# Schwarz Methods for Nonlocal Problems

Matthias Schuster  Universitaet Trier, D-54286 Trier, Germany; Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Christian Vollmann11footnotemark: 1 Volker Schulz11footnotemark: 1

Abstract. The first domain decomposition methods for partial differential
equations were already developed in 1870 by H. A. Schwarz. Here we consider a
nonlocal Dirichlet problem with variable coefficients, where a nonlocal
diffusion operator is used. We find that domain decomposition methods like the
so-called Schwarz methods seem to be a natural way to solve these nonlocal
problems. In this work we show the convergence for nonlocal problems, where
specific symmetric kernels are employed, and present the implementation of the
multiplicative and additive Schwarz algorithms in the above mentioned nonlocal
setting.
Keywords. Schwarz methods, nonlocal diffusion, finite element method, domain
decomposition.

##  1 Introduction

Nonlocal operators can be helpful to describe physical phenomena, where long
range interactions occur. Therefore, in these cases, nonlocal models are
better suited than the classical approach via partial differential equations.
Nonlocal operators have already been applied in many fields, such as image
denoising[3, 11], anomalous diffusion[2, 27, 6], peridynamics[26, 14] and
stochastic processes[21, 8], just to name a few.
Moreover, nonlocal interface problems as well as models that contain domain-
dependent parameters have become popular in recent years, see e.g., [23, 4,
12]. These domain-dependent parameters may naturally suggest a domain
decomposition which leads to domain decomposition methods that are widely used
in the case of partial differential equations and have been investigated in,
e.g., [20, 28]. In the matter of nonlocal equations the approach of finite
element tearing and interconnecting (FETI) has been successfully applied as
described in [15, 31]. Another class of domain decomposition methods are
Schwarz algorithms, that were initially developed in [24] for partial
differential equations. An abstract framework for the Schwarz approach can be
found in [19, 17, 18].
Although FETI is a very efficient way to solve nonlocal equations, Schwarz
methods have several advantages. They are easier to implement and it is
straightforward to incorporate black-box solvers, whereas for FETI the finite
element formulation on overlapping domains needs to be changed. Moreover, we
could observe that Schwarz methods can also converge in case of a nonsymmetric
kernel as shown in an experiment in Chapter 7.3. Lastly, if the parameter
$\delta$, which determines the the range of nonlocal interaction and thus the
width of the nonlocal boundary regarding one (sub-)domain (see Section 2), is
quite huge, FETI cannot decompose the domain in many subdomains. In this case,
GMRES with a Schwarz preconditioner might be faster, especially if we include
the set-up times of both solvers.
Schwarz methods have also been utilized to solve an energy-based Local-to-
Nonlocal(LtN) coupling in [1]. Additionally, in [1] they showed the
convergence of the multiplicative Schwarz method for this energy-based LtN
coupling by applying [19, Theorem I.1]. In Section 4 we present the
convergence of the multiplicative Schwarz method for two cases of nonlocal
problems in a similar fashion by referring to [19, Theorem I.2]. Furthermore,
we point out that the Schwarz formulation solves linear, quadratic and cubic
patch tests. These patch tests often serve as a criterion to evaluate and
therefore compare Local-to-Nonlocal couplings. We refer to the review paper
[10] for more information on LtN couplings. Additionally we test GMRES against
two preconditioned versions of GMRES that result from the multiplicative and
additive Schwarz method combined with the finite element method.
We start by introducing nonlocal Dirichlet problems and nonlocal problems that
involve Neumann boundary conditions in Chapter 2. After that we describe the
multiplicative and additive Schwarz method for the previously mentioned
nonlocal problems in Chapter 3. Next, in Chapter 4, we prove the convergence
of this multiplicative Schwarz method to the unique solution under the
assumption that the kernel is symmetric. Furthermore, we present the finite
element method as a way to numerically solve nonlocal problems in Chapter 5.
Here, we also illustrate how Schwarz methods are applied for this discrete
framework. In Chapter 6 we shortly introduce patch tests, which are used to
evaluate Local-to-Nonlocal or Nonlocal-to-Nonlocal couplings. In the last
Chapter 7 we start by testing the Schwarz methods on two examples. The first
one is a nonlocal Dirichlet problem, where we use a singular symmetric kernel,
and the second example is a nonlocal problem with Neumann boundary, where we
employ a constant kernel. Then, we investigate an example of a nonlocal
Dirichlet problem with a nonsymmetric kernel that additionally satisfies the
patch test. Lastly, we study two versions of preconditioned GMRES, that are
connected to the Schwarz method for the finite element formulation, and we
compare these two approaches to a GMRES without preconditioning.

##  2 Problem Formulation

Let $\Omega\subset\mathbb{R}^{d}$ be an open and bounded domain and
$\\{\Omega_{i}\\}_{i=1}^{n}$ be a partition of $\Omega$, where
$\Omega_{i}\subset\Omega$ is open for $i=1,...,n$, and that satisfies

$\displaystyle\overline{\Omega}=\bigcup_{i=1}^{n}\overline{\Omega}_{i}\text{
and }\Omega_{i}\cap\Omega_{j}=\emptyset\text{ for }i,j=1,...,n\text{ and
}i\neq j.$ (1)

From now on we denote by
$\gamma:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow[0,\infty)$ a nonnegative
(interaction) kernel and define the nonlocal boundary $\mathcal{I}$ of
$\Omega$ as follows

$\displaystyle\mathcal{I}:=\\{\mathbf{y}\in\mathbb{R}^{d}\setminus\Omega:\int_{\Omega}\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{x}>0\\}.$

Based on the partition $\\{\Omega_{i}\\}_{i=1}^{n}$ we can express the kernel
$\gamma$ as

$\displaystyle\gamma=\sum_{i,j=1}^{n}\gamma_{ij}\chi_{\Omega_{i}\times\Omega_{j}}+\sum_{i=1}^{n}\gamma_{i\mathcal{I}}\chi_{\Omega_{i}\times\mathcal{I}}+\sum_{i=1}^{n}\gamma_{\mathcal{I}i}\chi_{\mathcal{I}\times\Omega_{i}},$
(2)

with potentially different nonnegative kernels

$\displaystyle\gamma_{ij}:\Omega_{i}\times\Omega_{j}\rightarrow[0,\infty),\gamma_{i\mathcal{I}}:\Omega_{i}\times\mathcal{I}\rightarrow[0,\infty)\text{
and
}\gamma_{\mathcal{I}i}:\mathcal{I}\times\Omega_{i}\rightarrow[0,\infty),\text{
for }i,j=1,...,n.$

Then, we define the nonlocal diffusion operator $-\mathcal{L}$, which can be
found in, e.g., [8, 5, 7, 29, 30], as

$\displaystyle-\mathcal{L}u(\mathbf{x})=\int\limits_{\mathbb{R}^{d}}\left(u(\mathbf{x})-u(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}.$
(3)

###  2.1 Nonlocal Dirichlet Problems

One class of problems, that we focus on, are nonhomogeneous steady-state
Dirichlet problems with nonlocal boundary constraints given by

$\displaystyle\begin{split}-\mathcal{L}u&=f\text{ on }\Omega\\\ u&=g\text{ on
}\mathcal{I},\end{split}$ (4)

where the forcing term $f$ may vary across the various parts of the
decomposition (1), i.e., there exist possibly mutually different
$f_{i}:\Omega_{i}\rightarrow\mathbb{R},\text{ for }i=1,...,n,$ such that

$\displaystyle f=\sum_{i=1}^{n}f_{i}\chi_{\Omega_{i}}.$ (5)

Similar types of nonlocal problems can also be found in, e.g., [5, 23].
Moreover, the kernel $\gamma$ is assumed to be symmetric and is supposed to
fulfill the following conditions

  * (K1)

There exist $\delta>0$ and
$\phi:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow[0,\infty)$ with
$\gamma(\mathbf{x},\mathbf{y})=\phi(\mathbf{x},\mathbf{y})\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y})$.

  * (K2)

There exist $0<\gamma_{0}<\infty$ and $\epsilon\in(0,\delta)$ such that
$\gamma_{0}\leq\gamma(\mathbf{x},\mathbf{y})$ for $\mathbf{y}\in
B_{\epsilon}(\mathbf{x})$ and $\mathbf{x}\in\Omega$.

In order to derive a weak formulation, we multiply the first equation of the
left-hand side of (4) by a test function
$v:\Omega\cup\mathcal{I}\rightarrow\mathbb{R}$ and then integrating over
$\Omega$ yields

$\displaystyle\left(-\mathcal{L}u,v\right)_{L^{2}(\Omega)}$
$\displaystyle=\left(f,v\right)_{L^{2}(\Omega)}$
$\displaystyle\Leftrightarrow\int_{\Omega}v(\mathbf{x})\int_{\mathbb{R}^{d}}(u(\mathbf{x})$
$\displaystyle-u(\mathbf{y}))\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}=\int_{\Omega}f(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}.$
(6)

By applying Fubini’s theorem and by considering $v(\mathbf{y})=0$ on
$\mathcal{I}$ the first integral of the left-hand side can be expressed as

$\displaystyle\int_{\Omega}v(\mathbf{x})\int_{\Omega\cup\mathcal{I}}(u(\mathbf{x})-u(\mathbf{y}))\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle=\frac{1}{2}\iint_{(\Omega\cup\mathcal{I})^{2}}(v(\mathbf{x})-v(\mathbf{y}))(u(\mathbf{x})-u(\mathbf{y}))\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}.$

Thus we now define the nonlocal bilinear form

$\displaystyle A(u,v)$
$\displaystyle:=\frac{1}{2}\iint_{(\Omega\cup\mathcal{I})^{2}}(v(\mathbf{x})-v(\mathbf{y}))(u(\mathbf{x})-u(\mathbf{y}))\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle=\sum_{i,j=1}^{n}\frac{1}{2}\int_{\Omega_{i}}\int_{\Omega_{j}}\left(v(\mathbf{x})-v(\mathbf{y})\right)\left(u(\mathbf{x})-u(\mathbf{y})\right)\gamma_{ij}(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle~{}~{}~{}~{}+\sum_{i=1}^{n}\int_{\Omega_{i}}\int_{\mathcal{I}}(v(\mathbf{x})-v(\mathbf{y}))(u(\mathbf{x})-u(\mathbf{y}))\gamma_{i\mathcal{I}}(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$

and the linear functional

$\displaystyle F(v)$
$\displaystyle:=\int_{\Omega}f(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}=\sum_{i=1}^{n}\int_{\Omega_{i}}f_{i}(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x},$

where we used the definition of the forcing term (5) and that the kernel can
be expressed as a sum of kernels as in (2). Moreover, with $A$ we can define
the following semi-norm and norm, respectively,

$\displaystyle\begin{split}|u|_{V(\Omega\cup\mathcal{I})}&:=\sqrt{A(u,u)},\\\
||u||_{V(\Omega\cup\mathcal{I})}&:=|u|_{V(\Omega\cup\mathcal{I})}+||u||_{L^{2}(\Omega\cup\mathcal{I})}.\end{split}$
(7)

This enables us to define the following spaces

$\displaystyle V(\Omega\cup\mathcal{I})$ $\displaystyle:=\\{u\in
L^{2}(\Omega\cup\mathcal{I}):||u||_{V(\Omega\cup\mathcal{I})}<\infty\\}\text{
and}$ $\displaystyle V_{c}(\Omega\cup\mathcal{I})$ $\displaystyle:=\\{u\in
V(\Omega\cup\mathcal{I}):u=0\text{ on }\mathcal{I}\\}.$

We now present a variational formulation of problem (4):

######  Definition 2.1.

Given $f\in L^{2}(\Omega)$ and $g\in V(\Omega\cup\mathcal{I})$, if $u\in
V(\Omega\cup\mathcal{I})$ solves

$\displaystyle\begin{split}A(u,v)&=F(v)\text{ for all }v\in
V_{c}(\Omega\cup\mathcal{I})\text{ and}\\\ u&-g\in
V_{c}(\Omega\cup\mathcal{I}),\end{split}$ (8)

then $u$ is called weak solution of (4).

Since $A(u,v)=A(u-g,v)+A(g,v)$ and $u-g\in V_{c}(\Omega\cup\mathcal{I})$,
problem (8) can also be reformulated as a homogeneous Dirichlet problem

$\displaystyle\begin{split}&\textit{Given $f\in L^{2}(\Omega)$ and $g\in
V(\Omega\cup\mathcal{I})$, find }\tilde{u}\in V_{c}(\Omega\cup\mathcal{I}):\\\
&A(\tilde{u},v)=F(v)-A(g,v)\textit{ for all }v\in
V_{c}(\Omega\cup\mathcal{I}),\\\ &\textit{then }u:=\tilde{u}+g\in
V(\Omega\cup\mathcal{I})\textit{ is a weak solution of
\eqref{weak_formulation}.}\end{split}$ (9)

######  Remark 2.2.

In this work, we show the convergence of the multiplicative Schwarz method for
nonlocal Dirichlet problems for symmetric kernels, that fulfill (K1) and (K2).
Additionally, the kernels are assumed to satisfy the requirements of one of
the following two classes, where the well-posedness of the corresponding
nonlocal Dirichlet problem (8) is shown, e.g., in [7]:

  * •

Integrable kernels:
There exist constants $0<\gamma_{1},\gamma_{2}<\infty$ with

$\displaystyle\gamma_{1}\leq\inf_{\mathbf{x}\in\Omega}\int_{\Omega\cup\mathcal{I}}\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}\text{
and
}\sup_{\mathbf{x}\in\Omega}\int_{\Omega\cup\mathcal{I}}\left(\gamma(\mathbf{x},\mathbf{y})\right)^{2}~{}d\mathbf{y}\leq\left(\gamma_{2}\right)^{2}.$

Then the solution $u\in V(\Omega\cup\mathcal{I})$ for (8) exists, is unique
and in this case the spaces
$\left(V_{c}(\Omega\cup\mathcal{I}),|\cdot|_{V(\Omega\cup\mathcal{I})}\right)$
and
$\left(L_{c}^{2}(\Omega\cup\mathcal{I}),||\cdot||_{L^{2}(\Omega\cup\mathcal{I})}\right)$
as well as
$\left(V(\Omega\cup\mathcal{I}),||\cdot||_{V(\Omega\cup\mathcal{I})}\right)$
and
$\left(L^{2}(\Omega\cup\mathcal{I}),||\cdot||_{L^{2}(\Omega\cup\mathcal{I})}\right)$
are equivalent.

  * •

Singular kernels:
There exist constants $0<\gamma_{*},\gamma^{*}<\infty$ and $s\in(0,1)$ with

$\displaystyle\gamma_{*}<\gamma(\mathbf{x},\mathbf{y})||\mathbf{x}+\mathbf{y}||_{2}^{d+2s}<\gamma^{*}\text{
for all }\mathbf{y}\in B_{\delta}(\mathbf{x})\text{ and }\mathbf{x}\in\Omega.$

Then the solution to (8) exists, is unique and the spaces
$\left(V_{c}(\Omega\cup\mathcal{I}),|\cdot|_{V(\Omega\cup\mathcal{I})}\right)$
and
$\left(H^{s}_{c}(\Omega\cup\mathcal{I}),|\cdot|_{H^{s}(\Omega\cup\mathcal{I})}\right)$
as well as
$\left(V(\Omega\cup\mathcal{I}),||\cdot||_{V(\Omega\cup\mathcal{I})}\right)$
and
$\left(H^{s}(\Omega\cup\mathcal{I}),||\cdot||_{H^{s}(\Omega\cup\mathcal{I})}\right)$
are equivalent.

###  2.2 Nonlocal Problems with Robin Boundary Conditions

Besides solving a nonlocal Dirichlet problem, we are also interested in
solving nonlocal problems that involve a Neumann boundary condition, i.e.,
find a solution $u$ for

$\displaystyle\begin{split}-\mathcal{L}u(\mathbf{x})+\kappa(\mathbf{x})u(\mathbf{x})&=f(\mathbf{x})\text{
for }\mathbf{x}\in\Omega,\\\ \mathcal{N}u(\mathbf{y})&=g^{N}(\mathbf{y})\text{
for }\mathbf{y}\in\mathcal{I}^{N},\\\ u(\mathbf{y})&=g^{D}(\mathbf{y})\text{
for }\mathbf{y}\in\mathcal{I}^{D},\end{split}$ (10)

where

$\displaystyle\mathcal{I}=\mathcal{I}^{N}\dot{\cup}\mathcal{I}^{D},\ \kappa$
$\displaystyle:\Omega\rightarrow[\alpha,\beta]\text{ with
}0<\alpha\leq\beta<\infty,\
\gamma:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow[0,\infty)\text{
measurable and}$ $\displaystyle\mathcal{N}(\mathbf{y})$
$\displaystyle:=\int_{\Omega\cup\mathcal{I}}\left(u(\mathbf{x})-u(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}\text{
for }\mathbf{y}\in\mathcal{I}^{N}.$ (11)

Here, $\mathcal{N}$ is called nonlocal Neumann operator. For problems of type
(10), we only assume
$\gamma:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow[0,\infty)$ to be a
measurable and symmetric function. Furthermore, if
$\gamma(\mathbf{x},\mathbf{y})=0$ for $\mathbf{x},\mathbf{y}\in\mathcal{I}$,
the Neumann operator reduces to

$\displaystyle\mathcal{N}(\mathbf{y})=\int_{\Omega}\left(u(\mathbf{x})-u(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}\text{
for }\mathbf{y}\in\mathcal{I}^{N}.$ (12)

In the case of $\mathcal{I}^{N}=\mathcal{I}$ and (12) there exists a unique
weak solution to problem (10) according to [30]. In the following we show
existence of a unique solution to problem (10) for the more general case (10)
by following the approach in [30]. Therefore, in order to get a variational
formulation for the problem (10), we multiply the first two equations of (10)
by a test function $v:\Omega\cup\mathcal{I}\rightarrow\mathbb{R}$, integrate
over $\Omega$, or $\mathcal{I}^{N}$ respectively, and add up the two
equations. Then we get

$\displaystyle\left(-\mathcal{L}u+\kappa
u,v\right)_{L^{2}(\Omega)}+\left(\mathcal{N}u,v\right)_{L^{2}(\mathcal{I}^{N})}=\left(f,v\right)_{L^{2}(\Omega)}+\left(g^{N},v\right)_{L^{2}(\mathcal{I}^{N})}.$
(13)

By assuming $v=0$ on $\mathcal{I}^{D}$ and by using similar computations as in
Section 2 we can reformulate (13) as

$\displaystyle\begin{split}&\frac{1}{2}\int_{\Omega\cup\mathcal{I}}\int_{\Omega\cup\mathcal{I}}\left(v(\mathbf{x})-v(\mathbf{y})\right)\left(u(\mathbf{x})-u(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}+\int_{\Omega}\kappa(\mathbf{x})v(\mathbf{x})u(\mathbf{x})~{}d\mathbf{x}\\\
&=\int_{\Omega}f(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}+\int_{\mathcal{I}^{N}}g^{N}(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}.\end{split}$
(14)

Next, we define the spaces

$\displaystyle\mathcal{V}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$
$\displaystyle:=\\{v\in
L^{2}(\Omega\cup\mathcal{I}):||v||_{\mathcal{V}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}<\infty\\},\text{
and}$ $\displaystyle\mathcal{V}_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$
$\displaystyle:=\\{v\in\mathcal{V}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})\text{
with }v=0\text{ on }\mathcal{I}^{D}\\},\text{ where}$
$\displaystyle||v||^{2}_{\mathcal{V}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}$
$\displaystyle:=\int_{\Omega}v(\mathbf{x})^{2}~{}d\mathbf{x}+\int_{\Omega\cup\mathcal{I}}\int_{\Omega\cup\mathcal{I}}\left(v(\mathbf{x})-v(\mathbf{y})\right)^{2}\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}.$

Moreover, set

$\displaystyle\langle
u,v\rangle_{\mathcal{V}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}:=\int_{\Omega}u(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}$
$\displaystyle+\int_{\Omega\cup\mathcal{I}}\int_{\Omega\cup\mathcal{I}}\left(u(\mathbf{x})-u(\mathbf{y})\right)\left(v(\mathbf{x})-v(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x},$

which is a semi-inner product for
$\mathcal{V}_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$. We further define
the quotient space

$\displaystyle V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$
$\displaystyle:=\mathcal{V}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})/Z,\text{
and
}V_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D}):=\mathcal{V}_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})/Z,\text{
where}$ $\displaystyle Z$
$\displaystyle:=\\{v\in\mathcal{V}_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D}):v(\mathbf{x})=0\text{
for a.e. }\mathbf{x}\in\Omega\cup\mathcal{I}\\}\text{ and set}$
$\displaystyle\langle[u],[v]\rangle_{V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}$
$\displaystyle:=\langle
u,v\rangle_{\mathcal{V}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}\text{ and
}||[v]||_{V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}=||v||_{\mathcal{V}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}.$

In the following we write $u$ instead of $[u]$ for ease of presentation.
The first line of (LABEL:neumann_var_Form_end) yields the bilinear form
$\mathcal{B}:V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})\times
V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})\rightarrow\mathbb{R},$ where

$\displaystyle\mathcal{B}(u,v):=\frac{1}{2}\int\limits_{\Omega\cup\mathcal{I}}\int\limits_{\Omega\cup\mathcal{I}}\left(u(\mathbf{x})-u(\mathbf{y})\right)\left(v(\mathbf{x})-v(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}.$

Additionally, we define the linear functional
$F^{N}:V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})\rightarrow\mathbb{R}$ as

$\displaystyle
F^{N}(v)=\int_{\Omega}f(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}+\int_{\mathcal{I}^{N}}g^{N}(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}.$

######  Definition 2.3.

Given functions $g^{D}\in V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$, $f\in
L^{2}(\Omega)$, $g^{N}\in L^{2}(\mathcal{I}^{N})$ and
${\kappa:\Omega\rightarrow[\alpha,\beta]}$, where $0<\alpha\leq\beta<\infty$,
then, if a function $u\in V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$ solves

$\displaystyle\begin{split}\mathcal{B}(u,v)+\int_{\Omega}\kappa(\mathbf{x})u(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}&=F^{N}(v)\text{
for all }v\in V_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})\text{ and}\\\
u-g^{D}&\in V_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D}),\end{split}$ (15)

the function $u$ is called weak solution to problem (10).

Again, this weak formulation can also be reformulated as a problem with
homogeneous Dirichlet boundary conditions:

Given functions $g^{D}\in V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$, $f\in
L^{2}(\Omega)$, $g^{N}\in L^{2}(\mathcal{I}^{N})$ and
$\kappa:\Omega\rightarrow[\alpha,\beta]$, $\displaystyle\textit{where
$0<\alpha\leq\beta<\infty$, find a function }\tilde{u}\in
V_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})\textit{ such that}$
$\displaystyle\mathcal{B}(\tilde{u},v)+\int_{\Omega}\kappa(\mathbf{x})\tilde{u}(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}=F^{N}(v)-\left(\mathcal{B}(g^{D},v)+\int_{\Omega}\kappa(\mathbf{x})g^{D}(\mathbf{x})v(\mathbf{x})~{}d\mathbf{x}\right)$
(16) $\displaystyle\textit{ for all }v\in
V_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D}),\textit{then
}u:=\tilde{u}+g^{D}\in V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})\textit{
solves \eqref{neumannVarFormulation}.}$

Now, we can show the following theorem, which is similar to [30, Theorem 3.1].

######  Theorem 2.4.

The space $\mathcal{V}_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$ is
complete regarding
$||\cdot||_{\mathcal{V}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}$. As a
consequence, $V_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$ is a Hilbert
space with respect to $||\cdot||_{V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}$.

###### Proof.

Let $(v_{n})_{n\in\mathbb{N}}$ be a Cauchy sequence in
$\mathcal{V}_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$. Then, due to the
completeness of $L^{2}(\Omega)$, respectively
$L^{2}((\Omega\cup\mathcal{I}^{N})\times(\Omega\cup\mathcal{I}^{N}))$, there
exist functions $\tilde{v}\in L^{2}(\Omega)$ and $w\in
L^{2}((\Omega\cup\mathcal{I}^{N})\times(\Omega\cup\mathcal{I}^{N}))$, such
that

$\displaystyle\lim_{k\rightarrow\infty}\int_{\Omega}\left(v_{k}(\mathbf{x})-\tilde{v}(\mathbf{x})\right)^{2}~{}d\mathbf{x}=0\text{
and}$
$\displaystyle\lim_{k\rightarrow\infty}\int_{\Omega\cup\mathcal{I}^{N}}\int_{\Omega\cup\mathcal{I}^{N}}\left((v_{k}(\mathbf{x})-v_{k}(\mathbf{y}))\sqrt{\gamma(\mathbf{x},\mathbf{y})}-w(\mathbf{x},\mathbf{y})\right)^{2}~{}d\mathbf{y}d\mathbf{x}=0.$

Analogously to [30, Theorem 3.1] one can show that there exists a subsequence
$\left(v_{n_{l}}\right)_{l\in\mathbb{N}}$ of $(v_{n})_{n\in\mathbb{N}}$ and an
extension $v:\Omega\cup\mathcal{I}\rightarrow\mathbb{R}$ of $\tilde{v}$, i.e.,
$v=\tilde{v}$ on $\Omega$, with $v=0$ on $\mathcal{I}^{D}$ and

$\displaystyle\lim_{l\rightarrow\infty}v_{n_{l}}(\mathbf{x})=v(\mathbf{x})\text{
for a.e. }\mathbf{x}\in\Omega\cup\mathcal{I}.$

Consequently, we derive

$\displaystyle
w(\mathbf{x},\mathbf{y})=\lim_{l\rightarrow\infty}\left(v_{n_{l}}(\mathbf{x})-v_{n_{l}}(\mathbf{y})\right)\sqrt{\gamma(\mathbf{x},\mathbf{y})}=\left(v(\mathbf{x})-v(\mathbf{y})\right)\sqrt{\gamma(\mathbf{x},\mathbf{y})}\text{
for a.e. }(\mathbf{x},\mathbf{y})\in(\Omega\cup\mathcal{I})^{2}.$

Therefore, $v\in\mathcal{V}_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$ and
we can conclude

$\displaystyle\lim_{l\rightarrow\infty}||v_{n_{l}}-v||_{\mathcal{V}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}=0\text{
and
}\lim_{n\rightarrow\infty}||v_{n}-v||_{\mathcal{V}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}=0.$

∎

Set
$|||u|||_{V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}^{2}:=\mathcal{B}(u,u)+\int\limits_{\Omega}\kappa(\mathbf{x})u^{2}(\mathbf{x})~{}d\mathbf{x}$,
then the norms $|||\cdot|||_{V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}$ and
${||\cdot||_{V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}}$ are equivalent for
$u\in V_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$ since

$\displaystyle\min\\{\frac{1}{2},\alpha\\}||u||^{2}_{V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}\leq|||u|||_{V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}^{2}\leq\max\\{\frac{1}{2},\beta\\}||u||^{2}_{V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})}.$
(17)

By applying the Riesz representation theorem we directly get the following
conclusion.

######  Corollary 2.5.

Given $f\in L^{2}(\Omega)$, $g^{N}\in L^{2}(\mathcal{I}^{N})$ and $g^{D}\in
V(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$, there exists a unique function
$\tilde{u}\in V_{c}(\Omega,\mathcal{I}^{N},\mathcal{I}^{D})$ that satisfies
(16), i.e., $u=\tilde{u}+g^{D}$ is a weak solution for (10).

##  3 Schwarz Methods

The first domain decomposition method was already formulated by Hermann
Amandus Schwarz around 1869 and is now known as the Schwarz alternating
method[24, 25]. It is an iterative method which was originally used to solve
the Laplace equation on a domain by decomposing the domain into two
overlapping domains where the Laplace equation could be easily solved on each.
In every iteration the former solution regarding the problem on the other
domain was used as a boundary condition on the boundary that was part of the
other domain.
In this section we formulate the additive and multiplicative Schwarz method
for the nonlocal problem (4) in the same manner. Therefore, we denote the
nonlocal boundary regarding the subdomains $\Omega_{i}$ of the decomposition
(1) by

$\displaystyle\mathcal{I}_{i}:=\\{\mathbf{y}\in\mathbb{R}^{d}\setminus\Omega_{i}:\int_{\Omega_{i}}\gamma(\mathbf{x},\mathbf{y})d\mathbf{x}>0\\}.$

###  3.1 Schwarz Methods for Nonlocal Dirichlet Problems

Given the solution $u$ of the nonlocal problem (8), for $i=1,...,n$ we set
$u_{i}:=u|_{\Omega_{i}\cup\mathcal{I}_{i}},$ and observe that $u_{i}$ is a
solution of

$\displaystyle\textit{Find a weak solution }u_{i}$ $\displaystyle\in
V(\Omega_{i}\cup\mathcal{I}_{i})\textit{ subject to }$
$\displaystyle\begin{split}-\mathcal{L}u_{i}&=f_{i}\textit{ on }\Omega_{i},\\\
u_{i}&=u_{j}\textit{ on }\mathcal{I}_{i}\cap\Omega_{j},\textit{ for
}j=1,...,n\textit{ and }j\neq i,\\\ u_{i}&=g\textit{ on
}\mathcal{I}_{i}\cap\mathcal{I}.\end{split}$ (18)

In this case, $u|_{\Omega_{j}\cup\mathcal{I}_{j}}\in
V(\Omega_{j}\cup\mathcal{I}_{j})$ for $j=1,...,n$ and $j\neq i$ serve as the
boundary data.
On the other hand, as we will show in Section 4.1, if we have weak solutions
$\\{u_{i}\\}_{i=1}^{n}$ for (18), the function

$u:=\sum_{i=1}^{n}u_{i}\chi_{\Omega_{i}}+g\chi_{\mathcal{I}}$

solves the nonlocal problem (4).

######  Remark 3.1.

Since we derive (18) from (4) and (8), respectively, the boundary data of the
problem on $\Omega_{i}$ is an element of $V(\Omega_{i}\cup\mathcal{I}_{i})$
and can naturally be extended to a function of $V(\Omega\cup\mathcal{I})$.
Therefore, we expand the boundary $\mathcal{I}_{i}$ to
$\hat{\mathcal{I}}_{i}:=(\Omega\setminus\Omega_{i})\cup\mathcal{I}$ and
restrict ourselves to boundary data in
$V(\Omega_{i}\cup\hat{\mathcal{I}}_{i})$. Consequently, we will make use of
the spaces $V(\Omega_{i}\cup\hat{\mathcal{I}}_{i})=V(\Omega\cup\mathcal{I})$
and $V_{c}(\Omega_{i}\cup\hat{\mathcal{I}}_{i})=\\{u\in
V(\Omega\cup\mathcal{I}):u=0\text{ on
}\hat{\mathcal{I}}_{i}=\left(\Omega\setminus\Omega_{i}\right)\cup\mathcal{I}\\}$
in order to formulate a weak formulation for (18).

Thus, for $v_{i}\in V_{c}(\Omega_{i}\cup\hat{\mathcal{I}}_{i})$ and $u\in
V(\Omega\cup\mathcal{I})$ the bilinear form $A$ reduces to the corresponding
bilinear operator that only contains integrals over
$\left(\Omega_{i}\cup\mathcal{I}_{i}\right)\times\left(\Omega_{i}\cup\mathcal{I}_{i}\right)$
as follows

$\displaystyle A(u,v_{i})=$
$\displaystyle\sum_{l,j=1}^{n}\frac{1}{2}\int_{\Omega_{l}}\int_{\Omega_{j}}\left(u(\mathbf{x})-u(\mathbf{y})\right)\left(v_{i}(\mathbf{x})-v_{i}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\sum_{j=1}^{n}\int_{\Omega_{j}}u(\mathbf{x})v_{i}(\mathbf{x})\int_{\mathcal{I}}\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle=\frac{1}{2}\int_{\Omega_{i}}\int_{\Omega_{i}}\left(u(\mathbf{x})-u(\mathbf{y})\right)\left(v_{i}(\mathbf{x})-v_{i}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\int_{\Omega_{i}}\int_{\Omega_{j}}v_{i}(\mathbf{x})\left(u(\mathbf{x})-u(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}+\int_{\Omega_{i}}u(\mathbf{x})v_{i}(\mathbf{x})\int_{\mathcal{I}}\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x},$

where we used the symmetry of $\gamma$ in the last step. Analogously, $F$ can
in this case be written as an operator that only integrates over $\Omega_{i}$,
i.e.,

$\displaystyle F(v_{i})=$
$\displaystyle\int_{\Omega}fv_{i}~{}d\mathbf{x}=\int_{\Omega_{i}}f_{i}v_{i}~{}d\mathbf{x}.$

Then, we define the variational formulation for the subproblem on $\Omega_{i}$
as:

$\displaystyle\begin{split}\textit{Given }f\in L^{2}(\Omega)\textit{ and
}&\textit{boundary data }g\in
V(\Omega_{i}\cup\hat{\mathcal{I}}_{i})=V(\Omega\cup\mathcal{I}),\\\ \textit{
find }u_{i}\in
V(\Omega_{i}\cup\hat{\mathcal{I}}_{i})&=V(\Omega\cup\mathcal{I})\textit{ such
that }\\\ A(u_{i},v_{i})&=F(v_{i})\textit{ for all }v_{i}\in
V_{c}(\Omega_{i}\cup\hat{\mathcal{I}}_{i}),\\\ u_{i}-g&\in
V_{c}(\Omega_{i}\cup\hat{\mathcal{I}}_{i}).\end{split}$ (19)

In the following we show two popular Schwarz methods to solve the problem (18)
in an iterative manner. In both cases we start with initial guesses
$\\{u_{i}^{0}\\}_{i=1}^{n}$. In the multiplicative Schwarz method, as shown in
Algorithm 1, we use the most recent solutions in every iteration as the
boundary data, i.e., for the subproblem $i$ the solutions $u_{j}^{k+1}$ for
$0\leq j<i$ from the current iteration are employed.

input : $u_{i}^{0}$ for $i=1,...,n$

for  _$k=0,1,...$ until convergence_ do

for  _$i=1,...,n$_ do

Find $u_{i}^{k+1}$ s.t.  $\displaystyle-\mathcal{L}u_{i}^{k+1}$
$\displaystyle=f_{i}\quad$ $\displaystyle\text{on }\Omega_{i},$ $\displaystyle
u_{i}^{k+1}$ $\displaystyle=u_{j}^{k+1}\quad$ $\displaystyle\text{on
}\mathcal{I}_{i}\cap\Omega_{j}\text{ for }j=1,...,i-1,$ $\displaystyle
u_{i}^{k+1}$ $\displaystyle=u_{j}^{k}\quad$ $\displaystyle\text{on
}\mathcal{I}_{i}\cap\Omega_{j}\text{ for }j=i+1,...,n,$ $\displaystyle
u_{i}^{k+1}$ $\displaystyle=g\quad$ $\displaystyle\text{on
}\mathcal{I}_{i}\cap\mathcal{I}.$

end for

end for

Algorithm 1 Multiplicative Schwarz method

Another approach is to only use the solutions of the former outer iteration
$\\{u_{j}^{k}\\}_{j=1}^{n}$ instead of utilizing $u_{j}^{k+1}$ for the
subproblem $i$, if $0\leq j<i$. This leads to the so-called additive Schwarz
algorithm that is presented in Algorithm 2. This version of Schwarz iterative
methods is easily parallelizable. In some cases there is also a way to compute
the multiplicative Schwarz algorithm in parallel to some extend, which leads
to the multicolor Schwarz algorithm that is illustrated in [20, Algorithm
2.2.2] for a coercive elliptic partial differential equation.

input : $u_{i}^{0}$ for $i=1,...,n$

for  _$k=0,1,...$ until convergence_ do

for  _$i=1,...,n$_ do

Find $u_{i}^{k+1}$ s.t.  $\displaystyle-\mathcal{L}u_{i}^{k+1}$
$\displaystyle=f_{i}\quad$ $\displaystyle\text{on }\Omega_{i},$ $\displaystyle
u_{i}^{k+1}$ $\displaystyle=u_{j}^{k}\quad$ $\displaystyle\text{on
}\mathcal{I}_{i}\cap\Omega_{j}\text{ for }j=1,...,n,$ $\displaystyle
u_{i}^{k+1}$ $\displaystyle=g\quad$ $\displaystyle\text{on
}\mathcal{I}_{i}\cap\mathcal{I}.$

end for

end for

Algorithm 2 Additive Schwarz Method

######  Remark 3.2.

If we decompose $\Omega$ in only two domains $\Omega_{1}$ and $\Omega_{2}$ the
additive Schwarz method basically consists of two multiplicative Schwarz
methods, where one starts by solving the subproblem on $\Omega_{1}$ and the
second one begins by computing the solution on $\Omega_{2}$. Therefore, the
application of the additive Schwarz methods can only be faster, if the domain
$\Omega$ is decomposed in at least three subdomains.

In Chapter 5.3 we will present the finite element versions of the
multiplicative and additive Schwarz method. Then, the discretized
multiplicative Schwarz method coincides with the block-Gauß-Seidel algorithm
and the discretized additive Schwarz approach is equivalent to the block-
Jacobi method.

###  3.2 Schwarz Methods for Nonlocal Problems with Neumann Boundary
Conditions

In this section we apply the Schwarz Method on a nonlocal problem with Neumann
boundary conditions as introduced in Chapter 2.2 For ease of presentation, we
assume $\mathcal{I}^{N}=\mathcal{I}$ (and consequently
$\mathcal{I}^{D}=\emptyset$). The case, where
$\mathcal{I}=\mathcal{I}^{N}\dot{\cup}\mathcal{I}^{D}$ with
$\mathcal{I}^{D}\neq\emptyset$, can be handled analogously. Thus, we consider
the following problem.

$\displaystyle\begin{split}\textit{Find a weak solution }u&\in
V(\Omega,\mathcal{I},\emptyset)\textit{ to}\\\ -\mathcal{L}u+\kappa
u&=f\textit{ on }\Omega,\\\ \mathcal{N}u&=g^{N}\textit{ on
}\mathcal{I},\end{split}$ (20)

where $\gamma(\mathbf{x},\mathbf{y})=0$ for
$\mathbf{x},\mathbf{y}\in\mathcal{I}$, i.e., we use the nonlocal Neumann
operator (12).
Moreover, let $\\{\mathcal{I}_{i}^{N}\\}_{i=1}^{n}$ be a partition of the
nonlocal boundary $\mathcal{I}(=\mathcal{I}^{N})$ such that

$\displaystyle\overline{\mathcal{I}}=\left(\bigcup_{i=1}^{n}\overline{\mathcal{I}}_{i}^{N}\right),~{}\mathcal{I}_{i}^{N}\cap\mathcal{I}_{j}^{N}=\emptyset\text{
for }i,j=1,...,n,~{}i\neq j\text{ and}$
$\displaystyle\mathcal{I}_{i}^{N}\subset\mathcal{I}_{i}\cap\mathcal{I}=\\{\mathbf{y}\in\mathcal{I}:\int_{\Omega_{i}}\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{x}>0\\}.$

Here, it is possible that there exist a subset of indices
$\mathbb{S}\subset\\{1,...,n\\}$ with $|\mathbb{S}|\leq n-1$ such that

$\displaystyle\mathcal{I}^{N}_{j}=\emptyset\textit{ for }j\in\mathbb{S}.$

Thus, the partition $\\{\mathcal{I}_{i}^{N}\\}_{i=1}^{n}$ can contain the
empty set multiple times and includes at least one non-empty set. Moreover,
the boundary data $g^{N}$ may vary across the different parts of the
decomposition $\\{\mathcal{I}_{i}^{N}\\}_{i=1}^{n}$, i.e., there exist
mutually different $g_{i}^{N}:\mathcal{I}_{i}^{N}\rightarrow\mathbb{R}$, for
$i=1,...,n,$ such that
$g^{N}=\sum_{i=1}^{n}g_{i}^{N}\chi_{\mathcal{I}_{i}^{N}}$.
Given the weak solution $u$ of the problem (20), then the functions
$u_{i}:=u\chi_{\Omega_{i}\cup\mathcal{I}_{i}}$ for $i=1,...,n$ are solutions
of

$\displaystyle\begin{split}\textit{Find a weak solution }u_{i}&\in
V(\Omega_{i},\mathcal{I}_{i}^{N},\mathcal{I}_{i}\setminus\mathcal{I}_{i}^{N})\textit{~{}subject
to}\\\ -\mathcal{L}u_{i}+\kappa u_{i}&=f_{i}\textit{ on }\Omega_{i},\\\
\mathcal{N}u_{i}&=g_{i}^{N}\textit{ on }\mathcal{I}_{i}^{N},\\\
u_{i}&=u_{j}\textit{ on
}\mathcal{I}_{i}\cap\left(\Omega_{j}\cup\mathcal{I}_{j}^{N}\right),\textit{
for }j=1,...,n\textit{ and }j\neq i.\end{split}$ (21)

So, on every subproblem we are solving a nonlocal Robin problem. Additionally,
since the nonlocal boundary regarding $\Omega_{i}$ can be partially in another
subdomain $\Omega_{j}$, i.e., $\mathcal{I}_{i}\cap\Omega_{j}\neq\emptyset$,
the integration domain of the Neumann operator corresponding to the subproblem
on $\Omega_{i}$ can contain parts of the nonlocal Dirichlet Boundary and the
nonlocal Neumann operator is in this case of type (11) and not of type (12).
Again, if we have a weak solutions $(u_{i})_{i=1}^{n}$ of the problems (21),
we set

$\displaystyle u=\sum_{i=1}^{n}u_{i}\chi_{\Omega_{i}\cup\mathcal{I}_{i}^{N}}$

and then $u$ is weak solution to (21), which we show in Section 4.2. Again,
analogously to the nonlocal Dirichlet case as described in Remark 3.1, we
extend the Dirichlet boundary $\mathcal{I}_{i}\setminus\mathcal{I}_{i}^{N}$
corresponding to the subproblem on $\Omega_{i}\cup\mathcal{I}_{i}^{N}$ to the
domain ${\hat{\mathcal{I}}_{i}:=\bigcup\limits_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{n}\left(\Omega_{j}\cup\mathcal{I}_{j}^{N}\right)=\left(\Omega\cup\mathcal{I}\right)\setminus\left(\Omega_{i}\cup\mathcal{I}_{i}^{N}\right)}$.
Then, $V(\Omega,\mathcal{I},\emptyset)\subset
V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})$ and, due to the
symmetry of $\gamma$, we obtain for $u_{i}\in
V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})$ and $v_{i}\in
V_{c}(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})$ that

$\displaystyle\mathcal{B}(u_{i},v_{i})+\int_{\Omega}\kappa(\mathbf{x})u_{i}(\mathbf{x})v_{i}(\mathbf{x})~{}d\mathbf{x}=$
$\displaystyle\frac{1}{2}\int_{\Omega_{i}\cup\mathcal{I}_{i}^{N}}\int_{\Omega_{i}\cup\mathcal{I}_{i}^{N}}\left(u_{i}(\mathbf{x})-u_{i}(\mathbf{y})\right)\left(v_{i}(\mathbf{x})-v_{i}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\int_{\Omega_{i}\cup\mathcal{I}_{i}^{N}}\int_{\Omega_{j}\cup\mathcal{I}_{j}^{N}}v_{i}(\mathbf{x})\left(u_{i}(\mathbf{x})-u_{i}(\mathbf{y})\right)\gamma(\mathbf{x},\mathbf{y})~{}d\mathbf{y}d\mathbf{x}$
$\displaystyle+\int_{\Omega_{i}}\kappa(\mathbf{x})u_{i}(\mathbf{x})v_{i}(\mathbf{x})~{}d\mathbf{x}.$

Thus, we define the variational formulation of the Robin problem on
$\Omega_{i}\cup\mathcal{I}_{i}^{N}$ as:

$\displaystyle\textit{Given }f\in L^{2}(\Omega),g^{N}\in
L^{2}(\mathcal{I}^{N})\textit{ and }g^{D}\in V(\Omega_{i},\mathcal{I}_{i}^{N}$
$\displaystyle,\hat{\mathcal{I}}_{i}),\textit{ find }u_{i}\in
V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})\textit{ with }$
$\displaystyle\begin{split}\mathcal{B}(u_{i},v_{i})+\int_{\Omega_{i}}\kappa(\mathbf{x})v_{i}(\mathbf{x})u_{i}(\mathbf{x})~{}d\mathbf{x}&=F^{N}(v_{i})\textit{
for all }v_{i}\in
V_{c}(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})\\\ \textit{and
}u_{i}-g^{D}&\in
V_{c}(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i}).\end{split}$ (22)

The multiplicative Schwarz algorithm can then be formulated as follows:

input : $u_{i}^{0}$ for $i=1,...,n$

for  _$k=0,1,...$ until convergence_ do

for  _$i=1,...,n$_ do

Find weak solution $u_{i}^{k+1}$ to
$\displaystyle-\mathcal{L}u_{i}^{k+1}+\kappa u_{i}^{k+1}$
$\displaystyle=f_{i}\quad$ $\displaystyle\text{on }\Omega_{i},$
$\displaystyle\mathcal{N}u_{i}^{k+1}$ $\displaystyle=g_{i}^{N}$
$\displaystyle\text{on }\mathcal{I}_{i}^{N},$ $\displaystyle u_{i}^{k+1}$
$\displaystyle=u_{j}^{k+1}\quad$ $\displaystyle\text{on
}\mathcal{I}_{i}\cap\Omega_{j}\text{ for }j=1,...,i-1,$ $\displaystyle
u_{i}^{k+1}$ $\displaystyle=u_{j}^{k}\quad$ $\displaystyle\text{on
}\mathcal{I}_{i}\cap\Omega_{j}\text{ for }j=i+1,...,n.$

end for

end for

Algorithm 3 Multiplicative Schwarz Method for Nonlocal Problems with Neumann
Boundary Conditions

The additive Schwarz method can be analogously derived as in Section 3.1 by
only using the solution of the former outer iteration $(u_{i}^{k})_{i=1}^{n}$
as the boundary data in iteration $k$ to compute the new solutions
$(u_{i}^{k+1})_{i=1}^{n}$.

##  4 Well-posedness of the Multiplicative Schwarz Method

In this section we show that the multiplicative Schwarz methods as described
in Chapters 3.1 and 3.2 converge. Therefore we recall the first part of [19,
Theorem I.2]:

######  Theorem 4.1.

Given closed subspaces $\\{\mathbb{V}_{i}\\}_{i=1}^{n}$ of a Hilbert space
$\mathbb{V}:=\overline{\oplus_{i=1}^{n}\mathbb{V}_{i}}$, orthogonal
projections $\mathcal{P}_{i}:\mathbb{V}\rightarrow\mathbb{V}$ onto the
subspace $\mathbb{V}_{i}$ and a sequence $\left(u^{l}\right)_{l=1}^{\infty}$,
where $\mathcal{P}_{i}(u^{l-1})=u^{l}$ for $l=km+i$ given some data $u^{0}$.
Then $u^{l}$ converges to a unique $u\in\mathbb{V}$.

The version of Theorem 4.1 for two domains (see [19, Theorem I.1]) has also
been used to show the convergence of the multiplicative Schwarz method for an
energy-based Local-to-Nonlocal coupling (see [1]).

######  Remark 4.2.

The second part of [19, Theorem I.2] states that if
$\mathbb{V}=\oplus_{i=1}^{n}\mathbb{V}_{i}$ holds, then there exists a
constant $\epsilon\in(0,1)$ such that

$\displaystyle||u^{l}-u||_{\mathbb{V}}\leq\epsilon^{l}||u^{0}-u||_{\mathbb{V}}.$

Linear convergence of the Schwarz methods in the nonlocal framework of Chapter
3 can also be observed in the numerical experiments in Chapter 7 and is proven
in the discrete case(see Remark 5.3). However, the proof of the assumption
$\mathbb{V}=\oplus_{i=1}^{n}\mathbb{V}_{i}$ is left to future work.

###  4.1 Well-posedness of the Multiplicative Schwarz Method for Nonlocal
Dirichlet Problems

Here in this subsection, we assume the kernel $\gamma$ to be as described in
Remark 2.2, i.e., $\gamma$ fulfills conditions (K1) and (K2) and is either a
singular symmetric or an integrable symmetric kernel. As seen in (9) we only
need to consider the case $g=0$ on $\mathcal{I}$ since the case $g\neq 0$ on
$\mathcal{I}$ is a direct consequence. Given Algorithm 1 we can define for
every subproblem in (18) an operator

$\displaystyle\mathcal{S}^{f}_{i}:\left(V_{c}(\Omega\cup\mathcal{I}),|\cdot|_{V(\Omega\cup\mathcal{I})}\right)\rightarrow\left(V_{c}(\Omega\cup\mathcal{I}),|\cdot|_{V(\Omega\cup\mathcal{I})}\right),$

such that $\mathcal{S}^{f}_{i}(g)=u_{i}$ and
$u_{i}:\Omega_{i}\cup\hat{\mathcal{I}}_{i}\rightarrow\mathbb{R}$ is the weak
solution of the subproblem on $\Omega_{i}$ as described in (19) given $g$ and
$f$, i.e., $\mathcal{S}_{i}$ maps the boundary data $g$ onto the solution
$u_{i}$ on $\Omega_{i}$.
We start by showing the convergence for $f\equiv 0$. In this case, we set
$\mathcal{S}_{i}:=\mathcal{S}^{f}_{i}$.

######  Lemma 4.3.

The operator $\mathcal{S}_{i}$ is linear and bounded as follows

$\displaystyle||u_{i}||_{V(\Omega\cup\mathcal{I})}\leq
2|g|_{V(\Omega\cup\mathcal{I})}\text{ for }u_{i}=\mathcal{S}_{i}(g).$

###### Proof.

Since the linearity of $\mathcal{S}_{i}$ is trivially true, we only have to
proof the boundedness of $\mathcal{S}_{i}$. As a consequence of $f\equiv 0$
and (9) in the second step of the following calculation and by applying
Cauchy-Schwarz in the third step we derive

$\displaystyle|u_{i}-g|_{V(\Omega\cup\mathcal{I})}^{2}=A(u_{i}-g,u_{i}-g)\leq|A(g,u_{i}-g)|$
$\displaystyle\leq\sqrt{A(g,g)}\sqrt{A(u_{i}-g,u_{i}-g)}=|g|_{V(\Omega\cup\mathcal{I})}|u_{i}-g|_{V(\Omega\cup\mathcal{I})}\text{
and therefore
}|u_{i}-g|_{V(\Omega\cup\mathcal{I})}\leq|g|_{V(\Omega\cup\mathcal{I})}.$

Then, we conclude

$\displaystyle|u_{i}|_{V(\Omega\cup\mathcal{I})}\leq|u_{i}-g|_{V(\Omega\cup\mathcal{I})}+|g|_{V(\Omega\cup\mathcal{I})}\leq
2|g|_{V(\Omega\cup\mathcal{I})}.$

∎

Further, we define spaces

$\displaystyle\mathbb{V}_{i}(\Omega\cup\mathcal{I}):=\\{u_{i}\in
V_{c}(\Omega\cup\mathcal{I}):u_{i}=\mathcal{S}_{i}(u_{i})\\}~{}~{}\text{ and
}~{}~{}\mathbb{V}(\Omega\cup\mathcal{I}):=\overline{\oplus_{i=1}^{n}\mathbb{V}_{i}(\Omega\cup\mathcal{I})},$
(23)

that is equipped with the the inner product

$\displaystyle\langle
u,v\rangle_{\mathbb{V}(\Omega\cup\mathcal{I})}:=A\left(u,v\right)\text{ and
the norm }||u||_{\mathbb{V}(\Omega\cup\mathcal{I})}:=\sqrt{\langle
u,u\rangle}_{\mathbb{V}(\Omega\cup\mathcal{I})}.$

Then $\langle\cdot,\cdot\rangle_{\mathbb{V}(\Omega\cup\mathcal{I})}$ is linear
in the first argument, symmetric and positive-definite, see Remark 2.2. Here,
every $u_{i}\in\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ is a solution to (19) on
$\Omega_{i}$ given $u_{i}\in V(\Omega_{i}\cup\hat{\mathcal{I}}_{i})$ as
boundary data on $\hat{\mathcal{I}}_{i}$ and $f\equiv 0$.

######  Lemma 4.4.

Let the spaces $\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ and
$\mathbb{V}(\Omega\cup\mathcal{I})$ be as defined in (23) for $i=1,...,n$.
Then, the space $\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ is a closed subspace
for every $i=1,...,n$. Moreover, $\mathbb{V}(\Omega\cup\mathcal{I})$ is a
closed subspace of $V_{c}(\Omega\cup\mathcal{I})$ and therefore a Hilbert
space.

###### Proof.

  1. 1.

$\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ is closed under linear combinations:
Given $u_{i},\psi_{i}\in\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ and
$\lambda\in\mathbb{R}$, then $\lambda u_{i}+\psi_{i}\in
V_{c}(\Omega\cup\mathcal{I})$ and we get

$\displaystyle A(\lambda u_{i}+\psi_{i},v_{i})$ $\displaystyle=\lambda
A(u_{i},v_{i})+A(\psi_{i},v_{i})=0\text{ for all }v_{i}\in
V_{c}(\Omega_{i}\cup\hat{\mathcal{I}}_{i})$ $\displaystyle\Rightarrow\lambda
u_{i}+\psi_{i}\in\mathbb{V}_{i}(\Omega\cup\mathcal{I}).$

  2. 2.

$\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ is a closed subspace regarding
$||\cdot||_{\mathbb{V}(\Omega\cup\mathcal{I})}$: Given a sequence
$\\{u_{i}^{m}\\}_{m=1}^{\infty}$ such that
$u_{i}^{m}\in\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ for all $m\in\mathbb{N}$
and $\\{u_{i}^{m}\\}_{m=1}^{\infty}$ is a Cauchy sequence regarding
$||\cdot||_{\mathbb{V}(\Omega\cup\mathcal{I})}$. Then, there exists a unique
function $u_{i}\in\mathbb{V}(\Omega\cup\mathcal{I})$ with
$\lim\limits_{m\rightarrow\infty}|u_{i}^{m}-u_{i}|_{V(\Omega\cup\mathcal{I})}=0$,
because $\left(u_{i}^{m}\right)_{m=1}^{\infty}$ is a Cauchy sequence in the
Hilbert space
$\left(V_{c}(\Omega\cup\mathcal{I}),|\cdot|_{V(\Omega\cup\mathcal{I})}\right)$.
Consequently, the continuity of $\mathcal{S}_{i}$ implies
$u_{i}\in\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ since

$\displaystyle|u_{i}-\mathcal{S}_{i}(u_{i})|_{V_{c}(\Omega\cup\mathcal{I})}$
$\displaystyle\leq|u_{i}-u_{i}^{m}|_{V(\Omega\cup\mathcal{I})}+|\mathcal{S}_{i}(u_{i}^{m})-\mathcal{S}_{i}(u_{i})|_{V(\Omega\cup\mathcal{I})}$
$\displaystyle\leq 3|u_{i}-u_{i}^{m}|_{V(\Omega\cup\mathcal{I})}\rightarrow
0\quad\text{for }m\rightarrow\infty.$

  3. 3.

$\mathbb{V}(\Omega\cup\mathcal{I})$ is closed regarding linear combinations,
which follows directly from the definition of
$\mathbb{V}(\Omega\cup\mathcal{I})$ and the fact that every
$\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ is closed under linear combinations.
Then, $\mathbb{V}(\Omega\cup\mathcal{I})$ is a Hilbert space since it is
closed regarding $||\cdot||_{\mathbb{V}(\Omega\cup\mathcal{I})}$.

∎

We can now conclude the following:

######  Lemma 4.5.

For $i=1,...,n$ the function $\mathcal{S}_{i}$ is an orthogonal projection
onto $\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ regarding the inner product
$\langle\cdot,\cdot\rangle_{\mathbb{V}(\Omega\cup\mathcal{I})}$.

###### Proof.

Obviously $\mathcal{S}_{i}(g)\in\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ for all
$g\in\mathbb{V}(\Omega\cup\mathcal{I})$. In order to prove the orthogonality
condition
$g-\mathcal{S}_{i}(g)=\left(g-\mathcal{S}_{i}(g)\right)\chi_{\Omega_{i}}\perp\mathbb{V}_{i}(\Omega\cup\mathcal{I})$
we set $v_{i}:=g-\mathcal{S}_{i}(g)\in
V_{c}(\Omega_{i}\cup\hat{\mathcal{I}}_{i})$, then for all
$u_{i}\in\mathbb{V}_{i}(\Omega\cup\mathcal{I})$ we get

$\displaystyle\langle
u_{i},g-\mathcal{S}_{i}(g)\rangle_{\mathbb{V}(\Omega\cup\mathcal{I})}=A(u_{i},v_{i})=F(v_{i})=0,$

since $v_{i}\in V_{c}(\Omega_{i}\cup\hat{\mathcal{I}}_{i})$ can be interpreted
as a test function and $u_{i}$ with $u_{i}=\mathcal{S}_{i}(u_{i})$ solves the
subproblem on $\Omega_{i}$ as defined in (19) given $u_{i}$ as boundary data
and $f_{i}=0$. ∎

######  Corollary 4.6.

In the case of integrable symmetric or singular symmetric kernels for the
solution $u\in V_{c}(\Omega\cup\mathcal{I})$ of (9) trivially also holds
$u\in\mathbb{V}(\Omega\cup\mathcal{I})$. Therefore, the multiplicative Schwarz
method yields a sequence $\left(u^{l}\right)_{l=1}^{\infty}$, where
$u^{l}\in\mathbb{V}(\Omega\cup\mathcal{I})$ for $l\in\mathbb{N}$, that
converges to $u$ regarding
$||\cdot||_{\mathbb{V}(\Omega\cup\mathcal{I})}=|\cdot|_{V(\Omega\cup\mathcal{I})}$.

######  Remark 4.7.

For $f\nequiv 0$ the multiplicative Schwarz method yields a sequence
$\\{u^{l}\\}_{l=1}^{\infty}$ such that for $l=km+i$ the function $u^{l}$
fulfills $u^{l}=\mathcal{S}_{i}^{f}(u^{l-1})$ and $u^{l}-u^{l-1}\in
V_{c}(\Omega_{i}\cup\hat{\mathcal{I}}_{i})$, i.e., $u^{l}$ solves (19) on
$\Omega_{i}$ given $f$ and boundary data $u^{l-1}$. Let $u$ be the solution of
the problem on the whole domain. Then for $l=km+i$ the function
$\tilde{u}^{l}:=u^{l}-u$ fulfills

$\displaystyle
A(\tilde{u}^{l},v_{i})=A(u^{l}-u,v_{i})=F(v_{i})-F(v_{i})=0\text{ for all
}v_{i}\in V_{c}(\Omega_{i}\cup\hat{\mathcal{I}}_{i}).$

Therefore $\\{\tilde{u}^{l}\\}_{l=1}^{\infty}$ is a sequence generated by the
multiplicative Schwarz method which converges to zero, i.e.,
$\lim\limits_{l\rightarrow\infty}u^{l}-u=0$.

###  4.2 Well-posedness of the Multiplicative Schwarz Method for Nonlocal
Problems with Neumann Boundary Conditions

As mentioned in Chapters 2.2 and 3.2 we only consider the kernel $\gamma$ to
be symmetric in nonlocal problems of type (20). Next, we define

$\displaystyle\mathcal{S}_{i}^{N,f,g^{N}}:\left(V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i}),||\cdot||_{V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})}\right)\rightarrow\left(V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i}),||\cdot||_{V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})}\right)$

such that $\mathcal{S}_{i}^{N,f,g^{N}}\left(g\right)=u_{i}$ and
$u_{i}:\Omega\cup\mathcal{I}\rightarrow\mathbb{R}$ is the weak solution of the
subproblem regarding $\Omega_{i}$ as described in (22). We will only show the
convergence for $f,g^{N}\equiv 0$, since the convergence for $f\in
L^{2}(\Omega)$ and $g^{N}\in L^{2}(\mathcal{I})$ with $f,g^{N}\nequiv 0$ is a
direct conclusion, which can be shown analogously to Remark 4.7. In this case,
we set $\mathcal{S}_{i}^{N}:=\mathcal{S}_{i}^{N,f,g^{N}}$.

######  Lemma 4.8.

The operator $\mathcal{S}_{i}^{N}$ is linear and bounded by the boundary data,
i.e., there exists a constant $C>0$ with

$\displaystyle||u_{i}||_{V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})}\leq
C||g||_{V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})},\text{ if
}u_{i}=\mathcal{S}_{i}^{N}(g)\text{ and }g\in
V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i}).$ (24)

###### Proof.

Set $\widehat{C}:=\max\\{2,\frac{1}{\alpha}\\}$ and notice that $u_{i}-g\in
V_{c}(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})$. As a
consequence, we get by applying the norm equivalence (17) and formulation (16)
on the space $V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})$ that

$\displaystyle||u_{i}-g||^{2}_{V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})}$
$\displaystyle\leq\widehat{C}\left(\mathcal{B}(u_{i}-g,u_{i}-g)+\int_{\Omega_{i}}\kappa\left(u_{i}-g\right)^{2}~{}d\mathbf{x}\right)$
$\displaystyle\leq\widehat{C}\left|\mathcal{B}(g,u_{i}-g)\right|+\widehat{C}\left|\int_{\Omega_{i}}\kappa
g\left(u_{i}-g\right)~{}d\mathbf{x}\right|$
$\displaystyle\leq\widehat{C}\sqrt{\mathcal{B}(g,g)}\sqrt{\mathcal{B}(u_{i}-g,u_{i}-g)}+\widehat{C}\sqrt{\int_{\Omega_{i}}\kappa
g^{2}~{}d\mathbf{x}}\sqrt{\int_{\Omega_{i}}\kappa\left(u_{i}-g\right)^{2}~{}d\mathbf{x}}$
$\displaystyle\leq\widehat{C}||g||_{V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})}||u_{i}-g||_{V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})},$

where we used Cauchy-Schwarz in the third step. Then, the assertion follows
from

$\displaystyle||u_{i}||_{V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})}\leq||u_{i}-g||_{V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})}+||g||_{V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})}.$

∎

Additionally define

$\displaystyle\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset):=\\{u_{i}\in
V(\Omega,\mathcal{I},\emptyset):u_{i}=\mathcal{S}_{i}^{N}\left(u_{i}\right)\\}\text{
and
}\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset):=\overline{\oplus_{i=1}^{n}\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset)},$
(25)

which are equipped with the inner product $\langle
u,v\rangle_{\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)}:=\mathcal{B}(u,v)+\int_{\Omega}\kappa
uv~{}d\mathbf{x}$ and the norm
$||u||_{\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)}:=\sqrt{\langle
u,u\rangle}_{\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)}\left(=|||u|||_{V(\Omega,\mathcal{I},\emptyset)}\right)$.
Here, every function
$u_{i}\in\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset)$ is a solution to
(22) on $\Omega_{i}\cup\mathcal{I}_{i}^{N}$ given $u_{i}\in
V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})$ as boundary data on
$\hat{\mathcal{I}}_{i}$ and $f\equiv 0$.

######  Lemma 4.9.

Let the spaces $\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset)$ and
$\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)$ be as defined in (25) for every
${i=1,...,n}$. Then, the space
$\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset)$ is a closed subspace for
$i=1,...,n$ and $\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)$ is a closed
subspace of $V(\Omega,\mathcal{I},\emptyset)$ and therefore a Hilbert space.

###### Proof.

  1. 1.

$\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset)$ is closed under linear
combinations. Given
$u,\psi\in\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset)$ and
$\lambda\in\mathbb{R}$, then $\lambda u+\psi\in
V(\Omega,\mathcal{I},\emptyset)$ and we get

$\displaystyle\mathcal{B}(\lambda u+\psi,v_{i})+\int_{\Omega}\kappa
v_{i}\left(\lambda u+\psi\right)~{}d\mathbf{x}$
$\displaystyle=\lambda\left(\mathcal{B}(u,v_{i})+\int_{\Omega}\kappa
v_{i}u~{}d\mathbf{x}\right)$
$\displaystyle+\mathcal{B}(\psi,v_{i})+\int_{\Omega}\kappa
v_{i}\psi~{}d\mathbf{x}=0\text{ for all }v_{i}\in
V_{c}(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})$
$\displaystyle\Rightarrow\lambda u+\psi\in\mathbb{V}_{i}^{N}.$

  2. 2.

$\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset)$ is closed with regards to
$||\cdot||_{\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)}$. Given a Cauchy
sequence $\\{u_{i}^{m}\\}_{m=1}^{\infty}$ regarding
$||\cdot||_{\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)}$. Due to the
equivalence of
$||\cdot||_{\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)}=|||\cdot|||_{V(\Omega,\mathcal{I},\emptyset)}$
and $||\cdot||_{V(\Omega,\mathcal{I},\emptyset)}$, there exists a function
$u_{i}\in V(\Omega,\mathcal{I},\emptyset)$ such that

$\displaystyle\lim\limits_{m\rightarrow\infty}||u_{i}^{m}-u_{i}||_{V(\Omega,\mathcal{I},\emptyset)}=0.$

Then, by using the continuity of $\mathcal{S}_{i}^{N}$ we get
$u_{i}\in\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset)$ since

$\displaystyle||u_{i}-\mathcal{S}^{N}_{i}(u_{i})||_{V(\Omega,\mathcal{I},\emptyset)}$
$\displaystyle=||u_{i}-\mathcal{S}_{i}^{N}(u_{i})||_{V(\Omega_{i},\mathcal{I}^{N}_{i},\hat{\mathcal{I}}_{i})}$
$\displaystyle\leq||u_{i}-u_{i}^{m}||_{V(\Omega_{i},\mathcal{I}^{N}_{i},\hat{\mathcal{I}}_{i})}+||\mathcal{S}^{N}(u_{i}^{m})-\mathcal{S}_{i}^{N}(u_{i})||_{V(\Omega_{i},\mathcal{I}^{N}_{i},\hat{\mathcal{I}}_{i})}$
$\displaystyle\leq\left(1+C\right)||u_{i}-u_{i}^{m}||_{V(\Omega_{i},\mathcal{I}^{N}_{i},\hat{\mathcal{I}}_{i})}\rightarrow
0\quad m\rightarrow\infty,$

where we used that $u_{i}=\mathcal{S}_{i}^{N}(u_{i})$ on
$(\Omega\cup\mathcal{I})\setminus\Omega_{i}$ in the first step as well as (24)
and the fact that
${||\cdot||_{V(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})}\leq||\cdot||_{V(\Omega,\mathcal{I},\emptyset)}}$
in the last two step.

  3. 3.

Analogously to Lemma 4.4.

∎

Lastly, we can deduce the following:

######  Lemma 4.10.

The function $\mathcal{S}_{i}^{N}$ is an orthogonal projection onto
$\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset)$ regarding the inner product
$\langle\cdot,\cdot\rangle_{\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)}$ for
$i=1,...,n$.

###### Proof.

Obviously,
$\mathcal{S}_{i}^{N}(g)\in\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset)$
for all $g\in\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)$.
Set
$v_{i}:=g-\mathcal{S}_{i}^{N}(g)=\left(g-\mathcal{S}_{i}^{N}(g)\right)\chi_{\Omega_{i}\cup\mathcal{I}_{i}^{N}}\in
V_{c}(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})$. We now have to
proof

$\displaystyle
g-\mathcal{S}_{i}^{N}(g)=v_{i}\perp\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset).$

For all $u_{i}\in\mathbb{V}_{i}^{N}(\Omega,\mathcal{I},\emptyset)$ we get

$\displaystyle\langle
u_{i},g-\mathcal{S}_{i}^{N}(g)\rangle_{\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)}=\mathcal{B}(u_{i},v_{i})+\int_{\Omega_{i}}\kappa
u_{i}v_{i}=F^{N}(v_{i})=0,$

since $u_{i}$ with $u_{i}=\mathcal{S}_{i}^{N}(u_{i})$ solves the variational
formulation (22) on $\Omega_{i}\cup\mathcal{I}_{i}^{N}$ given the Dirichlet
boundary data $u_{i}$ and $v_{i}\in
V_{c}(\Omega_{i},\mathcal{I}_{i}^{N},\hat{\mathcal{I}}_{i})$ serves as a test
function. ∎

######  Corollary 4.11.

The weak solution $u\in V(\Omega,\mathcal{I},\emptyset)$ of (20) is also an
element of the space $\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)$. As a
consequence, the multiplicative Schwarz method yields a sequence
$\left(u^{l}\right)_{l=1}^{\infty}$, where
$u^{l}\in\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)$ for $l\in\mathbb{N}$,
which converges to $u$ regarding
${||\cdot||_{\mathbb{V}^{N}(\Omega,\mathcal{I},\emptyset)}=|||\cdot|||_{V(\Omega,\mathcal{I},\emptyset)}}$
(or equivalently, $||\cdot||_{V(\Omega,\mathcal{I},\emptyset)}$).

##  5 Schwarz Methods for the Finite Element Formulation

###  5.1 Finite Element Approximation

In order to solve problem (8) we use the finite element method. Therefore we
need a triangulation of finite (polyhedral) elements
$\mathcal{T}^{h}=\\{\mathcal{E}_{j}\\}_{j=1}^{J}$ with vertices
$\\{\mathbf{x}_{k}\\}_{k=1}^{K}$. We assume that
$\Omega=\bigcup_{j=1}^{J}\mathcal{E}_{j}$ or
$\Omega\approx\bigcup_{j=1}^{J}\mathcal{E}_{j}$ and, for ease of exposition,
that there exists a $K_{\Omega}\in\mathbb{N}$ such that
$\mathbf{x}_{k}\in\Omega$ for $k=1,...,K_{\Omega}$ and
$\mathbf{x}_{k}\in\mathcal{I}$ for $k=K_{\Omega}+1,...,K$. Furthermore, we
utilize continuous piecewise linear basis functions $\\{\phi_{k}\\}_{k=1}^{K}$
that satisfy $\phi_{k}(\mathbf{x}_{l})=\delta_{kl}$. In our tests we use a
slightly different set of basis functions, which we describe in Section 5.2.
Moreover, we define finite dimensional function spaces as

$\displaystyle V^{h}=span\\{\phi_{k}:k=1,...,K\\}\text{ and
}V_{c}^{h}=span\\{\phi_{k}:k=1,...,K_{\Omega}\\}.$

Thus, we can project $u\in V(\Omega\cup\mathcal{I})$ to a function
$\mathbf{u}^{h}\in V^{h}$ as follows

$\displaystyle\mathbf{u}_{i}^{h}:=u(\mathbf{x}_{i}),\ \text{for all
}i=1,...,K,$ and
$\displaystyle\mathbf{u}^{h}(\mathbf{x})=\sum_{i=1}^{K}\mathbf{u}_{i}^{h}\phi_{i}(\mathbf{x}).$

With the projection
$\mathbf{g}^{h}=\sum_{j=K_{\Omega}+1}^{K}\mathbf{g}_{i}\phi_{i}$ of the
boundary data $g$, we formulate a discretized version of the weak formulation
(9):
Find $\mathbf{u}_{\Omega}^{h}\in V_{c}^{h}$ such that

$\displaystyle A(\mathbf{u}_{\Omega}^{h},\phi_{i})$
$\displaystyle=F(\phi_{i})$ $\displaystyle\text{for all }i=1,...,K_{\Omega},$
$\displaystyle\Leftrightarrow\sum_{k=1}^{K_{\Omega}}A(\phi_{k},\phi_{i})\mathbf{u}_{k}^{h}$
$\displaystyle=\mathbf{f}_{i}^{h}-\sum_{j=K_{\Omega}+1}^{K}A(\phi_{j},\phi_{i})\mathbf{g}_{j}^{h},$
$\displaystyle\text{for all }i=1,...,K_{\Omega},$ (26)

where
$\mathbf{f}_{i}^{h}:=\int_{\Omega}f(\mathbf{x})\phi_{i}(\mathbf{x})~{}d\mathbf{x}$.
Then,
$\mathbf{u}^{h}=\sum_{i=1}^{K_{\Omega}}\mathbf{u}_{i}^{h}\phi_{i}+\sum_{j=K_{\Omega}+1}^{K}\mathbf{g}_{j}^{h}\phi_{j}$
is called finite element approximation of (9).
By defining a matrices $\mathbf{A},\mathbf{A}_{\mathcal{I}}$ and a forcing
vector $\mathbf{f}$ as

$\displaystyle\mathbf{f}=\left(\mathbf{f}_{i}^{h}\right)_{1\leq i\leq
K_{\Omega}},\mathbf{A}=\left(\mathbf{a}_{ij}\right)_{1\leq i,j\leq
K_{\Omega}},\text{ and
}\mathbf{A}_{\mathcal{I}}=\left(\mathbf{a}_{ij}\right)_{1\leq i\leq
K_{\Omega};K_{\Omega}+1\leq j\leq K},$

where $\mathbf{a}_{ij}:=A(\phi_{j},\phi_{i})$, we can write (26) as a system
of linear equations

$\displaystyle\mathbf{A}\mathbf{u}^{h}=\mathbf{f}-\mathbf{A}_{I}\mathbf{g}^{h}.$
(27)

A detailed description on how to assemble the stiffness matrices $\mathbf{A}$
and $\mathbf{A}_{\mathcal{I}}$ can be found in [9].

######  Remark 5.1.

The finite element formulation for the nonlocal problem with Neumann boundary
conditions (16) can be derived in a similar fashion by replacing the bilinear
operator $A(u,v)$ with the function
$\tilde{\mathcal{B}}(u,v):=\mathcal{B}(u,v)+\int_{\Omega}\kappa
uv~{}d\mathbf{x}$. Therefore, the Schwarz algorithms for the finite element
method, as described in Section 5.3, can be employed analogously in this case.

######  Remark 5.2.

In the case of a singular symmetric or an integrable symmetric kernel as
introduced in Remark 2.2 the corresponding variational operator $A$ is
symmetric and positive definite and, as a result, $\mathbf{A}$ is also
symmetric and positive definite. If we instead consider the nonlocal problem
with Neumann boundary condition, where the kernel $\gamma$ is only assumed to
be symmetric, we also derive that the corresponding bilinear operator
$\tilde{\mathcal{B}}(u,v)=\mathcal{B}(u,v)+\int_{\Omega}\kappa
uv~{}d\mathbf{x}$ and therefore the resulting finite element matrix is
positive definite and symmetric (see Chapter 2.2).

###  5.2 Splitting of Inner Boundary Basis Functions

For ease of exposition, we now assume
$\bar{\Omega}=\bar{\Omega}_{1}\cup\bar{\Omega}_{2}$ in this section. Since we
use the continuous Galerkin method in order to solve the variational
formulation, the support of a piecewise linear nodal function $\phi_{k}$
corresponding to a vertex $\mathbf{x}_{k}$, that lies on the boundary between
$\Omega_{1}$ and $\Omega_{2}$, intersects with both domains, i.e.,
$supp(\phi_{k})\cap\Omega_{i}\neq\emptyset$, for $i=1,2$. Therefore, we
replace every one of these basis function $\phi_{k}$, whose support intersects
with both domains, by two functions $\phi_{k}^{1}$ and $\phi_{k}^{2}$, that
fulfill

$\displaystyle\phi_{k}(\mathbf{x})=\phi_{k}^{1}(\mathbf{x})+\phi^{2}_{k}(\mathbf{x})\text{
and }supp(\phi_{k}^{i})\subset\bar{\Omega}_{i},\text{ for }i=1,2.$

Additionally, we replace $\mathbf{u}_{k}^{h}$ by two new degrees of freedom
$\mathbf{u}_{k}^{h,i}$, for $i=1,2$. If we now follow the derivation of the
finite element formulation analogously to Section 5.1 we get again a system of
linear equations, which contains a linear equation for each
$\mathbf{u}_{k}^{h,i}$.

###  5.3 Formulation of the Schwarz Methods

In this section we assume that the basis functions and the corresponding
degrees of freedom are constructed as described in the previous Section 5.2.
Then, in order to formulate the additive and multiplicative Schwarz methods
for the linear system (27) we now assign every node $\mathbf{x}_{k}$ to a
domain $\Omega_{i}$ according to the following index sets

$\displaystyle\mathfrak{J}_{i}$
$\displaystyle=\\{j_{1},...,j_{K_{i}}\\}\subset\\{1,...,K_{\Omega}\\},\text{
for }i=1,...,n,\text{ with}$ $\displaystyle\bigcup_{i=1}^{n}\mathfrak{J}_{i}$
$\displaystyle=\\{1,...,K_{\Omega}\\}\text{ and
}\mathfrak{J}_{i}\cap\mathfrak{J}_{j}=\emptyset,\text{ for }i,j=1,...,n\text{
and }i\neq j,$

where additionally the implication
$supp(\phi_{k})\subset\bar{\Omega}_{i}\Rightarrow k\in\mathfrak{J}_{i}$ is
fulfilled. So the index $k$ of every node $\mathbf{x}_{k}$ is in exactly one
$\mathfrak{J}_{i}$. With these index sets we now define several submatrices
and subvectors as follows

$\displaystyle\mathbf{A}_{ij}$
$\displaystyle:=\left(\mathbf{a}_{kl}\right)_{k\in\mathfrak{J}_{i},l\in\mathfrak{J}_{j}},\mathbf{A}_{i\mathcal{I}}:=\left(\mathbf{a}_{kl}\right)_{k\in\mathfrak{J}_{i},K_{\Omega}+1\leq
l\leq K},$ $\displaystyle\mathbf{f}_{i}$
$\displaystyle:=\left(\mathbf{f}_{k}^{h}\right)_{k\in\mathfrak{J}_{i}}\text{
and }\mathbf{u}_{i}:=\left(\mathbf{u}_{k}^{h}\right)_{k\in\mathfrak{J}_{i}}.$

Then, we can rewrite (26) as

$\displaystyle\sum_{j=1}^{n}\mathbf{A}_{ij}\mathbf{u}_{j}=\mathbf{f}_{i}-\mathbf{A}_{i\mathcal{I}}\mathbf{g}\quad\text{for
all }i=1,...,K_{\Omega},.$

Furthermore, in the Schwarz method below we denote by $\mathbf{u}_{i}^{k}$ the
solution for the subproblem on the domain $\Omega_{i}$ in iteration $k$. The
formulation of the additive Schwarz method is shown in Algorithm 4. In order
to compute solution $\mathbf{u}_{i}^{k+1}$ the solutions $\mathbf{u}_{j}^{k}$
regarding the other domains are known and serve as Dirichlet boundary
conditions. In contrast to (26) they are put on the right-hand side of the
equation. In this case the algorithm is equivalent to a block-Jacobi method to
solve the linear system (26).

input : $\mathbf{u}_{i}^{0}$ for $i=1,...,n$

for  _$k=0,1,...$ until convergence_ do

for  _$i=1,...,n$_ do

Find $\mathbf{u}_{i}^{k+1}$ s.t.
$\displaystyle\mathbf{A}_{ii}\mathbf{u}_{i}^{k+1}=\mathbf{f}_{i}-\mathbf{A}_{i\mathcal{I}}\mathbf{g}-\sum_{j\neq
i}\mathbf{A}_{ij}\mathbf{u}_{j}^{k}$ (28)

end for

end for

Algorithm 4 Additive Schwarz Method (Block-Jacobi Method)

The description of the multiplicative Schwarz method is depicted in Algorithm
5. As mentioned above, in the case of the multiplicative Schwarz formulation
the most recent solution $\mathbf{u}_{j}^{k+1}$ for $j=1,...,i-1$ is used to
solve the problem regarding $\Omega_{i}$. This alternating Schwarz method is
equivalent to the block-Gauß-Seidel method applied on the linear system (26).

input : $\mathbf{u}_{i}^{0}$ for $i=1,...,n$

for  _$k=0,1,...$ until convergence_ do

for  _$i=1,...,n$_ do

Find $\mathbf{u}_{i}^{k+1}$ s.t.
$\displaystyle\mathbf{A}_{ii}\mathbf{u}_{i}^{k+1}=\mathbf{f}_{i}-\mathbf{A}_{i\mathcal{I}}\mathbf{g}-\sum_{j=1}^{i-1}\mathbf{A}_{ij}\mathbf{u}_{j}^{k+1}-\sum_{j=i+1}^{n}\mathbf{A}_{ij}\mathbf{u}_{j}^{k}$

end for

end for

(Block-Gauss-Seidel Method)

Algorithm 5 Multiplicative Schwarz Method

(Block-Gauss-Seidel Method)

The description and analysis of the block-Jacobi and block-Gauß-Seidel method
for linear systems can be found in, e.g., [13, Chapter 3] or [22, Chapter 4].

######  Remark 5.3.

We recall from Remark 5.2 that we only consider nonlocal problems, where the
matrix $\mathbf{A}$ is symmetric and positive definite. Then, the
multiplicative Schwarz method converges linearly (see [13, Theorem 3.53 and
3.39]) and, if $n=2$, i.e., the domain $\Omega$ is decomposed in two
nonoverlapping subdomains $\Omega_{1}$ and $\Omega_{2}$, the additive Schwarz
method is also converging in a linear fashion to the unique solution, which
follows from [13, Corollary 3.52 and Theorem 3.36]. However, if $\Omega$ is
decomposed in more than two subdomains, the linear convergence of a relaxed
block-Jacobi algorithm as defined in [13, Chapter 12.5.3] can be shown, if the
dampening or Richardson parameter $\theta$ is small enough (see [22, Theorem
14.5 and 14.6]).

##  6 Patch Tests

In this section we show that Schwarz methods for nonlocal Dirichlet problems,
where we employ nonlocal diffusion operators, can be interpreted as a
Nonlocal-to-Nonlocal coupling method, that satisfies linear, quadratic and
cubic patch tests. Analogously to [10, Definition 1 and 2], we define the
linear and higher-order patch test for the coupling of two nonlocal operators
as follows:

######  Definition 6.1.

Given two nonlocal operators $-\mathcal{L}_{1}$ and $-\mathcal{L}_{2}$ and a
linear function
$u^{*}(\mathbf{x}):=c_{0}+c_{1}\mathbf{x}$, where
$c_{0},c_{1}\in\mathbb{R}^{d}$ and $u^{*}$ is a solution of

$\displaystyle-\mathcal{L}_{i}u=0\text{ on }\Omega,\quad
u(\mathbf{x})=c_{0}+c_{1}\mathbf{x}\text{ on }\mathcal{I}\text{ for }i=1,2.$
(29)

Then, a coupling method passes the linear patch test, if $u^{*}$ is also a
solution of the coupled problem with the same boundary condition.

######  Definition 6.2.

Given two nonlocal operators $-\mathcal{L}_{1}$ and $-\mathcal{L}_{2}$, and a
polynomial
$u^{*}(\mathbf{x}):=\sum_{|\boldsymbol{\alpha}|=0}^{p}c_{\boldsymbol{\alpha}}\mathbf{x}^{\boldsymbol{\alpha}}$
with degree $p\in\\{2,3\\}$, multi-indices
$\boldsymbol{\alpha}\in\mathbb{N}_{0}^{d}$ and
$c_{\boldsymbol{\alpha}}\in\mathbb{R}$. Additionally, suppose that $u^{*}$ is
a solution of

$\displaystyle-\mathcal{L}_{i}u=f^{poly}\text{ on }\Omega,\quad
u(\mathbf{x})=\sum_{|\boldsymbol{\alpha}|=0}^{p}c_{\boldsymbol{\alpha}}\mathbf{x}^{\boldsymbol{\alpha}}\text{
on }\mathcal{I}\text{ for }i=1,2,$ (30)

where $f^{poly}$ is a polynomial with degree $p-2$. Then, a coupling method
passes the quadratic(cubic) patch test, if $p=2$ ($p=3$) and $u^{*}$ is also a
solution of the coupled problem with the same boundary condition.

Given a solution $u^{*}$ of a nonlocal Dirichlet problem (29) or (30), and the
right-hand side $f^{poly}$ of the corresponding linear, quadratic or cubic
patch test, we can directly follow that $u^{*}$ is also the solution to the
Schwarz formulation (18) since the Dirichlet boundary conditions on every
$\mathcal{I}_{i}$ holds and

$\displaystyle-\mathcal{L}_{i}u^{*}=f^{poly}\text{ on }\Omega_{i}\text{ for
}i=1,...,n.$

Therefore the patch test is trivially fulfilled. Thus, we can also interpret
the Schwarz method as a coupling method. As we will demonstrate in Section 7,
we can, e.g., couple a singular and a constant kernel.
Next, we present a class of kernels where the corresponding nonlocal operator
is equivalent to the Laplace operator $-\Delta
u(\mathbf{x}):=\sum_{i=1}^{d}\partial_{i}^{2}u(\mathbf{x})$ in case of
polynomials up to degree three
$p(\mathbf{x})=\sum_{|\boldsymbol{\alpha}|=0}^{3}c_{\boldsymbol{\alpha}}\mathbf{x}^{\boldsymbol{\alpha}}$.
Hence, given a radially symmetric kernel
$\gamma_{\delta}(\mathbf{x},\mathbf{y}):=\gamma_{\delta}(|\mathbf{y}-\mathbf{x}|)\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y})$
that satisfies

$\displaystyle\int_{B_{\delta}(0)}\mathbf{z}_{i}^{2}\gamma_{\delta}(|\mathbf{z}|)~{}d\mathbf{z}=2,$
(31)

we derive for such polynomials $p$ that

$\displaystyle-\mathcal{L}p(\mathbf{x})$
$\displaystyle=\int_{B_{\delta}(\mathbf{x})}\left(p(\mathbf{x})-p(\mathbf{y})\right)\gamma_{\delta}(\mathbf{x},\mathbf{y})~{}d\mathbf{y}=\int_{B_{\delta}(0)}\left(p(\mathbf{x})-p(\mathbf{x}+\mathbf{z})\right)\gamma_{\delta}(|\mathbf{z}|)~{}d\mathbf{y}$
$\displaystyle=-\int_{B_{\delta}(0)}\sum_{\boldsymbol{\alpha}=1}^{3}\frac{D^{\boldsymbol{\alpha}}p(\mathbf{x})}{|\boldsymbol{\alpha}|!}\mathbf{z}^{\boldsymbol{\alpha}}\gamma_{\delta}(|\mathbf{z}|)~{}d\mathbf{z}=-\int_{B_{\delta}(0)}\frac{1}{2}\sum_{i=1}^{d}\partial_{i}^{2}p(\mathbf{x})\mathbf{z}_{i}^{2}\gamma_{\delta}(|\mathbf{z}|)~{}d\mathbf{z}=-\Delta
p(\mathbf{x}),$

where we used in the last step that the function
$\mathbf{z}^{\boldsymbol{\alpha}}$ is an odd function regarding one
$x_{i}$-axis, if $|\boldsymbol{\alpha}|$ is an odd number. Furthermore, if
$\boldsymbol{\alpha}=(\boldsymbol{\alpha}_{1},...,\boldsymbol{\alpha}_{d})$
with $|\boldsymbol{\alpha}|=2$,
$\boldsymbol{\alpha}_{i}=\boldsymbol{\alpha}_{j}=1$ and $i\neq j$, the
function $\mathbf{z}^{\boldsymbol{\alpha}}=\mathbf{z}_{i}\mathbf{z}_{j}$ is an
odd function regarding the $x_{j}$-axis. Since $B_{\delta}(0)$ is a symmetric
domain regarding every $x_{i}$-axis, the integrals, in which these functions
$\mathbf{z}^{\boldsymbol{\alpha}}$ appear, vanish. In Section 7.3 we use
different kernels that satisfy the conditions (31) in a numerical example,
where the nonlocal Schwarz formulation passes the linear patch test.

##  7 Numerical Experiments

In the following we numerically examine the Schwarz method for several
nonlocal problems. We start with an example for the nonlocal Dirchlet problem
and continue with a Neumann problem. After that we investigate the patch test
for one example. In the last section we study two preconditioned GMRES
versions that we compare with GMRES without preconditioner. We apply the
finite element method in all experiments as mentioned in Chapter 5. Moreover,
we consider the residual error regarding the Euclidean norm
$||\mathbf{A}\mathbf{u}^{k}-\mathbf{b}||_{2}$, where $\mathbf{u}^{k}$ is the
solution after the $k$-th iteration and
${\mathbf{b}:=\mathbf{f}-\mathbf{A}_{\mathcal{I}}\mathbf{g}}$. In our
experiments the energy error
${||\mathbf{u}^{k}-\mathbf{u}^{k-1}||_{\mathbf{A}}:=\left(\mathbf{u}^{k}-\mathbf{u}^{k-1}\right)^{\top}\mathbf{A}\left(\mathbf{u}^{k}-\mathbf{u}^{k-1}\right)}$
had the same behaviour as the residual error. For a more concise presentation,
we therefore only discuss results involving the residual error.

###  7.1 Nonlocal Dirichlet Problem

In our first experiment, we compute a nonlocal Dirichlet problem as described
in Section 2.1. Here, we choose $\Omega$ as depicted in Figure 7.1 and
$\delta=0.1$. Thus, we have
$\bar{\Omega}=\bar{\Omega}_{1}\cup\bar{\Omega}_{2}\cup\bar{\Omega}_{3}$.

$\Omega_{1}$
---
$\Omega_{2}$
---
$\Omega_{3}$
---
$\mathcal{I}$
---
Figure 7.1: The domain $\Omega=(0,1)^{2}$ is divided into $\Omega_{1}$,
$\Omega_{2}$ and $\Omega_{3}$. The nonlocal boundary $\mathcal{I}$ is depicted
in red.

Further, we use the following kernel, which satisfies the requirements of a
singular symmetric kernel of Remark 2.2, and we employ the subsequent
piecewise constant forcing term and boundary data

$\displaystyle\gamma(\mathbf{x},\mathbf{y})$
$\displaystyle=\begin{cases}\frac{4c_{\delta}}{||\mathbf{x}-\mathbf{y}||_{2}^{2+2s}}\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y})&\mathbf{x},\mathbf{y}\in\Omega_{1}\text{
or }\mathbf{x},\mathbf{y}\in\Omega_{2},\\\
\frac{7c_{\delta}}{||\mathbf{x}-\mathbf{y}||_{2}^{2+2s}}\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y})&\mathbf{x},\mathbf{y}\in\Omega_{3},\\\
\frac{5c_{\delta}}{||\mathbf{x}-\mathbf{y}||_{2}^{2+2s}}\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y})&\mathbf{x}\in\mathcal{I}\text{
or }\mathbf{y}\in\mathcal{I},\\\
\frac{10c_{\delta}}{||\mathbf{x}-\mathbf{y}||_{2}^{2+2s}}\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y})&\text{else},\end{cases}$
$\displaystyle c_{\delta}$ $\displaystyle=\frac{2-2s}{\pi\delta^{2-2s}},\
s=0.5,\
f(\mathbf{x})=5\chi_{\Omega_{1}\cup\Omega_{2}}(\mathbf{x})+1\chi_{\Omega_{3}}(\mathbf{x})\text{
and }g=0\text{ on }\mathcal{I}.$

Here, the kernel $\gamma$ can, e.g., be considered as a special case of the
kernels investigated in [5] or as a Case 1 kernel of [7]. The stiffness
matrices for every subproblem are computed by using the python package
nlfem[16], which has to be done only once at the beginning of the algorithm.
Then, every subproblem is solved with LGMRES with tolerance $10^{-12}$ and the
Schwarz method stops, when the residual error is below $10^{-9}$. The results
for the multiplicative and additive Schwarz method are shown in Figure 7.2 for
different choices of the mesh parameter $h$. Additionally, we can observe
quadratic $h$-convergence(see Figure 7.8).

Figure 7.2: Here, we can see the residual error of the multiplicative and
additive Schwarz method for the nonlocal Dirichlet problem with a singular
symmetric kernel regarding different choices for the mesh resolution $h$. In
all cases the error decreases in a linear fashion, which we expected at least
for the multiplicative Schwarz version due to Remark 5.3. Moreover, the
additive version needs roughly twice as many iterations as the multiplicative
Schwarz algorithm.

###  7.2 Nonlocal Problem with Neumann Boundary

In this section we define a nonlocal problem with Neumann boundary conditions
as described in 3.2, where we can have both Neumann and Dirichlet boundary
conditions for a subproblem, see (21). The setting is shown in Figure 7.3. By
applying the finite element method, we also have degrees of freedom on the
nonlocal boundary $\mathcal{I}$, which we therefore divide into
$\mathcal{I}_{1}^{N}$ and $\mathcal{I}_{2}^{N}$. Moreover, we choose an
integrable kernel $\gamma$ and $f$ as well as $\kappa$ to be piecewise
constant and dependent on $\Omega_{1}$ and $\Omega_{2}$ as follows

$\displaystyle\Omega$ $\displaystyle=\left(0,1\right)^{2},\ \delta=0.1,$
$\displaystyle\gamma(\mathbf{x},\mathbf{y})$
$\displaystyle=\frac{4}{\pi\delta^{4}}\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y}),$
$\displaystyle\kappa(\mathbf{x})$
$\displaystyle=1\chi_{\Omega_{1}}(\mathbf{x})+10\chi_{\Omega_{2}}(\mathbf{x})\text{
and }$ $\displaystyle f(\mathbf{x})$
$\displaystyle=10\chi_{\Omega_{1}}(\mathbf{x})+1\chi_{\Omega_{2}}(\mathbf{x}).$

For these tests we only consider the multiplicative Schwarz approach(see
Remark 3.2), and we take the same tolerances as in the previous Chapter, i.e.,
tolerance $10^{-12}$ for LGMRES and $10^{-9}$ regarding the residual error as
the termination criterion for the multiplicative Schwarz algorithm. As we can
observe in Figure 7.4 and as expected in Remark 5.3, the behavior of the
errors corresponding to the multiplicative Schwarz method stay the same
compared to Schwarz methods for nonlocal Dirichlet problems, i.e., the
residual error decreases in a linear way. However, the number of needed
iterations to fulfill the termination criterion stays roughly the same, which
can be explained by the fact, that the condition number of the stiffness
matrix in the continuous Galerkin approach(without splitting of inner boundary
basis functions) is constant for integrable kernels(see [7, Theorem 6.3]).
Lastly, we also notice quadratic $h$-convergence for this example, which is
illustrated in Figure 7.8.

$\Omega_{1}$
---
$\Omega_{2}$
---
$\mathcal{I}_{2}^{N}$
---
$\mathcal{I}_{1}^{N}$
---
Figure 7.3: In this example $\Omega$ is divided in two subdomains $\Omega_{1}$
and $\Omega_{2}$ and the nonlocal boundary $\mathcal{I}$ is decomposed in
$\mathcal{I}_{1}^{N}$ and $\mathcal{I}_{2}^{N}$. Figure 7.4: In this picture
the residual error regarding the multiplicative Schwarz method for the
nonlocal Problem with Neumann boundary condition w.r.t. a selection of mesh
parameter $h$ is depicted. Again, we can observe a linear decrease in the
residual error.

###  7.3 Patch Test

$\Omega_{1}$
---
$\Omega_{2}$
---
$\mathcal{I}$
---
Figure 7.5: Here, we see the decomposition of $\Omega$ in a turquoise area
$\Omega_{1}$ and in a gray domain $\Omega_{2}$ that we use for the patch test
and for testing the (preconditioned) GMRES in Chapter 7.4. The nonlocal
boundary $\mathcal{I}$ is again depicted in red.

Now, we conduct a linear patch regarding two nonlocal diffusion operators. We
solve in our case

$\displaystyle-\mathcal{L}_{1}u_{1}$ $\displaystyle=0\quad\quad$
$\displaystyle\text{on }\Omega_{1},$ $\displaystyle-\mathcal{L}_{2}u_{2}$
$\displaystyle=0\quad\quad$ $\displaystyle\text{on }\Omega_{2},$
$\displaystyle u_{1}$ $\displaystyle=u_{2}$ $\displaystyle\text{on
}\Omega_{2},\hskip 30.00005pt\text{and}\hskip 30.00005pt$ $\displaystyle
u_{2}$ $\displaystyle=u_{1}$ $\displaystyle\text{on }\Omega_{1},$
$\displaystyle u_{1}$ $\displaystyle=g$ $\displaystyle\text{on }\mathcal{I}$
$\displaystyle u_{2}$ $\displaystyle=g$ $\displaystyle\text{on }\mathcal{I},$

where

$\displaystyle g(\mathbf{x})$
$\displaystyle:=\mathbf{x}_{1}+\mathbf{x}_{2}\text{ for
}\mathbf{x}\in\Omega\cup\mathcal{I},\
-\mathcal{L}_{i}u(\mathbf{x}):=\int_{\Omega\cup\mathcal{I}}\left(u(\mathbf{x})-u(\mathbf{y})\right)\gamma_{i}(\mathbf{x},\mathbf{y})~{}d\mathbf{y}\text{
with}$ $\displaystyle\gamma_{1}(\mathbf{x},\mathbf{y})$
$\displaystyle:=\frac{4}{\pi\delta^{4}}\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y}),\
\gamma_{2}(\mathbf{x},\mathbf{y}):=\frac{c_{\delta}}{||\mathbf{x}-\mathbf{y}||^{2+2s}}\chi_{B_{\delta}(\mathbf{x})}(\mathbf{y}),~{}s=0.6\text{
and }c_{\delta}:=\frac{2-2s}{\pi\delta^{2-2s}}.$

Here, both kernels $\gamma_{1}$ and $\gamma_{2}$ are symmetric, but the
resulting kernel $\gamma$ on the complete domain
$\left(\Omega\cup\mathcal{I}\right)\times\left(\Omega\cup\mathcal{I}\right)$
with

$\displaystyle\gamma(\mathbf{x},\mathbf{y})=\gamma_{1}(\mathbf{x},\mathbf{y})\chi_{\Omega_{1}}(\mathbf{x})+\gamma_{2}(\mathbf{x},\mathbf{y})\chi_{\Omega_{2}}(\mathbf{x})$

is nonsymmetric. In Figure 7.5 the setup is the illustrated and the starting
and final solution are depicted in Figure 7.6. We again used LGMRES with
tolerance $10^{-12}$ and stopped after the residual error dropped under
$10^{-9}$.

Figure 7.6: On the left-hand side we see the starting solution, where we have
$u^{0}=g$ on the nonlocal boundary $\mathcal{I}$ and $u^{0}=0$ on the domain
$\Omega$. Additionally, the final solution is shown on the right-hand side.

The convergence results w.r.t. the residual error can be seen in Figure 7.7,
where we also observe linear convergence. Since we know the exact solution
$g$, we can examine the $L^{2}(\Omega\cup\mathcal{I})$-norm distance of
several solutions $u_{h}^{p}$ of the patch test, that depend on the choice of
the mesh parameter $h$, to the the function $g_{0.01}$, which is the
projection of $g$ onto the mesh with size $h=0.01$. The results are presented
in Figure 7.8. In this case, we again observe quadratic $h$-convergence.

Figure 7.7: In case of this linear patch test, we can observe linear
convergence regarding the residual error for the multiplicative Schwarz
approach. Figure 7.8: Here, we denote by $u_{h}^{D}$, $u_{h}^{N}$ and
$u_{h}^{p}$ the final solution of the Dirichlet problem, the Neumann problem
and the patch test in dependence of the mesh parameter $h$ as described in
Chapters 7.1, 7.2 and 7.3. In this Figure, we compare $u_{h}^{D}$ and
$u_{h}^{N}$ in the $L^{2}(\Omega\cup\mathcal{I})$-norm to the solution
$u^{D}_{0.01}$ and $u_{0.01}^{N}$, respectively, where $h=0.01$. Moreover, in
case of the patch test, we compute the $L^{2}(\Omega\cup\mathcal{I})$-norm
distance of $u_{k}^{p}$ to the projection of the exact solution $g$ onto the
space of continuous piecewise linear basis functions regarding the mesh with
size $h=0.01$, which is indicated by $g_{0.01}$. In all three cases, we can
observe a quadratic convergence.

###  7.4 Preconditioned GMRES

In the finite element setting the multiplicative Schwarz method is equivalent
to the block-Gauß-Seidel algorithm and the additive Schwarz method corresponds
to the block-Jacobi algorithm as seen in Section 5.3. Therefore, we make use
of the block-Gauß-Seidel preconditioner $\mathbf{M}_{BGS}^{-1}$ and the block-
Jacobi preconditioner $\mathbf{M}_{BJ}^{-1}$, that are defined as follows

$\displaystyle\mathbf{M}_{BGS}=\begin{pmatrix}\mathbf{A}_{11}&\cdots&0\\\
\vdots&\ddots&\vdots\\\
\mathbf{A}_{1n}&\cdots&\mathbf{A}_{nn}\end{pmatrix}\text{ and
}\mathbf{M}_{BJ}=diag(\mathbf{A}_{11},...,\mathbf{A}_{nn}),$

to precondition GMRES. For more information on these two preconditioner we
refer to [22, Chapter 4.1.2]. In the tests we compare GMRES equipped with a
left preconditioner $\mathbf{M}_{BGS}^{-1}$ or $\mathbf{M}_{BJ}^{-1}$ to GMRES
without any preconditioning.
In all cases we used the constant kernel $\gamma_{1}$ on $\Omega_{1}$ and the
fractional kernel $\gamma_{2}$ on $\Omega_{2}$ of Section 7.3. Moreover, we
set the boundary data $g=0$ on $\mathcal{I}$ and the forcing term $f=10$ on
$\Omega$. In every test the convergence tolerance of GMRES is chosen to be
$10^{-10}$. Additionally, we denote by $\kappa_{GMRES}$ the condition number
of the finite element matrix $\mathbf{A}$, by $\kappa_{GMRES+BJ}$ the
condition number of $\mathbf{M}_{BJ}^{-1}\mathbf{A}$ and by
$\kappa_{GMRES+BGS}$ the condition number of
$\mathbf{M}_{BGS}^{-1}\mathbf{A}$, respectively.

$h$ | GMRES | GMRES+BJ | GMRES+BGS | $\kappa_{\text{GMRES}}$ | $\kappa_{\text{GMRES+BJ}}$ | $\kappa_{\text{GMRES+BGS}}$
---|---|---|---|---|---|---
0.1 | 177 | 39 | 15 | 2715.41 | 39.79 | 18.85
0.05 | 365 | 30 | 15 | 4649.59 | 42.46 | 20.15
0.025 | 580 | 32 | 13 | 8110.49 | 45.72 | 20.53
Table 7.1: For $\delta=0.1$ and $s=0.5$ we tested different mesh sizes $h$ and documented the number of required iterations as well as the condition number of the (preconditioned) matrices. $s$ | GMRES | GMRES+BJ | GMRES+BGS | $\kappa_{\text{GMRES}}$ | $\kappa_{\text{GMRES+BJ}}$ | $\kappa_{\text{GMRES+BGS}}$
---|---|---|---|---|---|---
0.2 | 118 | 25 | 12 | 483.18 | 35.22 | 18.68
0.5 | 580 | 32 | 13 | 8110.49 | 45.72 | 20.53
0.8 | 2509 | 37 | 14 | 11687.19 | 59.13 | 22.18
Table 7.2: In these experiments we set $\delta=0.1$ and the mesh size $h=0.025$ was chosen. Then, we tested different values for the parameter $s$ of the singular kernel and recorded the condition number of the (preconditoned) system matrix and the number of iterations that each version of GMRES needed. $\delta_{2}$ | GMRES | GMRES+BJ | GMRES+BGS | $\kappa_{\text{GMRES}}$ | $\kappa_{\text{GMRES+BJ}}$ | $\kappa_{\text{GMRES+BGS}}$
---|---|---|---|---|---|---
0.1 | 580 | 32 | 13 | 8110.49 | 45.72 | 20.53
0.05 | 1089 | 35 | 14 | 18862.67 | 57.83 | 22.24
0.025 | 1675 | 36 | 14 | 39848.76 | 63.35 | 21.58
Table 7.3: In these tests we set the horizon $\delta_{1}=0.1$ for the constant
kernel and varied the parameter $\delta_{2}$ of the singular kernel. Moreover,
we chose $h=0.025$ and $s=0.5$ and noted the number of iterations that each
GMRES variation required as well as the condition number of the accompanying
(preconditioned) matrices.

As we can see in all tests of Tables 7.1 \- 7.3 the number of required
iterations and the condition number reduce significantly, if we use the block-
Jacobi or block-Gauß-Seidel preconditioner instead of GMRES with no
preconditioning. For the block-Gauß-Seidel preconditioner the number of
iterations as well as the condition number even stay roughly the same.

##  8 Conclusion

We have shown in this paper how the multiplicative and additive Schwarz method
can be applied to nonlocal Dirichlet problems and how they can be utilized to
solve nonlocal problems with Neumann boundary condition. In the first case we
showed the convergence of the multiplicative version of the Schwarz algorithm
for two widely used classes of symmetric kernels and in the second case we
only needed symmetric kernels to prove that the multiplicative Schwarz
approach converges. Additionally, coupling nonlocal operators with the Schwarz
method fulfills trivially nonlocal patch tests. In the last section we
provided examples for all discussed problems in this work. Additionally, we
observed in the patch test experiments in Chapter 7.3, that the Schwarz
approach can also work for nonsymmetric kernels in practice. Further, we
investigated preconditioned GMRES variants that resulted from the
multiplicative or additive Schwarz algorithm in the finite element setting,
where we noticed, that especially the block-Gauß-Seidel preconditioned GMRES
version only needs a small amount of iterations.

## Acknowledgements

This work has been supported by the German Research Foundation (DFG) within
the Research Training Group 2126: ’Algorithmic Optimization’.

## References

  * [1] G. Acosta, F. M. Bersetche, and J. D. Rossi.  A domain decomposition scheme for couplings between local and nonlocal equations.  Computational Methods in Applied Mathematics, (0), 2023.
  * [2] D. Brockmann.  Anomalous diffusion and the structure of human transportation networks.  The European Physical Journal Special Topics, 157:173–189, 2008\.
  * [3] A. Buades, B. Coll, and J.-M. Morel.  Image denoising methods. A new nonlocal principle.  SIAM review, 52(1):113–147, 2010.
  * [4] G. Capodaglio, M. D’Elia, P. Bochev, and M. Gunzburger.  An energy-based coupling approach to nonlocal interface problems.  Computers & Fluids, 207:104593, 2020.
  * [5] M. D’Elia and C. Glusa.  A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems.  Numerical Methods for Partial Differential Equations, 38(6):2084–2103, 2022.
  * [6] M. D’Elia and M. Gulian.  Analysis of anisotropic nonlocal diffusion models: Well-posedness of fractional problems for anomalous transport.  arXiv preprint arXiv:2101.04289, 2021.
  * [7] Q. Du, M. Gunzburger, R. B. Lehoucq, and K. Zhou.  Analysis and approximation of nonlocal diffusion problems with volume constraints.  SIAM review, 54(4):667–696, 2012.
  * [8] M. D’Elia, Q. Du, M. Gunzburger, and R. Lehoucq.  Nonlocal convection-diffusion problems on bounded domains and finite-range jump processes.  Computational Methods in Applied Mathematics, 17(4):707–722, 2017\.
  * [9] M. D’Elia, M. Gunzburger, and C. Vollmann.  A cookbook for approximating Euclidean balls and for quadrature rules in finite element methods for nonlocal problems.  Mathematical Models and Methods in Applied Sciences, 31(08):1505–1567, 2021.
  * [10] M. D’Elia, X. Li, P. Seleson, X. Tian, and Y. Yu.  A review of local-to-nonlocal coupling methods in nonlocal diffusion and nonlocal mechanics.  Journal of Peridynamics and Nonlocal Modeling, pages 1–50, 2021\.
  * [11] M. D’Elia, J. C. De Los Reyes, and A. Miniguano-Trujillo.  Bilevel parameter learning for nonlocal image denoising models.  Journal of Mathematical Imaging and Vision, 63(6):753–775, 2021\.
  * [12] C. Glusa, M. D’Elia, G. Capodaglio, M. Gunzburger, and P. Bochev.  An asymptotically compatible coupling formulation for nonlocal interface problems with jumps.  SIAM Journal on Scientific Computing, 45(3):A1359–A1384, 2023.
  * [13] W. Hackbusch.  Iterative Solution of Large Sparse Systems of Equations, volume 95.  Springer, 2016.
  * [14] A. Javili, R. Morasata, E. Oterkus, and S. Oterkus.  Peridynamics review.  Mathematics and Mechanics of Solids, 24(11):3714–3739, 2019.
  * [15] M. Klar, G. Capodaglio, M. D’Elia, C. Glusa, M. Gunzburger, and C. Vollmann.  A scalable domain decomposition method for FEM discretizations of nonlocal equations of integrable and fractional type.  arXiv preprint arXiv:2306.00094, 2023.
  * [16] M. Klar, C. Vollmann, and V. Schulz.  nlfem: A flexible 2d FEM Code for Nonlocal Convection-Diffusion and Mechanics, 2022.
  * [17] P.-L. Lions.  On the Schwarz alternating method. II.  Domain decomposition methods, 628:47–70, 1989.
  * [18] 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, volume 6, pages 202–223. SIAM Philadelphia, 1990.
  * [19] P.-L. Lions et al.  On the Schwarz alternating method. I.  In First international symposium on domain decomposition methods for partial differential equations, volume 1, page 42. Paris, France, 1988.
  * [20] T. Mathew.  Domain decomposition methods for the numerical solution of partial differential equations, volume 61.  Springer Science & Business Media, 2008.
  * [21] R. Metzler and J. Klafter.  The random walk’s guide to anomalous diffusion: a fractional dynamics approach.  Physics reports, 339(1):1–77, 2000.
  * [22] Y. Saad.  Iterative methods for sparse linear systems.  SIAM, 2003.
  * [23] M. Schuster, C. Vollmann, and V. Schulz.  Shape optimization for interface identification in nonlocal models, 2022\.
  * [24] H. A. Schwarz.  Ueber einige Abbildungsaufgaben.  1869\.
  * [25] H. A. Schwarz.  Ueber einen Grenzübergang durch alternirendes Verfahren.  Zürcher u. Furrer, 1870.
  * [26] S. A. Silling.  Reformulation of elasticity theory for discontinuities and long-range forces.  Journal of the Mechanics and Physics of Solids, 48(1):175–209, 2000\.
  * [27] J. L. Suzuki, M. Gulian, M. Zayernouri, and M. D’Elia.  Fractional modeling in action: A survey of nonlocal models for subsurface transport, turbulent flows, and anomalous materials.  Journal of Peridynamics and Nonlocal Modeling, pages 1–68, 2022\.
  * [28] A. Toselli and O. Widlund.  Domain decomposition methods-algorithms and theory, volume 34.  Springer Science & Business Media, 2004.
  * [29] C. Vollmann.  Nonlocal models with truncated interaction kernels - analysis, finite element methods and shape optimization.  PhD thesis, Universität Trier, 2019.
  * [30] M. Vu.  The Nonlocal Neumann Problem.  PhD thesis, Universität Trier, 2023.
  * [31] X. Xu, C. Glusa, M. D’Elia, and J. T. Foster.  A FETI approach to domain decomposition for meshfree discretizations of nonlocal problems.  Computer Methods in Applied Mechanics and Engineering, 387:114148, 2021.