitrary $\mu\in L^{2}(\Gamma)$ we have, due to
Corollary 7.13, that there exists a unique $\eta\in\Lambda_{i}$ such that
$(sJ+S_{i})\eta=J\mu$ in $\Lambda_{i}^{*}$, i.e.,

$S_{i}\eta=J(\mu-s\eta)\in L^{2}(\Gamma)^{*}.$

Hence, $\eta\in D(\mathcal{S}_{i})$ and $(sI+\mathcal{S}_{i})\eta=\mu$ in
$L^{2}(\Gamma)$. The operators $sI+\mathcal{S}_{i}:D(\mathcal{S}_{i})\to
L^{2}(\Gamma)$ are therefore bijective.

The Peaceman–Rachford splitting on $L^{2}(\Gamma)$ is now given by finding
$(\eta^{n}_{1},\eta^{n}_{2})\in D(\mathcal{S}_{1})\times D(\mathcal{S}_{2})$,
for $n=1,2,\ldots,$ such that

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

where $\eta_{2}^{0}\in D(\mathcal{S}_{2})$ is an initial guess. Lemma 8.5 then
directly yields the existence of the approximation.

######  Corollary 8.7.

If the Sections 2 and 3 hold and $\eta_{2}^{0}\in D(\mathcal{S}_{2})$, then
there exists a unique Peaceman–Rachford approximation
$(\eta^{n}_{1},\eta^{n}_{2})_{n\geq 1}\subset D(\mathcal{S}_{1})\times
D(\mathcal{S}_{2})$ given by Eq. 18 in $L^{2}(\Gamma)$.

######  Corollary 8.8.

Let the Sections 2 and 3 hold, $\eta_{2}^{0}\in D(\mathcal{S}_{2})$ and set
$u^{0}_{2}=F_{2}\eta_{2}^{0}$. The Peaceman–Rachford approximation
$(\eta^{n}_{1},\eta^{n}_{2})_{n\geq 1}\subset D(\mathcal{S}_{1})\times
D(\mathcal{S}_{2})$ also satisfies the weak formulation Eq. 14, and
$(u^{n}_{1},u^{n}_{2})_{n\geq 1}=(F_{1}\eta^{n}_{1},F_{2}\eta^{n}_{2})_{n\geq
1}$ is a weak Robin–Robin approximation Eq. 11.

######  Proof 8.9.

Assume that $(\eta^{n}_{1},\eta^{n}_{2})_{n\geq 1}\subset
D(\mathcal{S}_{1})\times D(\mathcal{S}_{2})$ is a Peaceman–Rachford
approximation in $L^{2}(\Gamma)$. Then,

$\displaystyle\bigl{(}(sI+\mathcal{S}_{1})\eta_{1}^{n+1},\mu\bigr{)}_{L^{2}(\Gamma)}$
$\displaystyle=\bigl{(}(sI-\mathcal{S}_{2})\eta_{2}^{n},\mu\bigr{)}_{L^{2}(\Gamma)},$
$\displaystyle\text{for all }\mu\in L^{2}(\Gamma),$
$\displaystyle\bigl{(}(sI+\mathcal{S}_{1})\eta_{1}^{n+1},\mu\bigr{)}_{L^{2}(\Gamma)}$
$\displaystyle=\langle(sJ+S_{1})\eta_{1}^{n+1},\mu\rangle,$
$\displaystyle\text{for all }\mu\in\Lambda_{1},\quad\text{and}$
$\displaystyle\bigl{(}(sI-\mathcal{S}_{2})\eta_{2}^{n},\mu\bigr{)}_{L^{2}(\Gamma)}$
$\displaystyle=\langle(sJ-S_{2})\eta_{2}^{n},\mu\rangle,$
$\displaystyle\text{for all }\mu\in\Lambda_{2}.$

This implies that

$\langle(sJ+S_{1})\eta_{1}^{n+1},\mu\rangle=\langle(sJ-
S_{2})\eta_{2}^{n},\mu\rangle,\quad\text{for all
}\mu\in\Lambda=\Lambda_{1}\cap\Lambda_{2},$

i.e., the first assertion of Eq. 14 holds. The same argumentation yields that
the second assertion of Eq. 14 is valid. As
$(\eta^{n}_{1},\eta^{n}_{2})_{n\geq 1}$ satisfies Eq. 14, Lemma 6.3 directly
implies that $(u^{n}_{1},u^{n}_{2})_{n\geq
1}=(F_{1}\eta^{n}_{1},F_{2}\eta^{n}_{2})_{n\geq 1}$ is a weak Robin–Robin
approximation Eq. 11.

######  Remark 8.10.

At a first glance, finding an initial guess satisfying $\eta^{0}_{2}\in
D(\mathcal{S}_{2})$ might seem limiting, as the domain is not explicitly
given. However, such an initial guess can, e.g., be found by solving $\langle
S_{2}\eta_{2}^{0},\mu\rangle=0$, for all $\mu\in\Lambda_{2}$.

With this $L^{2}(\Gamma)$-framework the key part of the convergence proof
follows by the abstract result [14, Proposition 1]. For sake of completeness
we state a simplified version of the short proof in the current notation.

######  Lemma 8.11.

Consider the solution of $\mathcal{S}\eta=0$ and the Peaceman–Rachford
approximation $(\eta^{n}_{1},\eta^{n}_{2})_{n\geq 1}$. If $\eta_{2}^{0}\in
D(\mathcal{S}_{2})$ and the Sections 2, 2 and 3 hold, then

(19)
$(\mathcal{S}_{i}\eta^{n}_{i}-\mathcal{S}_{i}\eta,\eta^{n}_{i}-\eta)_{L^{2}(\Gamma)}\to
0,\quad\text{as }n\to\infty,$

for $i=1,2$.

######  Proof 8.12.

By the hypotheses and Lemma 8.2, we obtain that $\eta\in
D(\mathcal{S}_{1})\cap D(\mathcal{S}_{2})$ and
$\mathcal{S}_{1}\eta=-\mathcal{S}_{2}\eta$. Furthermore,

$\eta^{n+1}_{1}=(sI+\mathcal{S}_{1})^{-1}(sI-\mathcal{S}_{2})\eta^{n}_{2}\in
D(\mathcal{S}_{1})\text{ and
}\eta^{n+1}_{2}=(sI+\mathcal{S}_{2})^{-1}(sI-\mathcal{S}_{1})\eta^{n+1}_{1}\in
D(\mathcal{S}_{2}).$

Next, we introduce the notation

$\mu^{n}=(sI+\mathcal{S}_{2})\eta^{n}_{2},\quad\mu=(sI+\mathcal{S}_{2})\eta,\quad\lambda^{n}=(sI-\mathcal{S}_{2})\eta^{n}_{2}\quad\text{and}\quad\lambda=(sI-\mathcal{S}_{2})\eta,$

which yields the representations

$\displaystyle\eta$ $\displaystyle=\frac{\mu+\lambda}{2s},$
$\displaystyle\mathcal{S}_{2}\eta$ $\displaystyle=\frac{\mu-\lambda}{2},$
$\displaystyle\mathcal{S}_{1}\eta=\frac{\lambda-\mu}{2},$
$\displaystyle\eta^{n}_{2}$ $\displaystyle=\frac{\mu^{n}+\lambda^{n}}{2s},$
$\displaystyle\mathcal{S}_{2}\eta^{n}_{2}$
$\displaystyle=\frac{\mu^{n}-\lambda^{n}}{2},$ $\displaystyle\eta^{n+1}_{1}$
$\displaystyle=\frac{\mu^{n+1}+\lambda^{n}}{2s},$
$\displaystyle\mathcal{S}_{1}\eta^{n+1}_{1}$
$\displaystyle=\frac{\lambda^{n}-\mu^{n+1}}{2}.$

The monotonicity of $\mathcal{S}_{i}$ then gives the bounds

$\displaystyle 0$
$\displaystyle\leq(\mathcal{S}_{2}\eta^{n}_{2}-\mathcal{S}_{2}\eta,\eta^{n}_{2}-\eta)_{L^{2}(\Gamma)}$
$\displaystyle=\frac{1}{4s}\bigl{(}(\mu^{n}-\mu)-(\lambda^{n}-\lambda),(\mu^{n}-\mu)+(\lambda^{n}-\lambda)\bigr{)}_{L^{2}(\Gamma)}$
$\displaystyle=\frac{1}{4s}\bigl{(}\|\mu^{n}-\mu\|^{2}_{L^{2}(\Gamma)}-\|\lambda^{n}-\lambda\|^{2}_{L^{2}(\Gamma)}\bigr{)},$

and

$\displaystyle 0$
$\displaystyle\leq(\mathcal{S}_{1}\eta^{n+1}_{1}-\mathcal{S}_{1}\eta,\eta^{n+1}_{1}-\eta)_{L^{2}(\Gamma)}$
$\displaystyle=\frac{1}{4s}\bigl{(}(\lambda^{n}-\lambda)-(\mu^{n+1}-\mu),(\lambda^{n}-\lambda)+(\mu^{n+1}-\mu)\bigr{)}_{L^{2}(\Gamma)}$
$\displaystyle=\frac{1}{4s}\bigl{(}\|\lambda^{n}-\lambda\|^{2}_{L^{2}(\Gamma)}-\|\mu^{n+1}-\mu\|^{2}_{L^{2}(\Gamma)}\bigr{)}.$

Putting this together yields that

$\|\mu^{n+1}-\mu\|^{2}_{L^{2}(\Gamma)}\leq\|\lambda^{n}-\lambda\|^{2}_{L^{2}(\Gamma)}\leq\|\mu^{n}-\mu\|^{2}_{L^{2}(\Gamma)},$

and we obtain the telescopic sum

$0\leq\sum_{n=0}^{N}\bigl{(}\|\mu^{n}-\mu\|^{2}_{L^{2}(\Gamma)}-\|\mu^{n+1}-\mu\|^{2}_{L^{2}(\Gamma)}\bigr{)}\leq\|\mu^{0}-\mu\|^{2}_{L^{2}(\Gamma)}-\|\mu^{N+1}-\mu\|^{2}_{L^{2}(\Gamma)},$

i.e.,
$\|\mu^{n}-\mu\|^{2}_{L^{2}(\Gamma)}-\|\mu^{n+1}-\mu\|^{2}_{L^{2}(\Gamma)}\to
0$ as $n\to\infty$. The latter together with the bounds above imply the sought
after limits Eq. 19.

######  Theorem 8.13.

Consider the Peaceman–Rachford approximation
$(\eta^{n}_{1},\eta^{n}_{2})_{n\geq 1}$, given by Eq. 18, of the
Steklov–Poincaré equation $\mathcal{S}\eta=0$ in $L^{2}(\Gamma)$, together
with the corresponding Robin–Robin approximation $(u^{n}_{1},u^{n}_{2})_{n\geq
1}=(F_{1}\eta^{n}_{1},F_{2}\eta^{n}_{2})_{n\geq 1}$ of the weak solution
$u=\\{F_{1}\eta\text{ on }\Omega_{1};F_{2}\eta\text{ on }\Omega_{2}\\}$ to the
nonlinear elliptic equation Eq. 5.
If $\eta_{2}^{0}\in D(\mathcal{S}_{2})$ and the Sections 2, 2 and 3 hold, then

(20)
$\|\eta^{n}_{1}-\eta\|_{\Lambda_{1}}+\|\eta^{n}_{2}-\eta\|_{\Lambda_{2}}\to
0,\quad\text{and}\quad\|u^{n}_{1}-u\|_{W^{1,p}(\Omega_{1})}+\|u^{n}_{2}-u\|_{W^{1,p}(\Omega_{2})}\to
0,$

as $n$ tends to infinity.

######  Proof 8.14.

By the monotonicity bound in Lemma 7.3, the property that
$\eta^{n}_{i},\eta\in D(\mathcal{S}_{i})$ and Lemma 8.11, we have the limits

$\displaystyle
c_{i}\bigl{(}\|\nabla(F_{i}\eta^{n}_{i}-F_{i}\eta)\|_{L^{p}(\Omega_{i})^{d}}^{p}$
$\displaystyle+\|F_{i}\eta^{n}_{i}-F_{i}\eta\|_{L^{r}(\Omega_{i})}^{r}\bigr{)}\leq\langle
S_{i}\eta^{n}_{i}-S_{i}\eta,\eta^{n}_{i}-\eta\rangle$
$\displaystyle=(\mathcal{S}_{i}\eta^{n}_{i}-\mathcal{S}_{i}\eta,\eta^{n}_{i}-\eta)_{L^{2}(\Gamma)}\to
0,\quad\text{as }n\to\infty,$

for $i=1,2$. Hence, each of the terms
$\|\nabla(F_{i}\eta^{n}_{i}-F_{i}\eta)\|_{L^{p}(\Omega_{i})^{d}}$ and
$\|F_{i}\eta^{n}_{i}-F_{i}\eta\|_{L^{r}(\Omega_{i})}$ tend to zero, which
yields that

$\|\eta^{n}_{i}-\eta\|_{\Lambda_{i}}\leq
C\|F_{i}\eta^{n}_{i}-F_{i}\eta\|_{V_{i}}\to 0,\quad\text{as }n\to\infty,$

for $i=1,2$. The desired convergence Eq. 20 is then proven, as
$\|\cdot\|_{V_{i}}$ and $\|\cdot\|_{W^{1,p}(\Omega_{i})}$ are equivalent
norms.

## References

  * [1] V. I. Agoshkov and V. I. Lebedev, Variational algorithms of the domain decomposition method [translation of Preprint 54, Akad. Nauk SSSR, Otdel. Vychisl. Mat., Moscow, 1983], vol. 5, 1990, pp. 27–46.  Soviet Journal of Numerical Analysis and Mathematical Modelling.
  * [2] H. Berninger, R. Kornhuber, and O. Sander, Convergence behaviour of Dirichlet-Neumann and Robin methods for a nonlinear transmission problem, in Domain decomposition methods in science and engineering XIX, vol. 78 of Lect. Notes Comput. Sci. Eng., Springer, Heidelberg, 2011, pp. 87–98, https://doi.org/10.1007/978-3-642-11304-8_8, https://doi.org/10.1007/978-3-642-11304-8_8.
  * [3] W. Chen, X. Xu, and S. Zhang, On the optimal convergence rate of a Robin-Robin domain decomposition method, J. Comput. Math., 32 (2014), pp. 456–475, https://doi.org/10.4208/jcm.1403-m4391, https://doi-org.ludwig.lub.lu.se/10.4208/jcm.1403-m4391.
  * [4] D. L. Cohn, Measure theory, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser/Springer, New York, second ed., 2013, https://doi.org/10.1007/978-1-4614-6956-8, https://doi.org/10.1007/978-1-4614-6956-8.
  * [5] M. Discacciati, A. Quarteroni, and A. Valli, Robin-Robin domain decomposition methods for the Stokes-Darcy coupling, SIAM J. Numer. Anal., 45 (2007), pp. 1246–1268, https://doi.org/10.1137/06065091X, https://doi.org/10.1137/06065091X.
  * [6] M. Dryja and W. Hackbusch, On the nonlinear domain decomposition method, BIT, 37 (1997), pp. 296–311, https://doi.org/10.1007/BF02510214, https://doi.org/10.1007/BF02510214.
  * [7] M. J. Gander, L. Halpern, and F. Nataf, Optimized Schwarz methods, in Domain decomposition methods in sciences and engineering (Chiba, 1999), DDM.org, Augsburg, 2001, pp. 15–27.
  * [8] V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, vol. 749 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1979.
  * [9] W. Guo and L. S. Hou, Generalizations and accelerations of Lions’ nonoverlapping domain decomposition method for linear elliptic PDE, SIAM J. Numer. Anal., 41 (2003), pp. 2056–2080, https://doi.org/10.1137/S0036142902407150, https://doi-org.ludwig.lub.lu.se/10.1137/S0036142902407150.
  * [10] A. Kufner, O. John, and S. Fučík, Function spaces, Noordhoff International Publishing, Leyden; Academia, Prague, 1977.  Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis.
  * [11] P. D. Lax, Functional analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002.
  * [12] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), pp. 1203–1219, https://doi.org/10.1016/0362-546X(88)90053-3, https://doi.org/10.1016/0362-546X(88)90053-3.
  * [13] P.-L. Lions, On the Schwarz alternating method. III. A variant for nonoverlapping subdomains, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), SIAM, Philadelphia, PA, 1990, pp. 202–223.
  * [14] P.-L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), pp. 964–979, https://doi.org/10.1137/0716071, https://doi.org/10.1137/0716071.
  * [15] S. H. Lui, A Lions non-overlapping domain decomposition method for domains with an arbitrary interface, IMA J. Numer. Anal., 29 (2009), pp. 332–349, https://doi.org/10.1093/imanum/drm011, https://doi-org.ludwig.lub.lu.se/10.1093/imanum/drm011.
  * [16] J. Naumann and C. G. Simader, Measure and Integration on Lipschitz-Manifolds, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik, 2011, https://doi.org/http://dx.doi.org/10.18452/2773.
  * [17] J. Nečas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012, https://doi.org/10.1007/978-3-642-10455-8, https://doi.org/10.1007/978-3-642-10455-8.  Translated from the 1967 French original by Gerard Tronel and Alois Kufner, Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader.
  * [18] A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1999.  Oxford Science Publications.
  * [19] T. Roubíček, Nonlinear partial differential equations with applications, vol. 153 of International Series of Numerical Mathematics, Birkhäuser/Springer Basel AG, Basel, second ed., 2013, https://doi.org/10.1007/978-3-0348-0513-1, https://doi.org/10.1007/978-3-0348-0513-1.
  * [20] S. Schreiber and R. Hochmuth, On the equivalence of transmission problems in nonoverlapping domain decomposition methods for quasilinear PDEs, Numer. Funct. Anal. Optim., 31 (2010), pp. 596–615, https://doi.org/10.1080/01630563.2010.490625, https://doi.org/10.1080/01630563.2010.490625.
  * [21] X.-C. Tai and M. Espedal, Rate of convergence of some space decomposition methods for linear and nonlinear problems, SIAM J. Numer. Anal., 35 (1998), pp. 1558–1570, https://doi.org/10.1137/S0036142996297461, https://doi.org/10.1137/S0036142996297461.
  * [22] X.-C. Tai and J. Xu, Global and uniform convergence of subspace correction methods for some convex optimization problems, Math. Comp., 71 (2002), pp. 105–124, https://doi.org/10.1090/S0025-5718-01-01311-4, https://doi.org/10.1090/S0025-5718-01-01311-4.
  * [23] A. Toselli and O. Widlund, Domain decomposition methods—algorithms and theory, vol. 34 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2005, https://doi.org/10.1007/b137868, https://doi-org.ludwig.lub.lu.se/10.1007/b137868.
  * [24] X. Xu and L. Qin, Spectral analysis of Dirichlet-Neumann operators and optimized Schwarz methods with Robin transmission conditions, SIAM J. Numer. Anal., 47 (2010), pp. 4540–4568, https://doi.org/10.1137/080713653, https://doi-org.ludwig.lub.lu.se/10.1137/080713653.
  * [25] E. Zeidler, Nonlinear functional analysis and its applications. II/B, Springer-Verlag, New York, 1990, https://doi.org/10.1007/978-1-4612-0985-0, https://doi.org/10.1007/978-1-4612-0985-0.  Nonlinear monotone operators, Translated from the German by the author and Leo F. Boron.