# Validated integration of semilinear parabolic PDEs

Jan Bouwe van den Berg Maxime Breden Ray Sheombarsing VU Amsterdam, Department
of Mathematics, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands.
<EMAIL_ADDRESS>CNRS, École polytechnique, Institut Polytechnique de
Paris, 91120 Palaiseau, France<EMAIL_ADDRESS>Amsterdam,
Department of Mathematics, De Boelelaan 1081, 1081 HV Amsterdam, The
Netherlands<EMAIL_ADDRESS>
###### Abstract

Integrating evolutionary partial differential equations (PDEs) is an essential
ingredient for studying the dynamics of the solutions. Indeed, simulations are
at the core of scientific computing, but their mathematical reliability is
often difficult to quantify, especially when one is interested in the output
of a given simulation, rather than in the asymptotic regime where the
discretization parameter tends to zero. In this paper we present a computer-
assisted proof methodology to perform rigorous time integration for scalar
semilinear parabolic PDEs with periodic boundary conditions. We formulate an
equivalent zero-finding problem based on a variations of constants formula in
Fourier space. Using Chebyshev interpolation and domain decomposition, we then
finish the proof with a Newton–Kantorovich type argument. The final output of
this procedure is a proof of existence of an orbit, together with guaranteed
error bounds between this orbit and a numerically computed approximation. We
illustrate the versatility of the approach with results for the Fisher
equation, the Swift–Hohenberg equation, the Ohta–Kawasaki equation and the
Kuramoto–Sivashinsky equation. We expect that this rigorous integrator can
form the basis for studying boundary value problems for connecting orbits in
partial differential equations.

Keywords

Parabolic PDEs $\cdot$ Initial value problems $\cdot$ Computer-assisted proofs
$\cdot$ A posteriori error estimates

Mathematics Subject Classification (2020)

35K15 $\cdot$ 35K30 $\cdot$ 35K58 $\cdot$ 37L65 $\cdot$ 65G20 $\cdot$ 65M15
$\cdot$ 65M70

##  1 Introduction

During the last decades, computer-assisted proofs have become an increasingly
effective tool in the study of nonlinear ordinary differential equations
(ODEs) and dynamical systems in general. The rapid progress of the development
of computer hardware has made it possible to put numerical simulations on a
rigorous footing through the construction of theorems whose hypotheses can be
verified with the aid of a computer. Today, there exists a large variety of
rigorous numerical methods for studying invariant objects in systems of ODEs,
such as equilibria, periodic orbits, connecting orbits, invariant manifolds,
etc. In particular, we mention the prominent software packages CAPD [24], COSY
[1] and Intlab [8, 33]. Detailed knowledge of invariant objects can provide
deep insight into the global structure of a dynamical system, which for
nonlinear systems is typically difficult to obtain solely from pen-and-paper
analysis.

While evolutionary partial differential equations (PDEs) and the associated
_infinite dimensional dynamical systems_ are much more challenging, computer-
assisted proofs are also gradually being developed for this setting. In
particular, going back to the early works [25, 30], many computer-assisted
proofs techniques have been developed for elliptic problems, corresponding to
stationary solutions of associated parabolic PDEs), see for instance [2, 13,
17, 29, 35, 44, 47, 48]. A complete review of these works is beyond the scope
of this paper, but we refer the interested readers to the recent book [27] and
the references therein. In addition to steady states, traveling waves
solutions have been studied with computer-assistance [4, 6, 43], as well as
periodic solutions of PDEs [3, 19, 20, 39, 50], even for some ill-posed
problems [9]. More recently, rigorous enclosures for local stable and unstable
manifolds in parabolic PDEs have been developed as well [31, 41]. We also
mention the works [7, 10], where other state-of-the-art computer-assisted
proofs for PDEs are described.

In this article, we introduce a rigorous initial value solver for semilinear
parabolic PDEs. One main motivation behind this work is to later use this
rigorous solver to validate boundary value problems for parabolic PDEs, and to
combine it with the above mentioned techniques for local stable and unstable
manifolds [31, 41], in order to validate connecting orbits in parabolic PDEs,
potentially between two saddle equilibria. However, since rigorous integration
of PDEs is already a relatively involved task, we focus in this work on IVPs
only. More precisely, in this paper we develop a rigorous computational method
for validating solutions of initial value problems for scalar semilinear
parabolic equations, with periodic boundary conditions, of the form

$\displaystyle\begin{cases}\dfrac{\partial u}{\partial
t}=(-1)^{R+1}\dfrac{\partial^{2R}u}{\partial
x^{2R}}+\displaystyle\sum_{j=0}^{2R-1}\dfrac{\partial^{j}g^{(j)}(u)}{\partial
x^{j}},&t\in(0,2\tau],\ x\in[0,2\pi],\\\\[8.61108pt]
u(t,0)=u(t,2\pi),&t\in[0,2\tau],\\\\[8.61108pt]
u(0,x)=f(x),&x\in[0,2\pi],\end{cases}$ (1)

where $R\in\mathbb{N}_{\geq 1}$ is a positive integer, $\tau>0$ determines the
integration time, $g^{(j)}:\mathbb{R}\to\mathbb{R}$ are given polynomial
functions for $j=0,\ldots,2R-1$, and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a
$2\pi$-periodic function.

######  Remark 1.1.

The factor $2$ in the time interval $(0,2\tau]$ is of course inconsequential,
and only introduced to simplify the expressions we obtain after rescaling time
in Section 3.

To sketch the context and be able to compare with the existing literature, we
start with a more conceptional discussion rather than diving into details
immediately. Conceptually, one can think of (1) as being simply an equation of
the form

$\displaystyle\dfrac{\partial u}{\partial t}=H(u),$ (2)

and we assume to have an approximate solution $\bar{u}$ at our disposal. Our
strategy for validating this approximate solution starts by using a well
chosen splitting of the righthand side as

$\displaystyle H(u)=\mathcal{L}u+G(u),$ (3)

were $\mathcal{L}$ is a linear operator which may or may not depend on time.
When $\mathcal{L}$ is time _independent_ , as it essentially is in the current
work (although it is taken piecewise constant in time at a later stage), we
can rewrite (2) as a fixed point problem, using the semigroup associated to
$\mathcal{L}$ and the fixed point operator $\tilde{T}$ defined by

$\displaystyle\tilde{T}(u)(t)=e^{t\mathcal{L}}f+\int_{0}^{t}e^{(t-s)\mathcal{L}}G(u(s))\mbox{d}s,\quad
t\in[0,2\tau].$ (4)

Our validation procedure for proving the existence of an orbit near $\bar{u}$
will be based on the Banach fixed-point theorem. Therefore, we would like
$\tilde{T}$ to be a contraction on a neighborhood of $\bar{u}$, preferably for
long integration times $2\tau$.

######  Remark 1.2.

We rely heavily on the structure of equation (1), and more precisely on the
fact that the highest order term $(-1)^{R+1}\dfrac{\partial^{2R}}{\partial
x^{2R}}$ is dissipative. Indeed, this ensures that the associated semi-group
is contracting on the higher order Fourier modes. Yet, we want our approach to
be applicable for equations showcasing interesting dynamics, hence we must
also be able to handle the appearance of some unstable modes, which means we
allow the semigroup generated by $\mathcal{L}$ to not be contracting for
(finitely many) low order modes.

We note that, of course, using the splitting $H(u)=\mathcal{L}u+G(u)$, one
could always artificially choose $\mathcal{L}$ so that the associated
semigroup is contracting, but that does not mean that $\tilde{T}$ itself will
be contracting for larger integration times, since this is also influenced by
$G(u)$. For instance, the scalar ODE $u^{\prime}=u$ can be split as
$u^{\prime}=-ku+(k+1)u$, with $k>0$, which gives rise to

$\displaystyle\tilde{T}(u)(t)=e^{-tk}f+(k+1)\int_{0}^{t}e^{-(t-s)k}u(s)\mbox{d}s,$

which is not contracting in $C^{0}$, except for short times.

The pivotal choice of $\mathcal{L}$ that we make in this work is meant to help
us getting a contracting operator $\tilde{T}$, even when the semigroup
generated by $\mathcal{L}$ is not contracting for low order modes. To that
end, we take for $\mathcal{L}$ an approximation of $DH(\bar{u})$, the Fréchet
derivative of $H$ at the approximate solution, in order to make
$DG(\bar{u})=\mathcal{L}-DH(\bar{u})$ small. However, by imposing that
$\mathcal{L}$ does not depend on time, in general we cannot hope to get
$\mathcal{L}$ arbitrarily close to $DH(\bar{u})$. Therefore, we further
reformulate the problem by first turning it into a zero finding problem

$\displaystyle
F(u)(t)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,e^{t\mathcal{L}}f+\int_{0}^{t}e^{(t-s)\mathcal{L}}G(u(s))\mbox{d}s-u(t)=0,\quad\text{for
all }t\in[0,2\tau],$

which itself is then turned into another fixed point problem, given by a
Newton-like operator

$\displaystyle T:u\mapsto u-AF(u),$ (5)

where $A$ is a suitable approximate inverse of $DF(\bar{u})$. Notice that, for
$A=I$, we get $T=\tilde{T}$. Since, as mentioned previously, we already expect
$\tilde{T}$ to be contracting for sufficiently high order Fourier modes, we
only need $A$ to act nontrivially on finitely many modes. Once a suitable
approximate inverse $A$ is defined, we then want to prove that $T$ is a
contraction in a neighborhood of $\bar{u}$, and this is what most of this
paper is devoted to.

Before we proceed, let us again emphasize that the issue of whether $T$ is
contracting or not, and the issue of whether or not we are actually able to
prove it (with computer assistance), both depend greatly on three main
(related) ingredients:

  * •

the choice of $\mathcal{L}$,

  * •

the choice of $A$,

  * •

the choice of the Banach space in which we try to prove the contraction.

In light of the above discussion, we now review some of the existing
literature regarding rigorous integrators for parabolic PDEs, and describe how
they relate to or differ from our proposed approach.

One class of rigorous PDE integrators, which use completely different
techniques compared to the ones discussed in this paper, is the one developed
in [50, 51, 52]. It is not centered around a fixed point argument, but rather
based on the validated integration of a finite dimensional system of ODEs
using Lohner-type algorithms developed in [49] and on the notion of self-
consisted bounds introduced in [52]. This methodology was further developed in
[12] and has been used to rigorously study, among other things, globally
attracting solutions in the one dimensional Burgers equation [16],
heteroclinic connections in the one-dimensional Ohta-Kawasaki model [15] and
very recently to prove chaos in the Kuramoto-Sivashinky equation [46].

A second family of rigorous PDE integrators, to which our approach belongs, is
based on a fixed point reformulation which allows to prove, a posteriori, that
a true solution exists in a small neighborhood of a numerically computed
approximation. Within this class, we first mention the foundational work [3],
where numerical solutions are validated using a fixed point operator similar
to $\tilde{T}$ in (4), and where $\mathcal{L}$ is in some sense chosen to be
simple, namely $\mathcal{L}=DH(0)$. In the context of (1), the approach from
[3] would amount to taking111To be more precise, the operator $\mathcal{L}$
used in [3] is sometimes slightly modified, the parts that would contribute to
eigenvalues with positive real part being moved to $G$.

$\displaystyle\mathcal{L}=(-1)^{R+1}\dfrac{\partial^{2R}}{\partial
x^{2R}}+\displaystyle\sum_{j=0}^{2R-1}\alpha_{j}\dfrac{\partial^{j}}{\partial
x^{j}},$ (6)

where $\alpha_{j}\in\mathbb{R}$ is the derivative of $g^{(j)}$ at $0$. This
choice for $\mathcal{L}$ is enticing for making it easier to control the
associated semigroup, which is needed when trying to prove that $\tilde{T}$ is
a contraction. This is especially true if the domain and the boundary
conditions allow for an explicit diagonalization of such an $\mathcal{L}$, as
is the case in [3] and in the current paper. However, while this choice of
$\mathcal{L}$ simplifies the analysis of $\tilde{T}$, it also comes at the
cost of making the operator $\tilde{T}$ potentially _less contracting_
($DG(\bar{u})$ will not be arbitrarily small), and since [3] does not use a
subsequent Newton-like reformulation, the fixed point operator can generally
only be contracting for relatively small values of the integration time
$2\tau$. A similar approach, at least at the level of choosing the fixed point
reformulation, is used in [14]222Formally, in [14] Equation (2) is first
rewritten as
$\tilde{F}(u)=\left(\partial_{t}\right)^{-1}\left(\partial_{t}u-H(u)\right)=0$,
and then turned into the fixed point problem
$\tilde{T}(u)=\left(D\tilde{F}(0)\right)^{-1}\left(D\tilde{F}(0)u-\tilde{F}(u)\right)=u$.
However, noticing that
$\left(D\tilde{F}(0)\right)^{-1}=\left(\partial_{t}-DH(0)\right)^{-1}\partial_{t}$,
one finds that $\tilde{T}(u)$ is nothing but
$\left(\partial_{t}-DH(0)\right)^{-1}\left(H(u)-DH(0)u\right)$, which
corresponds to (4) with $\mathcal{L}=DH(0)$., although the main point of the
latter paper is different and revolves around a Fourier-Chebyshev (fully)
spectral approach, see also Remark 1.4.

On the other end of the spectrum, at least with regards to $\mathcal{L}$, are
the techniques introduced in [34, 36]. There, in terms of the splitting (3),
$\mathcal{L}$ is chosen as a very accurate time dependent approximation of
$DH(\bar{u})$. This means that the mild solution reformulation then reads

$\displaystyle u(t)=U(t,0)f+\int_{0}^{t}U(t,s)G(u(s))\mbox{d}s,\quad
t\in[0,2\tau],$

where $U(t,s)$ is the evolution operator associated to $\mathcal{L}$, i.e.
$U(t,s)\varphi$ is the solution at time $t$ of

$\displaystyle\begin{cases}\dfrac{\partial u}{\partial
t}=\mathcal{L}(t)u,&t\in(s,2\tau],\ x\in[0,2\pi],\\\\[8.61108pt]
u(t,0)=u(t,2\pi),&t\in[s,2\tau],\\\\[8.61108pt]
u(s,x)=\varphi(x),&x\in[0,2\pi].\end{cases}$

By allowing $\mathcal{L}$ to depend on time, one can in principle make
$DG(\bar{u})$ arbitrarily small, hence ensuring that the fixed point operator

$\displaystyle\tilde{T}(u)(t)=U(t,0)f+\int_{0}^{t}U(t,s)G(u(s))\mbox{d}s,\quad
t\in[0,2\tau],$ (7)

is contracting in a neighborhood of $\bar{u}$. Nevertheless, when actually
having to prove that $\tilde{T}$ is a contraction in this setting, one then
has to explicitly control the evolution operator $U(t,s)$, which can be much
more challenging and resource consuming than only having to control the
semigroup associated to a time _independent_ $\mathcal{L}$.

Yet another approach, which starts with a different viewpoint but for which
the fixed point reformulation ends up being very similar, is the one in [23],
see also [27, Chapter 5]. In the context of (1), assuming $R=1$, the approach
in [23] can be interpreted as taking $\mathcal{L}$ to be exactly equal to
$DH(\bar{u})$, and the validation is then based on applying Schauder’s fixed
point theorem to the operator (7), where now $U(t,s)$ is the evolution
operator associated to $DH(\bar{u})$. This choice of $\mathcal{L}$ leads to
$DG(\bar{u})=0$, hence one should be able to find a small neighborhood of
$\bar{u}$ being mapped into itself by $\tilde{T}$, but doing so now requires
controlling the evolution operator associated to $\mathcal{L}=DH(\bar{u})$. In
the end, this is accomplished by introducing some approximation of
$DH(\bar{u})$, see e.g. [22] for the details. We point out that some of the
ideas in this approach but also in the other ones, can be traced back to the
early work [26], in which $\mathcal{L}$ was simply taken to be the Laplacian.

All these works share some similarities regarding the fixed point operator
used for the validation, or, at the very least, they can all be reformulated
into this common framework. However, we emphasize that they then differ
significantly when it comes to how this fixed point operator is analyzed, see
also Remark 1.4. Regarding the choice of $\mathcal{L}$, our own work can be
seen as a compromise between the different approaches just mentioned, as we
try to take for $\mathcal{L}$ an approximation of $DH(\bar{u})$ which is _as
good as we can while remaining time independent_ , thereby avoiding the
problem of having to deal with a solution operator of a non-autonomous system.

######  Remark 1.3.

In all of the approaches discussed above, as well as in the current work, when
trying to validate solutions for larger and larger time intervals, it proves
more efficient (and it is often mandatory in order to obtain a proof) to split
the interval $[0,2\tau]$ into shorter pieces. Of course, one has to glue these
subproblems together, and to do so there are two alternatives, which we call
here _time stepping_ (also sometimes referred to as _concatenation scheme_)
and _domain decomposition_.

The idea of time stepping is to start with some initial condition and to
rigorously integrate the associated initial value problem on a small time
interval. If the proof is successful, we compute a rigorous enclosure for the
endpoint of the orbit and try to integrate the _enclosure_ forward in time.
This process is repeated as long as necessary (and possible). In order for
long time integration to be feasible, the size of the enclosures should be
“managed” properly. In particular, they should not grow too fast. This is
where the dissipativity of the system can prove very helpful, if correctly
leveraged.

An alternative to time stepping is domain decomposition. The idea of this
approach is to still split the time domain into smaller subintervals, but then
to validated everything at once, via a single fixed point operator coupling
all pieces together. This solution can be computationally more expensive, but
has the advantage of being directly adaptable to solving boundary value
problems in time, which is not as straightforward for the time stepping
approach. In this work, we mainly focus on the domain decomposition approach,
which will be presented in detail in Section 6, but time stepping is also
covered in Appendix A.

######  Remark 1.4.

Up to now, while reviewing the literature concerning rigorous integration of
PDEs, we stayed at a rather general level of exposition, and in particular
have not yet mentioned discretization choices. We purposely postponed this
discussion until now, in order to first better emphasize the conceptual
similarities between these different techniques, but of course the way the
approximate solutions are represented (and the often related choices of
function spaces that are made) are crucial when actually having to derive the
precise estimates needed for the validation.

Regarding the spatial variable, Fourier series and associated sequences spaces
are used in [3], whereas results using finite elements and Sobolev spaces can
be found in [23, 27]. Regarding the time variable, most of the existing works
make use of approximations based on low order 1D finite elements, the notable
exception being [14], where a setup allowing for a spectral approach in both
space and time is introduced.

In this work, we make use of a Fourier decomposition in space. Roughly
speaking, dissipativity then tells us that the long time behavior of the
dynamical system should mostly be governed by only a finite number of Fourier
modes, see Section 3. This is why it makes sense to study (1) via a Fourier
expansion in space in the first place, on both paper and in the computer, and
it influences the way we define $\mathcal{L}$. Specifically, for the high
frequency Fourier modes, we take a time and space average of $DH(\bar{u})$ as
a starting point for defining $\mathcal{L}$, which yields a diagonal operator
in Fourier space. We then also incorporate a time independent approximation of
$DH(\bar{u})$ projected onto finitely many (low frequency) Fourier modes.
Regarding the time variable, we approximate the solution using polynomial
interpolation at Chebyshev nodes, which allows us to easily obtain high order
approximations when needed, e.g. for accuracy purposes.

To summarize, the key features of this work are:

  * •

A new and deliberate choice of the operator $\mathcal{L}$ used for the Duhamel
reformulation (4), specifically: time independent but spatially dependent.
This greatly improves the applicability of the validation procedure compared
to more naive choices like
$\mathcal{L}=(-1)^{R+1}\dfrac{\partial^{2R}}{\partial x^{2R}}$ or
$\mathcal{L}$ as in (6), because we allow $\mathcal{L}$ to depend on the
approximate solution $\bar{u}$.

  * •

The incorporation of the domain decomposition technique, which will facilitate
a future extension to boundary value problems in time, and which comes with an
interesting twist in the way the approximate inverse $A$ used for the Newton-
Kantorovich reformulation (5) is defined (see Section 6.1).

  * •

In the context of domain decomposition, the operator $\mathcal{L}$ can be
interpreted as being time dependent in the sense that it is piecewise constant
in time. The Newton-Kantorovich contraction theorem is then applied to a
function space with a (time-dependent) weighted norm, where the weight are
piecewise constant in time, see (19).

  * •

The usage of Chebyshev interpolation in time, whose order can be adapted to
the problem at hand.

  * •

General estimates and associated code (available at [40]) for equations of the
form (1). That is, the user only has to provide $R\in\mathbb{N}_{\geq 1}$,
$\tau>0$, polynomial functions $g^{(j)}$ for $j=0,\ldots,2R-1$, and initial
data $f$; and the code then automatically generates an approximate solution
and tries to validate it using the estimates developed in this paper. This
does involve tweaking of computational parameters, which is partly automated
and partly manual.

Below is an example of the type of results that can be obtained with our
approach.

######  Theorem 1.5.

Consider the Swift-Hohenberg equation

$\displaystyle\begin{cases}\dfrac{\partial u}{\partial
t}=-\left(\dfrac{\partial^{2}}{\partial x^{2}}+1\right)^{2}u+\alpha
u-u^{3},&(t,x)\in(0,t_{\text{{end}}}]\times[0,L],\\\\[8.61108pt]
u(t,0)=u(t,L),&t\in[0,t_{\text{{end}}}],\\\\[8.61108pt]
u(0,x)=f(x),&x\in[0,L]\end{cases}$ (8)

with $\alpha=5$, $L=6\pi$, $t_{\text{{end}}}=3/2$, and
$f(x)=0.4\cos\left(\frac{2\pi x}{L}\right)-0.3\cos\left(\frac{4\pi
x}{L}\right)$. Let $\bar{u}=\bar{u}(t,x)$ be the function represented in
Figure 1, and whose precise description in terms of Chebyshev$\times$Fourier
coefficients can be downloaded at [40]. Then, there exists a smooth solution
$u$ of (8) such that

$\displaystyle\sup_{t\in[0,t_{\text{{end}}}]}\sup_{x\in[0,L]}|u(t,x)-\bar{u}(t,x)|\leq
4\times 10^{-8}.$ (9)

Figure 1: The approximate solution $\bar{u}$ of (8), which has been validated
in Theorem 1.5, depicted twice with different views.

######  Remark 1.6.

The norm used in (9) differs from the one used in the proof, which is stronger
and controls the one in (9). In particular, the solution $u$ obtained in the
above theorem is analytic in the space variable $x$, and the norm used in the
paper reflects that. Furthermore, the error bound obtained from the proof is
not uniform in time, but we report the somewhat weaker resulting uniform bound
in (9). We refer to Section 2, Section 3 and Section 6 for details.

Finally, let us mention a few possible extensions. The theory presented in
this paper can be extended in a straightforward manner to deal with _systems_
of parabolic PDEs, to higher dimensional rectangular spatial domains, to any
boundary conditions allowing for smooth Fourier series representation, and to
complex valued solutions. Non-constant coefficients in the lower order terms
(that is, everywhere except in the $\dfrac{\partial u}{\partial
t}-(-1)^{R+1}\dfrac{\partial^{2R}u}{\partial x^{2R}}$ part) could also be
handled. Regarding nonlinearities, we made some choices in (1) which are not
fully general, for the sake of unifying the presentation. For instance, an
equation like

$\displaystyle\dfrac{\partial u}{\partial t}=-\dfrac{\partial^{4}u}{\partial
x^{4}}-\dfrac{\partial^{2}u}{\partial x^{2}}-\left(\dfrac{\partial u}{\partial
x}\right)^{2},$ (10)

which is one version of the Kuramoto-Sivashinsky equation, cannot be written
the form (1), at least not without changing variables (in this case, the
equation satisfied by $\partial_{x}u$ can in fact be written in the form (1)).
If one wanted to deal directly with (10), or more generally with equations of
the form

$\displaystyle\dfrac{\partial u}{\partial
t}=(-1)^{R+1}\dfrac{\partial^{2R}u}{\partial x^{2R}}+g\left(u,\dfrac{\partial
u}{\partial x},\ldots,\dfrac{\partial^{2R-1}u}{\partial x^{2R-1}}\right),$

where $g$ is a (multivariate) polynomial, some of the estimates derived in
this paper would have to be adapted, but in principle such equations could
also be handled. The extension from initial to boundary value problems is a
topic of ongoing research.

The paper is organized as follows. In Section 2, we give some definitions and
introduce notation related to sequences spaces and Chebyshev interpolation
which will be useful throughout the paper, and state the fixed point Theorem
2.12 that is at the heart of our validation procedure. In Section 3, we give a
precise definition of the crucial operator $\mathcal{L}$ that we use in this
work, and describe the reformulation of (1) leading to the fixed point problem
to which we want to apply Theorem 2.12. Here we initially restrict attention
to a _single_ time domain, postponing the domain decomposition to Section 6.
In Section 4, we collect several technical lemmas, which are then used in
Section 5, where we derive all the estimates required to apply Theorem 2.12 to
the reformulation introduced in Section 3. In Section 6, we describe how to
modify the setup and the already obtained bounds in order to incorporate the
domain decomposition approach in time. Finally, in Section 7 we present
several examples for the Fisher-KPP equation, the Swift-Hohenberg equation,
the Otha-Kawasaki equation and the Kuramoto-Sivashinsky equation, which
illustrate the advantages provided by our choice of $\mathcal{L}$ and by the
domain decomposition. Appendix A contains details about time stepping,
Appendix B describes new sharp interpolation error estimates, Appendix C
specifies how we rigorously enclose some integrals, and Appendix D provides an
algorithm to explicitly compute some constants introduced in Section 4.

All the computations presented in this paper have been implemented in Matlab,
using the phenomenal Intlab package [33] for interval arithmetic. The
computer-assisted parts of the proofs can be reproduced using the code
available at [40].

##  2 Preliminaries

In this section we introduce notation and provide the necessary background for
the tools used in this paper.

###  2.1 Sequence spaces

The functional analytic reformulation of (1) in terms of the Fourier
coefficients is posed on a space of continuous functions from $[-1,1]$ into a
space of geometrically decaying sequences

$\displaystyle\ell^{1}_{\nu}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\biggl{\\{}a\in\mathbb{C}^{\mathbb{Z}}:\sum_{n\in\mathbb{Z}}\left|a_{n}\right|\nu^{|n|}<\infty\biggr{\\}},$

endowed with the norm
$\left\|a\right\|_{\ell^{1}_{\nu}}=\sum_{n\in\mathbb{Z}}\left|a_{n}\right|\nu^{|n|}$,
where $\nu\geq 1$ is some decay rate to be chosen later. We define
$\mathcal{X}_{\nu}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,C\left(\left[-1,1\right],\ell^{1}_{\nu}\right)$
to be the space of continuous functions from $[-1,1]$ into $\ell^{1}_{\nu}$.

Recall that the Fourier coefficients of the product of two Fourier expansions
is given by the discrete convolution: when $u$ and $v$ are $2\pi$-periodic
functions given by $u(x)=\sum_{n\in\mathbb{Z}}\hat{u}_{n}e^{inx}$ and
$v(x)=\sum_{n\in\mathbb{Z}}\hat{v}_{n}e^{inx}$, then

$\displaystyle\left(uv\right)(x)=\sum_{n\in\mathbb{Z}}\left(\hat{u}\ast\hat{v}\right)_{n}e^{inx},\qquad\text{where}\qquad\left(\hat{u}\ast\hat{v}\right)_{n}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\sum_{m\in\mathbb{Z}}\hat{u}_{m}\hat{v}_{n-m},\quad
n\in\mathbb{Z},$

and these formal computations are justified as soon as $u$ and $v$ are smooth
enough, say Lipschitz-continuous, which will be the case in this work.

######  Remark 2.1.

In order to simplify the notation, in the rest of this paper we will use the
same symbol to denote a function and its (discrete) Fourier transform, that is
we write

$u(x)=\sum_{n\in\mathbb{Z}}u_{n}e^{inx}.$

It should be clear from the context whether $u$ denotes the function $x\mapsto
u(x)$ or the sequence of Fourier coefficients
$u=\left(u_{n}\right)_{n\in\mathbb{Z}}$.

We also recall that the discrete convolution gives $\ell^{1}_{\nu}$ a Banach
algebra structure.

######  Lemma 2.2.

For all $u,v\in\ell^{1}_{\nu}$, $u\ast v\in\ell^{1}_{\nu}$ and $\left\|u\ast
v\right\|_{\ell^{1}_{\nu}}\leq\left\|u\right\|_{\ell^{1}_{\nu}}\left\|v\right\|_{\ell^{1}_{\nu}}$.

###  2.2 Chebyshev interpolation

In this section we recall the basics of Chebyshev interpolation. The reader is
referred to [11, 32, 37] for a comprehensive introduction into the theory of
Chebyshev approximation.

######  Definition 2.3 (Chebyshev points).

Let $K\in\mathbb{N}_{\geq 1}$. The $K$-th order Chebyshev points
$\left(t^{K}_{k}\right)_{k=0}^{K}$ are defined by
$t^{K}_{k}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\cos(\frac{\pi
k}{K})$.

######  Remark 2.4.

We shall omit the superscript $K$ from the notation whenever it can be easily
inferred from the context. We have ordered the Chebyshev points from $1$ to
$-1$, i.e., $t_{0}=1$ and $t_{K}=-1$. In the literature the points
$\left(t_{k}\right)_{k=0}^{K}$ are often referred to as the Chebyshev points
of the second kind.

We will refer to the $K$-th order polynomial which interpolates a continuous
function $f:[-1,1]\rightarrow\mathbb{C}$ at the Chebyshev points
$\left(t_{k}\right)_{k=0}^{K}$ as the $K$-th order Chebyshev interpolant of
$f$. Furthermore, we shall denote the operator which sends a continuous
function $f$ to its $K$-th order Chebyshev interpolant by
$P_{K}:C([-1,1],\mathbb{C})\rightarrow C([-1,1],\mathbb{C})$. We note that
$f-P_{K}(f)$ vanishes at $t=\pm 1$ for any $K$ and any $f$, because we have
chosen a family of Chebyshev nodes containing the endpoints of the interval
$[-1,1]$. We denote the space $C([-1,1],\mathbb{C})$ with the $\sup$-norm by
$C^{0}$ throughout. Our approximate solutions and our analysis makes use of
Chebyshev interpolation in time, and we therefore need to control the
associated interpolation errors.

######  Definition 2.5 ($C^{l}$ interpolation error).

For $K\in\mathbb{N}_{\geq 1}$ and $l\in\\{0,1,\ldots,K\\}$ we denote by
$\sigma_{K,l}$ a real constant such that, for all $f\in
C^{l+1}([-1,1],\mathbb{C})$,

$\displaystyle\left\|f-P_{K}(f)\right\|_{C^{0}}\leq\sigma_{K,l}\left\|f^{(l+1)}\right\|_{C^{0}}.$

In practice, we need an explicit value for this constant. One can easily find
explicit values in the literature, at least for real-valued functions, but not
necessarily optimal ones. In Appendix B, we recall some known results about
this constant, and provide almost optimal values for $\sigma_{K,0}$ and $K$
small, which is a crucial case for this work.

Another case which we will encounter frequently is when the function to be
interpolated is analytic in an open set of the complex plane containing the
segment $[-1,1]$. In such a situation, the following error estimate applies.

######  Definition 2.6.

For $\rho\geq 1$, we denote by $\mathcal{E}_{\rho}\subset\mathbb{C}$ the open
ellipse with foci $\pm 1$ such that the length of the semi-major and of the
semi-minor axes sum up to $\rho$, or equivalently, the set of all complex
numbers $z$ such that $|z-1|+|z+1|<\rho+\rho^{-1}$. The set
$\mathcal{E}_{\rho}$ is sometimes referred to as a _Bernstein ellipse_. Notice
that $\rho=1$ is a degenerate case: $\mathcal{E}_{1}=(-1,1)$.

######  Theorem 2.7.

Let $\rho>1$. Suppose $f:\left[-1,1\right]\rightarrow\mathbb{C}$ can be
analytically extended to $\mathcal{E}_{\rho}$, and is bounded on
$\mathcal{E}_{\rho}$. Then

$\displaystyle\left\|f-P_{K}(f)\right\|_{C^{0}}\leq\frac{4\rho^{-K}}{\rho-1}\sup_{z\in\mathcal{E}_{\rho}}\left|f(z)\right|,$

for any $K\in\mathbb{N}$.

The proof can be found in [37, Theorem $8.2$].

In practice, we compute and represent interpolation polynomials using the
Chebyshev basis $\left(T_{k}\right)_{k\in\mathbb{N}_{0}}$.

######  Definition 2.8.

The Chebyshev polynomials $T_{k}:\left[-1,1\right]\rightarrow\mathbb{R}$ are
defined by the relation
$T_{k}\left(\cos\theta\right)=\cos\left(k\theta\right)$, where
$k\in\mathbb{N}_{0}$ and $\theta\in\left[0,\pi\right]$.

######  Remark 2.9.

In the literature the polynomials $\left(T_{k}\right)_{k\in\mathbb{N}_{0}}$
are often referred to as the Chebyshev polynomials of the first kind. The
$K$-th order Chebyshev points $\left(t^{K}_{k}\right)_{k=0}^{K}$ are the
points in $\left[-1,1\right]$ at which $T_{K}$ attains its extrema.
Furthermore, note that $T_{k}\left(t^{K}_{j}\right)=\cos(\frac{\pi jk}{K})$
for $k,j\in\mathbb{N}_{0}$.

The polynomials $\left(T_{k}\right)_{k=0}^{K}$ constitute a basis for the
space of $K$-th order polynomials $\mathbb{P}_{K}$. Hence, any polynomial $P$
of degree at most $K$, and in particular any interpolation polynomial
$P_{K}(f)$, can be uniquely written as

$\displaystyle P=P_{0}+2\sum_{k=1}^{K}P_{k}T_{k},$ (11)

for some Chebyshev coefficients $P_{0},\ldots,P_{K}\in\mathbb{C}$.

For any given polynomial written in the Chebyshev basis, we can easily control
both its $C^{0}$ norm and its supremum on a Bernstein ellipse
$\mathcal{E}_{\rho}$, in terms of its Chebyshev coefficients.

######  Lemma 2.10.

Let $K\in\mathbb{N}_{\geq 0}$ and $P=P_{0}+2\sum_{k=1}^{K}P_{k}T_{k}$, then

$\displaystyle\left\|P\right\|_{C^{0}}\leq\left|P_{0}\right|+2\sum_{k=1}^{K}\left|P_{k}\right|,$
(12)

and

$\displaystyle\sup_{z\in\mathcal{E}_{\rho}}\left|P(z)\right|\leq\left|P_{0}\right|+\sum_{k=1}^{K}\left|P_{k}\right|\left(\rho^{k}+\rho^{-k}\right).$
(13)

###### Proof.

Any element $z$ of $\mathcal{E}_{\rho}$ can be written
$z=\frac{\omega+\omega^{-1}}{2}$ for some $\omega\in\mathbb{C}$ such that
$1\leq|\omega|\leq\rho$, and we have the identity
$T_{k}\bigl{(}\frac{\omega+\omega^{-1}}{2}\bigr{)}=\frac{\omega^{k}+\omega^{-k}}{2}$,
which holds because both sides are analytic and coincide for $\omega$ on the
unit circle. Hence
$\sup_{z\in\mathcal{E}_{\rho}}\left|T_{k}(z)\right|=\frac{\rho^{k}+\rho^{-k}}{2}$,
which yields (13). ∎

######  Remark 2.11.

Note that (11) is, up to the coordinate transformation $x=\cos\theta$, a
Fourier cosine series. This is the motivation for using the factor $2$ in
front of the coefficients $(P_{k})_{k=1}^{K}$. In particular, with this
convention the Chebyshev coefficients of the product of two Chebyshev
expansions can be computed directly (i.e without a rescaling factor) via the
discrete convolution.

###  2.3 A fixed point Theorem

Let $(X^{m},\|\cdot\|_{X^{m}})_{m=1}^{M}$ be Banach spaces,
$X=\Pi_{m=1}^{M}X^{m}$ the product space, and $\pi^{m}:X\to X^{m}$ the
projections onto the components. Let
$r_{*}=(r_{*}^{m})_{m=1}^{M}\in\mathbb{R}_{>0}^{M}$ and $\bar{x}\in X$. For
any $r\in\mathbb{R}_{>0}^{M}$, we define
$\operatorname{Box}(\bar{x},r)=\\{x\in
X:\left\|\pi^{m}(x-\bar{x})\right\|_{X^{m}}\leq r^{m}\text{ for }1\leq m\leq
M\\}$. We consider a map $T\in C^{1}(\operatorname{Box}(\bar{x},r),X)$. For
$r,r_{*}\in\mathbb{R}_{>0}^{M}$ we say that $r\leq r_{*}$ if $r^{m}\leq
r_{*}^{m}$ for all $1\leq m\leq M$. Finally, we denote partial Fréchet
derivatives by $D_{i}$.

The following statement, based on the Banach fixed point Theorem, provides
explicit conditions under which $T$ has a unique fixed point in
$\operatorname{Box}(\bar{x},r)$ (for some explicit $r$). Many similar versions
of this theorem have been used in the last decades for computer-assisted
proofs, see e.g. [3, 17, 21, 27, 29, 42, 47]. The one we use here, with
possibly different radii in each component, originates from [38].

######  Theorem 2.12.

Assume that $Y^{m}\geq 0$, $Z^{m}_{i}\geq 0$, $W^{m}_{ij}\geq 0$ for $1\leq
i,j,m\leq M$ satisfy

$\displaystyle\left\|\pi^{m}(T(\bar{x})-\bar{x})\right\|_{X^{m}}$
$\displaystyle\leq Y^{m},$ (14)
$\displaystyle\left\|\pi^{m}D_{i}T(\bar{x})\right\|_{B(X^{i},X^{m})}$
$\displaystyle\leq Z^{m}_{i},$ (15)
$\displaystyle\left\|\pi^{m}(D_{i}T(x)-D_{i}T(\bar{x}))\right\|_{B(X^{i},X^{m})}$
$\displaystyle\leq\sum_{j=1}^{M}W^{m}_{ij}\left\|\pi^{j}(x-\bar{x})\right\|_{X^{j}}$
$\displaystyle\text{for all }x\in\operatorname{Box}(\bar{x},r_{*}).$ (16)

If $r,\eta\in\mathbb{R}_{>0}^{M}$ with $r\leq r_{*}$ satisfy

$\displaystyle
Y^{m}+\sum_{i=1}^{M}Z^{m}_{i}r^{i}+\frac{1}{2}\sum_{i,j=1}^{M}W^{m}_{ij}r^{i}r^{j}$
$\displaystyle\leq r^{m}$ (17)
$\displaystyle\sum_{i=1}^{M}Z^{m}_{i}\eta^{i}+\sum_{i,j=1}^{M}W^{m}_{ij}\eta^{i}r^{j}$
$\displaystyle<\eta^{m}$ (18)

for $1\leq m\leq M$, then $T$ has a unique fixed point in
$\operatorname{Box}(\bar{x},r)$.

###### Proof.

Writing

$\displaystyle
T(x)-\bar{x}=T(\bar{x})-\bar{x}+DT(\bar{x})(x-\bar{x})+\int_{0}^{1}\left[DT\left(\bar{x}+s(x-\bar{x})\right)-DT\left(\bar{x}\right)\right](x-\bar{x})\,\mbox{d}s,$

the inequalities (17) imply that $T$ maps $\operatorname{Box}(\bar{x},r)$ into
itself. Then, writing

$\displaystyle
DT(x)=DT(\bar{x})+\left[DT\left(x\right)-DT\left(\bar{x}\right)\right],$

and considering the weighted maximum norm

$\left\|x\right\|_{X}=\max_{1\leq m\leq
M}\frac{\left\|\pi^{m}x\right\|_{X^{m}}}{\eta^{m}},$ (19)

we get, for all $x$ in $\operatorname{Box}(\bar{x},r_{*})$,

$\displaystyle\left\|DT(x)\right\|_{B(X,X)}\leq\max_{1\leq m\leq
M}\frac{1}{\eta^{m}}\sum_{i=1}^{M}\biggl{(}Z^{m}_{i}+\sum_{j=1}^{M}W^{m}_{ij}r^{j}\biggr{)}\eta^{i},$

and (18) yields that $T$ is a contraction on
$\operatorname{Box}(\bar{x},r)\subset X$ (which is a closed set for
$\left\|\cdot\right\|_{X}$). ∎

######  Remark 2.13.

If $M=1$, we recover a more _classical_ form of this statement, since $\eta$
plays no role and can therefore be taken equal to 1. In that case,
$\left\|\cdot\right\|_{X}=\left\|\cdot\right\|_{X^{1}}$, and the set
$\operatorname{Box}(\bar{x},r)$ is simply a ball for this norm.

If $M>1$, the weights $\eta$ provide some flexibility which makes it easier to
prove that $T$ is a contraction. Instead of working with the set
$\operatorname{Box}(\bar{x},r)$, one could also start with a proper ball for
the $\left\|\cdot\right\|_{X}$ norm defined in (19), centered at $\bar{x}$ and
of radius say $\varrho$, and try to prove that this ball is mapped into itself
by $T$. One would then end up with a system of inequalities in terms of
$(\varrho,\eta)$ equivalent to (17)-(18), but fully coupled. The main
advantage of working with $\operatorname{Box}(\bar{x},r)$ is that these
inequalities are immediately decoupled: in practice we can first try to find
the smallest $r$ satisfying (17), and then look for $\eta$ satisfying (18).

In the sequel, we use this theorem with $M=1$ in the case where the time
interval $[0,2\tau]$ is not split, and with $M>1$ when using domain
decomposition. In Section 3, we introduce in full detail the setup in which we
apply this theorem to validate solutions of (1). The remainder of the paper is
then devoted to the derivation of (computable) bounds $Y^{m}$, $Z^{m}_{i}$ and
$W^{m}_{ij}$ satisfying (14), (15) and (16) for this problem.

##  3 Functional analytic setup

In this section we construct a fixed point map whose fixed points correspond
to solutions of (1). Here we consider a _single_ time domain. The
generalization to domain decomposition is presented in Section 6. First, we
recast (1) into an infinite set of coupled ODEs on the sequence space
$\ell^{1}_{\nu}$ by using a Fourier transformation in the spatial variable. We
then reformulate the system of ODEs as an equivalent zero finding problem on
$\mathcal{X}_{\nu}$ by using the variation of constants formula. Next, we
perform a finite dimensional reduction by approximating a finite number of
(time varying) Fourier modes with the aid of Chebyshev interpolation. This
reduction is used to set up a Newton-like map $T$ based at an approximate
(numerically computed) zero, which is the one on which Theorem 2.12 will be
applied.

###  3.1 An equivalent zero-finding problem

In this section we set up a zero finding problem for (1) by using the
variation of constants formula. First, rewrite (1) as

$\displaystyle\dfrac{\partial u}{\partial t}-\mathcal{L}u$
$\displaystyle=(-1)^{R+1}\dfrac{\partial^{2R}u}{\partial
x^{2R}}+\displaystyle\sum_{j=0}^{2R-1}\dfrac{\partial^{j}g^{(j)}(u)}{\partial
x^{j}}-\mathcal{L}u,$ (20)

where $\mathcal{L}$ is the linear operator which is going to be the generator
of the semigroup used in the Duhamel formula. The choice of $\mathcal{L}$ is
actually crucial in our approach: we want $\mathcal{L}$ to make the
linearization of the right-hand side of (20) around the approximation solution
as a small as possible, but we also want to keep $\mathcal{L}$ relatively
simple, so that we can easily get explicit estimates on the associated
semigroup. The precise definition of $\mathcal{L}$ is given in Section 3.4.

Next, let $\left(u_{n}\left(t\right)\right)_{n\in\mathbb{Z}}$ and
$\left(f_{n}\right)_{n\in\mathbb{Z}}$ denote the Fourier coefficients of
$u\left(t,\cdot\right)$ and $f$, respectively, that is

$\displaystyle u(t,x)=\sum_{n=-\infty}^{\infty}u_{n}(t)e^{inx},\quad
f(x)=\sum_{n=-\infty}^{\infty}f_{n}e^{inx},$

for all $t\in[0,2\tau]$ and $x\in\left[0,2\pi\right]$. Slightly abusing the
notations, we also use $(g^{(j)}_{n}(u)\left(t\right))_{n\in\mathbb{Z}}$ to
denote the Fourier coefficients of $g^{(j)}(u)\left(t,\cdot\right)$, i.e.

$\displaystyle
g^{(j)}(u)(t,x)=\sum_{n=-\infty}^{\infty}g^{(j)}_{n}(u)(t)e^{inx},$

for all $j=0,\ldots,2R-1$.

Since the approximation of the solutions will be made using Chebyshev
interpolation in time, we rescale the time domain $[0,2\tau]$ to $[-1,1]$, on
which the theory of Chebyshev approximations is developed (see Section 2.2).
Substitution of the above Fourier expansions into (20) yields an infinite
dimensional system of ODEs on $[-1,1]$ for the Fourier coefficients
$\left(u_{n}\right)_{n\in\mathbb{Z}}$:

$\displaystyle\begin{cases}\dfrac{du}{dt}(t)-\tau\mathcal{L}u(t)=\tau\biggl{(}(-1)^{R+1}\mathcal{D}^{2R}u(t)+\displaystyle\sum_{j=0}^{2R-1}\mathcal{D}^{j}g^{(j)}(u(t))-\mathcal{L}u(t)\biggr{)},&t\in[-1,1],\\\\[8.61108pt]
u\left(-1\right)=f,\end{cases}$ (21)

where $\mathcal{D}$ is the Fourier transform of $\dfrac{\partial}{\partial x}$
:

$\left(\mathcal{D}u\right)_{n}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,inu_{n},$

and where we abuse notation and denote the Fourier transform of $\mathcal{L}$
still by $\mathcal{L}$.

Finally, integration of (21) with the aid of variation of constants yields the
following map.

######  Definition 3.1.

The zero finding map $F:\mathcal{X}_{\nu}\rightarrow\mathcal{X}_{\nu}$ for (1)
is defined by

$\displaystyle F\left(u\right)\left(t\right)$
$\displaystyle\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,e^{\tau(t+1)\mathcal{L}}f+\tau\int_{-1}^{t}e^{\tau(t-s)\mathcal{L}}\gamma(u(s))\mbox{d}s-u(t),$

where, for any $v\in\ell^{1}_{\nu}$,

$\gamma(v)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,(-1)^{R+1}\mathcal{D}^{2R}v+\displaystyle\sum_{j=0}^{2R-1}\mathcal{D}^{j}g^{(j)}(v)-\mathcal{L}v.$
(22)

######  Remark 3.2.

While we have not defined $\mathcal{L}$ precisely yet (this will be done in
Section 3.4), we are assuming that $\mathcal{L}$ generates a $C^{0}$ semi-
group on $\ell^{1}_{\nu}$, so that $F$ is well defined.

###  3.2 Finite dimensional reduction

In this section we introduce a finite dimensional reduction of $F$. To
accomplish this we will need to truncate the phase space $\mathcal{X}_{\nu}$
and to discretize time.

######  Definition 3.3 (Truncation of phase space).

Let $N\in\mathbb{N}$ be a truncation parameter. The projection
$\Pi_{N}:\mathcal{X}_{\nu}\rightarrow\mathcal{X}_{\nu}$ is defined by

$\displaystyle\left(\Pi_{N}(u)\right)_{n}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\begin{cases}u_{n},&0\leq|n|\leq
N,\\\ 0,&|n|>N.\end{cases}$

Furthermore, we set
$\Pi_{\infty}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,I-\Pi_{N}$,
where $I$ is the identity on $\mathcal{X}_{\nu}$. We also introduce the
subspaces $\mathcal{X}^{N}_{\nu}=\Pi_{N}\left(\mathcal{X}_{\nu}\right)$ and
$\mathcal{X}^{\infty}_{\nu}=\Pi_{\infty}\left(\mathcal{X}_{\nu}\right)$, for
which we have

$\mathcal{X}_{\nu}=\mathcal{X}^{N}_{\nu}\oplus\mathcal{X}^{\infty}_{\nu}.$

######  Remark 3.4.

Henceforth we shall identify $\Pi_{N}\left(u\right)$ with the vector of
functions

$\displaystyle\begin{bmatrix}u_{-N}\\\ \vdots\\\ u_{N}\end{bmatrix}\in
C\left(\left[-1,1\right],\mathbb{C}^{2N+1}\right).$

######  Definition 3.5 (Time discretization).

Let $K\in\mathbb{N}$. The Fourier-Chebyshev projection
$\Pi_{KN}:\mathcal{X}_{\nu}\rightarrow
C\left(\left[-1,1\right],\mathbb{C}^{2N+1}\right)$ is defined by

$\displaystyle\Pi_{KN}(u)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\begin{bmatrix}P_{K}(u_{-N})\\\
\vdots\\\ P_{K}(u_{N})\end{bmatrix},$

where $P_{K}:C\left([-1,1],\mathbb{C}\right)\rightarrow
C\left([-1,1],\mathbb{C}\right)$ is the operator which sends a continuous
function to its Chebyshev interpolant (see Section 2.2). We sometimes abuse
notation by applying $P_{K}$ to a vector of functions, meaning we apply
$P_{K}$ to each component, e.g. $\Pi_{KN}(u)=P_{K}\Pi_{N}(u)$. Furthermore, we
set $\Pi_{\infty
N}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\Pi_{N}-\Pi_{KN}$.
We note that $\Pi_{\infty N}(u)$ vanishes at $t=\pm 1$.

The truncation of phase space and discretization of time give rise to the
decomposition

$\displaystyle\mathcal{X}_{\nu}=\mathcal{X}^{KN}_{\nu}\oplus\mathcal{X}^{\infty
N}_{\nu}\oplus\mathcal{X}^{\infty}_{\nu},$ (23)

where

$\displaystyle\mathcal{X}^{KN}_{\nu}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\Pi_{KN}\left(\mathcal{X}_{\nu}\right),\quad\mathcal{X}^{\infty
N}_{\nu}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\Pi_{\infty
N}\left(\mathcal{X}_{\nu}\right),\quad\mathcal{X}^{\infty}_{\nu}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\Pi_{\infty}\left(\mathcal{X}_{\nu}\right).$

We equip $\mathcal{X}^{KN}_{\nu}$ with the norm

$\displaystyle\left\|u\right\|_{\mathcal{X}^{KN}_{\nu}}$
$\displaystyle\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\Biggl{\|}\biggl{[}\left|u_{n0}\right|+2\sum_{k=1}^{K}\left|u_{nk}\right|\biggr{]}_{n=-N}^{N}\Biggr{\|}_{\ell^{1}_{\nu}}=\sum_{|n|\leq
N}\biggl{(}\left|u_{n0}\right|+2\sum_{k=1}^{K}\left|u_{nk}\right|\biggr{)}\nu^{|n|}.$
(24)

There are several reasons for choosing this particular norm over the more
obvious supremum norm. First of all, the norm in (24) is numerically easy to
compute, whereas the computation of a supremum norm is relatively complicated.
Furthermore, with this norm it is easy to compute operator norms which amounts
to computing weighted $l^{1}$-norms of finite dimensional matrices. This is to
be contrasted with the use of a supremum norm, where the analysis of linear
operators is much more complicated. Lastly, it follows from (12) that the norm
in (24) is stronger than the supremum norm, i.e.,

$\displaystyle\sup_{t\in[-1,1]}\left\|u(t)\right\|_{\ell^{1}_{\nu}}\leq\left\|\|u\|_{C^{0}}\right\|_{\ell^{1}_{\nu}}\leq\left\|u\right\|_{\mathcal{X}^{KN}_{\nu}},\quad\text{for
all }u\in\mathcal{X}^{KN}_{\nu},$

thereby allowing one to relate the two norms in a straightforward manner. Here
and throughout the paper, when we apply the $C^{0}$-norm to an element of
$\mathcal{X}_{\nu}$, it should be understood as applying the $C^{0}$-norm to
each Fourier mode.

The subspaces $\mathcal{X}^{\infty N}_{\nu}$ and $\mathcal{X}^{\infty}_{\nu}$
are both endowed with the
$\left\|\cdot\right\|_{\ell^{1}_{\nu}\left(C^{0}\right)}=\left\|\left\|\cdot\right\|_{C^{0}}\right\|_{\ell^{1}_{\nu}}$
norm, i.e.,

$\displaystyle\left\|u\right\|_{\ell^{1}_{\nu}\left(C^{0}\right)}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\sum_{n\in\mathbb{Z}}\left\|u_{n}\right\|_{C^{0}}\nu^{|n|}.$

Furthermore, the full space $\mathcal{X}_{\nu}$ is equipped with the norm

$\displaystyle\left\|u\right\|_{\mathcal{X}_{\nu}}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\left\|\Pi_{KN}u\right\|_{\mathcal{X}^{KN}_{\nu}}+\epsilon_{\infty
N}^{-1}\left\|\Pi_{\infty N}u\right\|_{\mathcal{X}^{\infty
N}_{\nu}}+\epsilon_{\infty}^{-1}\left\|\Pi_{\infty}u\right\|_{\mathcal{X}^{\infty}_{\nu}}$
(25)

where $\epsilon_{\infty N},\epsilon_{\infty}>0$ are weights to be chosen
later. The purpose of these weights is to provide some control over the
truncation errors in phase space and the interpolation errors in time. Over
the whole space, the norm $\left\|\cdot\right\|_{\mathcal{X}_{\nu}}$ still
controls the $C^{0}$ norm:

######  Lemma 3.6.

Let $\vartheta_{\epsilon}=\max(1,\epsilon_{\infty N},\epsilon_{\infty})$.
Then,

$\left\|u\right\|_{\ell^{1}_{\nu}\left(C^{0}\right)}\leq\vartheta_{\epsilon}\left\|u\right\|_{\mathcal{X}_{\nu}},\quad\text{for
all }u\in\mathcal{X}_{\nu}.$

Finally, we define a finite dimensional reduction of $F$.

######  Definition 3.7 (Finite dimensional reduction of $F$).

The finite dimensional reduction
$F_{KN}:\mathcal{X}^{KN}_{\nu}\rightarrow\mathcal{X}^{KN}_{\nu}$ of $F$ is
defined by
$F_{KN}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\Pi_{KN}\circ
F|_{\mathcal{X}^{KN}_{\nu}}$.

###  3.3 A posteriori analysis

In this section we construct a Newton-like map for $F$ by using the finite
dimensional reduction $F_{KN}$. To this end, suppose we have computed the
following:

  * $(i)$

An approximate zero $\bar{u}$ in
$\mathcal{X}^{KN}_{\nu}\simeq\mathbb{C}^{(2N+1)\left(K+1\right)}$ of $F_{KN}$.

  * $(ii)$

The derivative $DF_{KN}\left(\bar{u}\right)$.

  * $(iii)$

An approximate injective inverse $A_{KN}$ of $DF_{KN}\left(\bar{u}\right)$.

######  Remark 3.8.

One can check that $A_{KN}$ is injective by verifying that the bound in (34)
is strictly smaller than $1$. In fact, our computer-assisted proof can only be
successful when this inequality is satisfied, cf. (18). Therefore, if the
computer-assisted proof is successful, we may a posteriori conclude that
$A_{KN}$ is injective without any further ado.

We will use the finite dimensional data to construct an approximate inverse of
$DF\left(\bar{u}\right)$. We anticipate that

$\displaystyle\Pi_{\infty N}DF\left(\bar{u}\right)\approx-\Pi_{\infty
N}\qquad\text{and}\qquad\Pi_{\infty}DF\left(\bar{u}\right)\approx-\Pi_{\infty}$

in a small neighborhood of $\bar{u}$ provided $K$ and $N$ are sufficiently
large. These observations motivate the following definitions:

######  Definition 3.9 (Approximation of $DF\left(\bar{u}\right)$).

The approximate derivative
$\widehat{DF}:\mathcal{X}_{\nu}\rightarrow\mathcal{X}_{\nu}$ of $F$ at
$\bar{u}$ is defined by

$\displaystyle\widehat{DF}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,DF_{KN}\left(\bar{u}\right)\oplus\left(-\Pi_{\infty
N}\right)\oplus\left(-\Pi_{\infty}\right).$

######  Definition 3.10 (Approximate inverse of $DF\left(\bar{u}\right)$).

The approximate inverse $A:\mathcal{X}_{\nu}\rightarrow\mathcal{X}_{\nu}$ of
$DF\left(\bar{u}\right)$ is defined by

$\displaystyle
A\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,A_{KN}\oplus\left(-\Pi_{\infty
N}\right)\oplus\left(-\Pi_{\infty}\right).$

Next, we define a Newton-like operator $T$ for $F$ based at $\bar{u}$:

######  Definition 3.11 (Newton-like operator for $F$).

The Newton-like operator $T:\mathcal{X}_{\nu}\rightarrow\mathcal{X}_{\nu}$ for
$F$ based at $\bar{u}$ is defined by

$\displaystyle
T\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,I-AF.$

The idea is to seek fixed points of $T$ in a small neighborhood of $\bar{u}$.
To be more precise, let $B_{r,\epsilon}(0)$ denote the _closed_ ball of radius
$r>0$ centered at $0$ in $\mathcal{X}_{\nu}$, i.e.,

$\displaystyle
B_{r,\epsilon}(0)=\left\\{h\in\mathcal{X}_{\nu}:\left\|\Pi_{KN}\left(h\right)\right\|_{\mathcal{X}^{KN}_{\nu}}+\epsilon_{\infty
N}^{-1}\left\|\Pi_{\infty N}\left(h\right)\right\|_{\mathcal{X}^{\infty
N}_{\nu}}+\epsilon_{\infty}^{-1}\left\|\Pi_{\infty}\left(h\right)\right\|_{\mathcal{X}^{\infty}_{\nu}}\leq
r\right\\}.$ (26)

We shall prove the existence of a fixed point $u^{*}$ of $T$ in
$B_{r,\epsilon}\left(\bar{u}\right)=\bar{u}+B_{r,\epsilon}(0)$, where $r>0$ is
an unknown radius to be determined, by using Theorem 2.12, with $M=1$ and
$X=\mathcal{X}_{\nu}$.

######  Remark 3.12.

This fixed point $u^{*}$ of $T$ corresponds to a zero of $F$, since $A$ is
injective. We have thus found the solution of the initial value problem (1).
In view of Lemma 3.6, the error bound on the distance between the solution
$u^{*}$ and its approximation $\bar{u}$ is controlled by

$\max_{\begin{subarray}{c}t\in[0,2\tau],x\in[0,2\pi]\end{subarray}}|u^{*}(t,x)-\bar{u}(t,x)|\leq\|u^{*}-\bar{u}\|_{\ell^{1}_{\nu}(C^{0})}\leq\vartheta_{\epsilon}r.$

###  3.4 Construction of $\mathcal{L}$

For reasons already outlined in the introduction, and that will be made more
concrete later on, we want $\mathcal{L}$ to be a time _independent_
approximation of

$\displaystyle
h\mapsto(-1)^{(R+1)}\mathcal{D}^{2R}h+\sum_{j=0}^{2R-1}\mathcal{D}^{j}\left(\left(g^{(j)}\right)^{\prime}(\bar{u}(s))\ast
h\right),\quad h\in\ell^{1}_{\nu},$ (27)

where $\left(g^{(j)}\right)^{\prime}$ simply denotes the derivative of the
polynomial function $g^{(j)}$. Although we choose $\mathcal{L}$ to be uniform
in $s$, it does approximate the linear operator (27), hence we have that, with
$\gamma$ defined in (22),

$s\mapsto D\gamma(\bar{u}(s))\approx 0.$

To construct $\mathcal{L}$, we start by introducing vectors
$\bar{v}^{(j)}\in\Pi^{N}\left(\ell^{1}_{\nu}\right)$, $j=0,\ldots,2R-1$, which
in practice should be (constant in time) approximations of
$s\mapsto\left(g^{(j)}\right)^{\prime}(\bar{u})(s)$. Then, we define
$\tilde{\mathcal{L}}$ on
$\mathcal{X}_{\nu}=\Pi_{N}\mathcal{X}_{\nu}\oplus\Pi_{\infty}\mathcal{X}_{\nu}$
by

$\left\\{\begin{aligned}
&\tilde{\mathcal{L}}\Pi_{N}u\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,(-1)^{R+1}\mathcal{D}^{2R}\Pi_{N}u+\displaystyle\sum_{j=0}^{2R-1}\mathcal{D}^{j}\Pi_{N}\bigl{(}\bar{v}^{(j)}\ast\Pi_{N}u\bigr{)}\\\
&\left(\tilde{\mathcal{L}}\Pi_{\infty}u\right)_{n}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\biggl{(}-n^{2R}+\displaystyle\sum_{j=0}^{2R-1}(in)^{j}\bar{v}^{(j)}_{0}\biggr{)}u_{n}\qquad\text{for
all }|n|>N.\end{aligned}\right.$

Notice that $\tilde{\mathcal{L}}$ leaves $\mathcal{X}^{N}_{\nu}$ invariant,
and that it does not depend on time, therefore its restriction to
$\mathcal{X}^{N}_{\nu}$ can be represented as a finite dimensional matrix,
that we denote by $\tilde{\mathcal{L}}_{N}$.

Finally, we consider a diagonal matrix
$\Lambda_{N}=\text{diag}\left(\lambda_{-N},\ldots,\lambda_{N}\right)$ and an
invertible matrix $Q$ such that

$\tilde{\mathcal{L}}_{N}\approx Q\Lambda_{N}Q^{-1},$

and define

$\mathcal{L}_{N}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,Q\Lambda_{N}Q^{-1},$

and $\mathcal{L}$ on
$\mathcal{X}_{\nu}=\Pi_{N}\mathcal{X}_{\nu}\oplus\Pi_{\infty}\mathcal{X}_{\nu}$
by

$\left\\{\begin{aligned}
&\mathcal{L}\Pi_{N}u\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\mathcal{L}_{N}\Pi_{N}u\\\
&\left(\mathcal{L}\Pi_{\infty}u\right)_{n}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\biggl{(}-n^{2R}+\displaystyle\sum_{j=0}^{2R-1}(in)^{j}\bar{v}^{(j)}_{0}\biggr{)}u_{n}\quad\text{for
all }|n|>N.\end{aligned}\right.$

For further use, we denote, for all $|n|>N$,

$\lambda_{n}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,-n^{2R}+\displaystyle\sum_{j=0}^{2R-1}(in)^{j}\left(\bar{v}^{(j)}\right)_{0},$
(28)

and

$\mathcal{R}_{N}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\mathcal{L}_{N}-\tilde{\mathcal{L}}_{N}.$

Notice that with this notation we can write

$\Pi_{N}D\gamma(\bar{u})u=\sum_{j=0}^{2R-1}\mathcal{D}^{j}\Pi_{N}\left[\left(\left(g^{(j)}\right)^{\prime}(\bar{u})-\bar{v}^{(j)}\right)\ast\Pi_{N}u+\left(g^{(j)}\right)^{\prime}(\bar{u})\ast\Pi_{\infty}u\right]-\mathcal{R}_{N}\Pi_{N}u.$
(29)

Now that $A$ and $\mathcal{L}$ have been explicitly constructed, we are ready
to derive the estimates needed to prove that $T$ has a locally unique fixed
point around $\bar{u}$.

##  4 General estimates

We compile here a list of lemmas which will be used extensively in the
computation of the $Y$, $Z$ and $W$ bounds needed for Theorem 2.12. These are
essentially technical estimates, the reader more interested in an overview of
the proof is encouraged to jump directly to Section 5, and to only refer to
this section when needed.

###  4.1 An $\ell^{1}_{\nu}$ dual estimate

######  Lemma 4.1.

For a given $(2N+1)\times(2N+1)$ matrix $B$ with complex entries, define
$\Upsilon(B)\in\mathbb{C}^{2N+1}$ by

$\Upsilon_{n}(B)=\max_{|m|\leq N}\frac{|B_{n,m}|}{\nu^{|m|}},\quad|n|\leq N.$

Then, for any $v\in\Pi_{N}\ell^{1}_{\nu}\cong\mathbb{C}^{2N+1}$,

$\displaystyle\left|Bv\right|\leq\Upsilon(B)\left\|v\right\|_{\ell^{1}_{\nu}},$

where the absolute values apply component-wise.

###  4.2 Controlling convolution products

######  Lemma 4.2.

Let $u,v\in\ell^{1}_{\nu}$. Then, for all $n\in\mathbb{Z}$

$\left|(u\ast v)_{n}\right|\leq\Psi_{n}(u)\left\|v\right\|_{\ell^{1}_{\nu}},$

where

$\Psi_{n}(u)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\sup_{m\in\mathbb{Z}}\frac{|u_{m}|}{\nu^{|n-m|}}.$

######  Lemma 4.3.

Let $a,b,h\in\mathcal{X}_{\nu}$. Define, for all $n$ in $\mathbb{Z}$,

$\Phi^{\epsilon}_{n}(a,b)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\max\biggl{\\{}\epsilon_{\infty
N}\max_{|n-m|\leq N}\frac{\left\|a_{m}\right\|_{C^{0}}}{\nu^{|n-m|}},\
\epsilon_{\infty}\sup_{|n-m|>N}\frac{\left\|b_{m}\right\|_{C^{0}}}{\nu^{|n-m|}}\biggr{\\}},$
$\check{\Phi}^{\epsilon}_{n}(a,b)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\max\biggl{\\{}\max(1,\epsilon_{\infty
N})\max_{|n-m|\leq N}\frac{\left\|a_{m}\right\|_{C^{0}}}{\nu^{|n-m|}},\
\epsilon_{\infty}\sup_{|n-m|>N}\frac{\left\|b_{m}\right\|_{C^{0}}}{\nu^{|n-m|}}\biggr{\\}},$

and

$\displaystyle\tilde{\Phi}^{\epsilon}(a,b)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\max\biggl{\\{}$
$\displaystyle\max(1,\epsilon_{\infty N})\max_{|m|\leq N}\sum_{|n|\leq
N}\left\|a_{n-m}\right\|_{C^{0}}\nu^{|n|-|m|},$ $\displaystyle\hskip
85.35826pt\epsilon_{\infty}\sup_{|m|>N}\sum_{|n|\leq
N}\left\|b_{n-m}\right\|_{C^{0}}\nu^{|n|-|m|}\biggr{\\}}.$

Then, for all $n$ in $\mathbb{Z}$,

$\displaystyle\left\|\left(a\ast\Pi_{\infty
N}h+b\ast\Pi_{\infty}h\right)_{n}\right\|_{C^{0}}\leq\Phi^{\epsilon}_{n}(a,b)\left\|h\right\|_{\mathcal{X}_{\nu}},$
$\displaystyle\left\|\left(a\ast\Pi_{N}h+b\ast\Pi_{\infty}h\right)_{n}\right\|_{C^{0}}\leq\check{\Phi}^{\epsilon}_{n}(a,b)\left\|h\right\|_{\mathcal{X}_{\nu}},$

and

$\displaystyle\left\|\Pi_{N}\left[a\ast\Pi_{N}h+b\ast\Pi_{\infty}h\right]\right\|_{\ell^{1}_{\nu}(C^{0})}\leq\tilde{\Phi}^{\epsilon}(a,b)\left\|h\right\|_{\mathcal{X}_{\nu}}.$

###### Proof.

Let $h^{\infty N}=\Pi_{\infty N}h$ and $h^{\infty}=\Pi_{\infty}h$. For all
$s\in[-1,1]$, by using (25),

$\displaystyle\left|\left(a\ast h^{\infty N}+b\ast
h^{\infty}\right)_{n}(s)\right|$
$\displaystyle\leq\sum_{m\in\mathbb{Z}}\left(\left|a_{m}(s)h^{\infty
N}_{n-m}(s)\right|+\left|b_{m}(s)h^{\infty}_{n-m}(s)\right|\right)$
$\displaystyle\leq\sum_{|n-m|\leq
N}\frac{\left\|a_{m}\right\|_{C^{0}}}{\nu^{|n-m|}}\left\|h^{\infty
N}_{n-m}\right\|_{C^{0}}\nu^{|n-m|}$
$\displaystyle\quad+\sum_{|n-m|>N}\frac{\left\|b_{m}\right\|_{C^{0}}}{\nu^{|n-m|}}\left\|h^{\infty}_{n-m}\right\|_{C^{0}}\nu^{|n-m|}$
$\displaystyle\leq\max_{|n-m|\leq
N}\frac{\left\|a_{m}\right\|_{C^{0}}}{\nu^{|n-m|}}\left\|\Pi_{\infty
N}h\right\|_{\mathcal{X}^{\infty
N}_{\nu}}+\sup_{|n-m|>N}\frac{\left\|b_{m}\right\|_{C^{0}}}{\nu^{|n-m|}}\left\|\Pi_{\infty}h\right\|_{\mathcal{X}^{\infty}_{\nu}}$
$\displaystyle\leq\Phi^{\epsilon}_{n}(a,b)\left\|h\right\|_{\mathcal{X}_{\nu}}.$

The second estimate is obtained in the same way, using additionally that

$\displaystyle\sum_{|m|\leq N}\left\|h_{m}\right\|_{C^{0}}\nu^{|m|}$
$\displaystyle\leq\left\|\Pi_{KN}h\right\|_{\mathcal{X}^{KN}_{\nu}}+\left\|\Pi_{\infty
N}h\right\|_{\mathcal{X}^{\infty N}_{\nu}}$
$\displaystyle\leq\max(1,\epsilon_{\infty
N})\left(\left\|\Pi_{KN}h\right\|_{\mathcal{X}^{KN}_{\nu}}+\epsilon_{\infty
N}^{-1}\left\|\Pi_{\infty N}h\right\|_{\mathcal{X}^{\infty N}_{\nu}}\right).$

For the final estimate, introducing $h^{N}=\Pi_{N}h$, we proceed in a similar
fashion:

$\displaystyle\left\|\Pi_{N}\left[a\ast h^{N}+b\ast
h^{\infty}\right]\right\|_{\ell^{1}_{\nu}(C^{0})}$
$\displaystyle\leq\sum_{|n|\leq
N}\sum_{m\in\mathbb{Z}}\left(\left\|a_{n-m}\right\|_{C^{0}}\left\|h^{N}_{m}\right\|_{C^{0}}+\left\|b_{n-m}\right\|_{C^{0}}\left\|h^{\infty}_{m}\right\|_{C^{0}}\right)\nu^{|n|}$
$\displaystyle\leq\sum_{|m|\leq
N}\left\|h_{m}\right\|_{C^{0}}\nu^{|m|}\sum_{|n|\leq
N}\left\|a_{n-m}\right\|_{C^{0}}\nu^{|n|-|m|}$
$\displaystyle\qquad+\sum_{|m|>N}\left\|h_{m}\right\|_{C^{0}}\nu^{|m|}\sum_{|n|\leq
N}\left\|b_{n-m}\right\|_{C^{0}}\nu^{|n|-|m|}$
$\displaystyle\leq\left(\left\|\Pi_{KN}h\right\|_{\mathcal{X}^{KN}_{\nu}}+\epsilon_{\infty
N}^{-1}\left\|\Pi_{\infty N}h\right\|_{\mathcal{X}^{\infty N}_{\nu}}\right)$
$\displaystyle\qquad\times\max(1,\epsilon_{\infty N})\max_{|m|\leq
N}\sum_{|n|\leq N}\left\|a_{n-m}\right\|_{C^{0}}\nu^{|n|-|m|}$
$\displaystyle\qquad\qquad+\epsilon_{\infty}^{-1}\left\|\Pi_{\infty}h\right\|_{\mathcal{X}^{\infty}_{\nu}}\,\epsilon_{\infty}\sup_{|m|>N}\sum_{|n|\leq
N}\left\|b_{n-m}\right\|_{C^{0}}\nu^{|n|-|m|}.\qed$

######  Remark 4.4.

As soon as $b$ only has a finite number of non-zero Fourier modes, the suprema
in $\Phi^{\epsilon}_{n}(a,b)$, $\check{\Phi}^{\epsilon}_{n}(a,b)$ and
$\tilde{\Phi}^{\epsilon}(a,b)$ become maxima that can be computed.

######  Lemma 4.5.

Let $r\geq 0$ and $p$ be a polynomial, written in the canonical basis

$p(x)=\sum_{k=0}^{N_{p}}p_{k}x^{k}.$

If we denote by $|p|$ the polynomial

$|p|(x)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\sum_{k=0}^{N_{p}}|p_{k}|x^{k},$

then for all $\|v\|_{\mathcal{X}_{\nu}}\leq 1$ and
$\|h\|_{\mathcal{X}_{\nu}}\leq 1$ we have

$\left\|p\left(\bar{u}+rv\right)\ast v\ast
h\right\|_{\ell^{1}_{\nu}(C^{0})}\leq\vartheta_{\epsilon}^{2}\,|p|\left(\left\|\bar{u}\right\|_{\ell^{1}_{\nu}(C^{0})}+\vartheta_{\epsilon}r\right).$

###### Proof.

We use the Banach algebra properties of $\ell^{1}_{\nu}$ and
$\ell^{1}_{\nu}(C^{0})$ to get

$\displaystyle\left\|p\left(\bar{u}+rv\right)\ast v\ast
h\right\|_{\ell^{1}_{\nu}(C^{0})}$
$\displaystyle\leq\left\|p\left(\bar{u}+rv\right)\right\|_{\ell^{1}_{\nu}(C^{0})}\|v\|_{\ell^{1}_{\nu}(C^{0})}\|h\|_{\ell^{1}_{\nu}(C^{0})}$
$\displaystyle\leq\vartheta_{\epsilon}^{2}\sum_{k=0}^{N_{p}}|p_{k}|\left\|\bigl{(}\left\|\bar{u}\right\|_{C^{0}}+r\left\|v\right\|_{C^{0}}\bigr{)}^{k}\right\|_{\ell^{1}_{\nu}}$
$\displaystyle\leq\vartheta_{\epsilon}^{2}\sum_{k=0}^{N_{p}}|p_{k}|\left\|\bigl{(}\left\|\bar{u}\right\|_{C^{0}}+r\left\|v\right\|_{C^{0}}\bigr{)}\right\|_{\ell^{1}_{\nu}}^{k}$
$\displaystyle\leq\vartheta_{\epsilon}^{2}\sum_{k=0}^{N_{p}}|p_{k}|\bigl{(}\left\|\bar{u}\right\|_{\ell^{1}_{\nu}(C^{0})}+\vartheta_{\epsilon}r\bigr{)}^{k}.\qed$

###  4.3 Interpolation errors

######  Lemma 4.6.

Let $\lambda\in\mathbb{C}$, $\tau>0$ and $\rho>1$,

$\displaystyle\left\|\left(I-P_{K}\right)\left(t\mapsto
e^{\tau(t+1)\lambda}\right)\right\|_{C^{0}}$
$\displaystyle\leq\frac{4\rho^{-K}}{\rho-1}\exp\left(\tau\Bigl{(}\Re(\lambda)+\frac{1}{2}\sqrt{\Re(\lambda)^{2}(\rho+\rho^{-1})^{2}+\Im(\lambda)^{2}(\rho-\rho^{-1})^{2}}\Bigr{)}\right),$

where $\Re(\lambda)$ and $\Im(\lambda)$ denote the real part and the imaginary
part of $\lambda$, respectively.

###### Proof.

From Theorem 2.7 we get that

$\displaystyle\left\|\left(I-P_{K}\right)\left(t\mapsto
e^{\tau(t+1)\lambda}\right)\right\|_{C^{0}}$
$\displaystyle\leq\frac{4\rho^{-K}}{\rho-1}\sup_{z\in\mathcal{E}_{\rho}}e^{\tau\Re\left(\lambda(z+1)\right)}.$

Any $z\in\mathcal{E}_{\rho}$ can be written
$z=\frac{1}{2}\left(re^{i\theta}+r^{-1}e^{-i\theta}\right)$, for some
$(r,\theta)\in[1,\rho]\times[0,2\pi]$, and we then have

$\displaystyle\Re\left(\lambda\left(z+1\right)\right)$
$\displaystyle=\Re(\lambda)\left(1+\frac{1}{2}(r+r^{-1})\cos\theta\right)-\Im(\lambda)\frac{1}{2}(r-r^{-1})\sin\theta$
$\displaystyle=\Re(\lambda)+\frac{1}{2}\sqrt{\Re(\lambda)^{2}(r+r^{-1})^{2}+\Im(\lambda)^{2}(r-r^{-1})^{2}}\cos(\theta-\theta_{r,\lambda}),$

for some $\theta_{r,\lambda}\in[0,2\pi]$, hence

$\sup_{z\in\mathcal{E}_{\rho}}\Re\left(\lambda(z+1)\right)=\Re(\lambda)+\frac{1}{2}\sqrt{\Re(\lambda)^{2}(\rho+\rho^{-1})^{2}+\Im(\lambda)^{2}(\rho-\rho^{-1})^{2}}.\hfill\qed$

######  Remark 4.7.

For a given $\lambda$, the quality of the above estimate depends on the choice
of $\rho$, therefore in practice we approximately optimize over $\rho$.

######  Lemma 4.8.

Let $\varphi_{N}\in C^{0}\left([-1,1],\mathbb{C}^{2N+1}\right)$,
$j\in\mathbb{N}$, $\tau>0$, $\sigma_{K,0}>0$ satisfying Definition 2.5, and
consider the $(2N+1)\times(2N+1)$ diagonal matrix

$D_{N}(t)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\Re(\Lambda_{N})^{-1}\left(e^{\tau(1+t)\Re(\Lambda_{N})}-I_{2N+1}\right)=\tau\int_{-1}^{t}\left|e^{\tau(t-s)\Lambda_{N}}\right|\mbox{d}s,$
(30)

where $\Re$ and $|\cdot|$ applied to matrices must be understood component-
wise. Then

$\displaystyle\left\|(I-P_{K})\left(t\mapsto\tau\int_{-1}^{t}e^{\tau(t-s)\mathcal{L}_{N}}\mathcal{D}^{j}\varphi_{N}(s)\mbox{d}s\right)\right\|_{\mathcal{X}^{\infty
N}_{\nu}}$
$\displaystyle\qquad\leq\tau\sigma_{K,0}\left(\left\||\mathcal{D}^{j}|+|Q||\Lambda_{N}|D_{N}(1)|Q^{-1}||\mathcal{D}^{j}|\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}\right)\left\|\varphi_{N}\right\|_{\ell^{1}_{\nu}\left(C^{0}\right)}.$

###### Proof.

Using Definition 2.5, we get

$\displaystyle\left\|(I-P_{K})\left(t\mapsto\tau\int_{-1}^{t}e^{\tau(t-s)\mathcal{L}_{N}}\mathcal{D}^{j}\varphi_{N}(s)\mbox{d}s\right)\right\|_{\mathcal{X}^{\infty
N}_{\nu}}$
$\displaystyle\qquad=\left\|\left\|(I-P_{K})\left(t\mapsto\tau\int_{-1}^{t}e^{\tau(t-s)\mathcal{L}_{N}}\mathcal{D}^{j}\varphi_{N}(s)\mbox{d}s\right)\right\|_{C^{0}}\right\|_{\ell^{1}_{\nu}}$
$\displaystyle\qquad\leq\tau\sigma_{K,0}\left\|\left\|\frac{d}{dt}\left(t\mapsto\int_{-1}^{t}e^{\tau(t-s)\mathcal{L}_{N}}\mathcal{D}^{j}\varphi_{N}(s)\mbox{d}s\right)\right\|_{C^{0}}\right\|_{\ell^{1}_{\nu}}$
$\displaystyle\qquad=\tau\sigma_{K,0}\left\|\left\|\mathcal{D}^{j}\varphi_{N}(t)+\tau\mathcal{L}_{N}\int_{-1}^{t}e^{\tau(t-s)\mathcal{L}_{N}}\mathcal{D}^{j}\varphi_{N}(s)\mbox{d}s\right\|_{C^{0}}\right\|_{\ell^{1}_{\nu}}$
$\displaystyle\qquad=\tau\sigma_{K,0}\left\|\left\|\mathcal{D}^{j}\varphi_{N}(t)+\tau
Q\Lambda_{N}\int_{-1}^{t}e^{\tau(t-s)\Lambda_{N}}Q^{-1}\mathcal{D}^{j}\varphi_{N}(s)\mbox{d}s\right\|_{C^{0}}\right\|_{\ell^{1}_{\nu}}$
$\displaystyle\qquad\leq\tau\sigma_{K,0}\left\||\mathcal{D}^{j}|\left\|\varphi_{N}\right\|_{C^{0}}+|Q||\Lambda_{N}|D_{N}(1)|Q^{-1}||\mathcal{D}^{j}|\left\|\varphi_{N}\right\|_{C^{0}}\right\|_{\ell^{1}_{\nu}}\qed$

######  Lemma 4.9.

Let $\varphi:[-1,1]\to\mathbb{C}$ be an infinitely differentiable function,
$\lambda\in\mathbb{C}$, $\tau>0$, $q\in\mathbb{N}$ and

$\tilde{C}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\sigma_{K,q}\left(\left|\tau\lambda\right|^{q+1}\frac{e^{2\tau\Re(\lambda)}-1}{\Re(\lambda)}\left\|\varphi\right\|_{C^{0}}+\tau\sum_{i=0}^{q}\left|\tau\lambda\right|^{i}\left\|\varphi^{(q-i)}\right\|_{C^{0}}\right),$

where $\varphi^{(i)}$ is the $i$-th derivative of $\varphi$ and $\sigma_{K,q}$
satisfies Definition 2.5. Then

$\displaystyle\left\|(I-P_{K})\left(t\mapsto\tau\int_{-1}^{t}e^{\tau(t-s)\lambda}\varphi(s)\mbox{d}s\right)\right\|_{C^{0}}\leq\tilde{C}.$

Assume further that $\varphi$ can be analytically extended onto
$\mathcal{E}_{\rho}$ for some $\rho>1$, and let

$\displaystyle\zeta(x)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\left\\{\begin{aligned}
&\dfrac{e^{x}-1}{x}\qquad&x\neq 0,\\\ &1\qquad&x=0,\end{aligned}\right.$

and

$\displaystyle\check{C}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\frac{2\tau(\rho+\rho^{-1}+2)}{\rho^{K}(\rho-1)}\zeta\left(\tau\left(\Re(\lambda)+\frac{1}{2}\sqrt{\Re(\lambda)^{2}(\rho+\rho^{-1})^{2}+\Im(\lambda)^{2}(\rho-\rho^{-1})^{2}}\right)\right)\sup_{z\in\mathcal{E}_{\rho}}\left|\varphi(z)\right|.$

Then

$\displaystyle\left\|(I-P_{K})\left(t\mapsto\tau\int_{-1}^{t}e^{\tau(t-s)\lambda}\varphi(s)\mbox{d}s\right)\right\|_{C^{0}}\leq\check{C}.$

###### Proof.

The first estimate is a direct consequence of the interpolation error estimate
described in Definition 2.5, and of the fact that

$\displaystyle\left\|\frac{d^{q+1}}{dt^{q+1}}\left(t\mapsto\tau\int_{-1}^{t}e^{\tau(t-s)\lambda}\varphi(s)\mbox{d}s\right)\right\|_{C^{0}}$
$\displaystyle\qquad\qquad\qquad\leq\left(\left|\tau\lambda\right|^{q+1}\frac{e^{2\tau\Re(\lambda)}-1}{\Re(\lambda)}\left\|\varphi\right\|_{C^{0}}+\tau\sum_{i=0}^{q}\left|\tau\lambda\right|^{i}\left\|\varphi^{(q-i)}\right\|_{C^{0}}\right).$

The second estimate follows from Theorem 2.7. More precisely, for any
$z\in\mathcal{E}_{\rho}$ we consider $\gamma(\tilde{s})=-1+\tilde{s}(z+1)$,
$\tilde{s}\in[0,1]$, so that $\gamma$ parametrizes the segment $[-1,z]$, and
we estimate

$\displaystyle\tau\left|\int_{\gamma}e^{\tau(z-y)\lambda}\varphi(y)\mbox{d}y\right|$
$\displaystyle\leq\tau\left(\sup_{z\in\mathcal{E}_{\rho}}|\varphi(z)|\right)\left(\frac{\rho+\rho^{-1}}{2}+1\right)\int_{0}^{1}\left|e^{\tau(z+1)(1-s)\lambda}\right|\text{d}s$
$\displaystyle=\left(\sup_{z\in\mathcal{E}_{\rho}}|\varphi(z)|\right)\left(\frac{\rho+\rho^{-1}}{2}+1\right)\left(\frac{e^{\tau\Re((z+1)\lambda)}-1}{\Re((z+1)\lambda)}\right).$

Since the map $\zeta$ is increasing on $\mathbb{R}$,

$\displaystyle\sup_{z\in\mathcal{E}_{\rho}}\tau\left|\int_{\gamma}e^{\tau(z-y)\lambda}\varphi(y)\mbox{d}y\right|\leq\left(\sup_{z\in\mathcal{E}_{\rho}}|\varphi(z)|\right)\left(\frac{\rho+\rho^{-1}}{2}+1\right)\tau\zeta\left(\tau\sup_{z\in\mathcal{E}_{\rho}}\Re((z+1)\lambda)\right),$

where, as already established in the proof of Lemma 4.6,

$\displaystyle\sup_{z\in\mathcal{E}_{\rho}}\Re((z+1)\lambda)=\Re(\lambda)+\frac{1}{2}\sqrt{\Re(\lambda)^{2}(\rho+\rho^{-1})^{2}+\Im(\lambda)^{2}(\rho-\rho^{-1})^{2}}.$

From Theorem 2.7, we get

$\displaystyle\left\|(I-P_{K})\left(t\mapsto\tau\int_{-1}^{t}e^{\tau(t-s)\lambda}\varphi(s)\mbox{d}s\right)\right\|_{C^{0}}\leq\frac{4}{\rho^{K}(\rho-1)}\sup_{z\in\mathcal{E}_{\rho}}\tau\left|\int_{\gamma}e^{\tau(z-y)\lambda}\varphi(y)\mbox{d}y\right|,$

which finishes the proof. ∎

######  Remark 4.10.

Assume we need to control an interpolation error of the form $(I-P_{K})g$,
where the function $g$ is known, and one of the above lemmas is applicable.
Rather than directly using the appropriate lemma, one may get a sharper
estimate by first splitting the error as follows

$\displaystyle\left\|(I-P_{K})g\right\|_{C^{0}}\leq\left\|(P_{K_{0}}-P_{K})g\right\|_{C^{0}}+\left\|(I-P_{K_{0}})g\right\|_{C^{0}},$

where $K_{0}$ is larger than $K$. The rational behind this splitting is that,
by taking $K_{0}$ large enough, $P_{K_{0}}g$ becomes an extremely accurate
approximation of $g$, and the first term
$\left\|(P_{K_{0}}-P_{K})g\right\|_{C^{0}}$ is essentially the interpolation
error one is interested in. The main difference is now that
$(P_{K_{0}}-P_{K})g$ is merely a polynomial, hence its $C^{0}$ norm can very
easily be estimated via (12), and we only have to use one of the above lemmas
to control the second term $\left\|(I-P_{K_{0}})g\right\|_{C^{0}}$, which is
expected to be much smaller since $K_{0}$ is larger than $K$.

###  4.4 Truncation errors in Fourier space

######  Lemma 4.11 (Truncation-error).

Define the linear operators
$K^{(j)}_{\infty}:\mathcal{X}^{\infty}_{\nu}\rightarrow\mathcal{X}^{\infty}_{\nu}$,
$j=0,\ldots,2R-1$, by

$\displaystyle\left[K^{(j)}_{\infty}\left(u\right)\left(t\right)\right]_{n}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\begin{cases}0,&0\leq|n|\leq
N,\\\\[8.61108pt]
\displaystyle\tau\int_{-1}^{t}e^{\tau\lambda_{n}\left(t-s\right)}(in)^{j}u_{n}\left(s\right)\mbox{d}s,&|n|>N.\end{cases}$
(31)

Then $K^{(j)}_{\infty}$ is bounded and

$\displaystyle\left\|K^{(j)}_{\infty}\right\|_{\mathcal{B}\left(\mathcal{X}^{\infty}_{\nu},\mathcal{X}^{\infty}_{\nu}\right)}\leq\chi^{(j)}_{N}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\sup_{n>N}n^{j}\frac{1-e^{2\tau\Re(\lambda_{n})}}{-\Re(\lambda_{n})}.$

###### Proof.

Let $u\in\mathcal{X}^{\infty}_{\nu}$. Then, using that
$\Re(\lambda_{n})=\Re(\lambda_{-n})$ for all $|n|>N$,

$\displaystyle\left\|K^{(j)}_{\infty}\left(u\right)\right\|_{\mathcal{X}^{\infty}_{\nu}}$
$\displaystyle\leq\tau\sum_{n=N+1}^{\infty}\left(\left\|u_{n}\right\|_{C^{0}}+\left\|u_{-n}\right\|_{C^{0}}\right)n^{j}\left\|t\mapsto\int_{-1}^{t}e^{\tau\Re(\lambda_{n})\left(t-s\right)}\mbox{d}s\right\|_{C^{0}}\nu^{n}$
$\displaystyle=\sum_{n=N+1}^{\infty}\left(\left\|u_{n}\right\|_{C^{0}}+\left\|u_{-n}\right\|_{C^{0}}\right)n^{j}\frac{1-e^{2\tau\Re(\lambda_{n})}}{-\Re(\lambda_{n})}\nu^{n}$
$\displaystyle\leq\chi^{(j)}_{N}\left\|u\right\|_{\ell^{1}_{\nu}(C^{0})}.\qed$

######  Remark 4.12.

Note that $\chi_{N}^{(j)}\sim N^{-2R+j}$ as $N\to\infty$, by (28). An easy way
to get an explicit value for $\chi_{N}^{(j)}$ is given in Appendix D.

##  5 Bounds

In this section, we derive computable bounds $Y$, $Z$ and $W$ satisfying
assumptions (14)-(16) of Theorem 2.12, for the operator $T$, the space
$X=\mathcal{X}_{\nu}$ and the approximate solution $\bar{u}$ introduced in
Section 3. Since we use here the case $M=1$, we drop the indices $i,j$ and the
exponent $m$ from $Y$, $Z$ and $W$.

###  5.1 Y-bound

In this subsection, we deal with the residual bound $Y$ satisfying assumption
(14) of Theorem 2.12. First, observe that

$\displaystyle T(\bar{u})-\bar{u}$ $\displaystyle=-AF(\bar{u})$
$\displaystyle=\left(-A_{KN}F_{KN}(\bar{u})\right)\oplus\Pi_{\infty
N}F(\bar{u})\oplus\Pi_{\infty}F(\bar{u}).$

We are going to estimate each of the three terms separately, i.e. we derive
bounds $Y_{KN}$, $Y_{\infty N}$ and $Y_{\infty}$ such that

$\displaystyle\left\|A_{KN}F_{KN}(\bar{u})\right\|_{\mathcal{X}^{KN}_{\nu}}\leq
Y_{KN},\quad\left\|\Pi_{\infty N}F(\bar{u})\right\|_{\mathcal{X}^{\infty
N}_{\nu}}\leq Y_{\infty
N},\quad\left\|\Pi_{\infty}F(\bar{u})\right\|_{\mathcal{X}^{\infty}_{\nu}}\leq
Y_{\infty},$

so that

$\displaystyle
Y\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,Y_{KN}+\epsilon_{\infty
N}^{-1}Y_{\infty N}+\epsilon_{\infty}^{-1}Y_{\infty},$

satisfies (14).

####  5.1.1 Finite dimensional projection

A bound for the finite dimensional part can be obtained by simply evaluating

$\displaystyle
Y_{KN}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\left\|A_{KN}F_{KN}(\bar{u})\right\|_{\mathcal{X}^{KN}_{\nu}}.$

In order to make this bound computable, we need to be able to rigorously
enclose each entry of $F_{KN}(\bar{u})$, that is, to get enclosure for the
vectors

$\displaystyle\Pi_{N}F(\bar{u})(t_{k})=e^{\tau(t_{k}+1)\mathcal{L}}\Pi_{N}f+\tau\Pi_{N}\left(\int_{-1}^{t_{k}}e^{\tau(t_{k}-s)\mathcal{L}}\gamma(\bar{u}(s))\mbox{d}s\right)-\bar{u}\left(t_{k}\right),$

for all $0\leq k\leq K$.

######  Remark 5.1.

There are at least two convenient ways of representing (each component of) an
element $\bar{u}\in\mathcal{X}^{KN}_{\nu}$, namely via its values at the
Chebyshev nodes, or via its coefficients in the Chebyshev basis. The Chebyshev
coefficients representation simplifies all the norm computations, whereas the
representation with the values at the nodes is the one we naturally end up
with when computing the projection of $\Pi_{N}F(\bar{u})$ onto
$\mathcal{X}^{KN}_{\nu}$. In practice, we therefore chose to implement
$F_{KN}$ as going from Chebyshev coefficients to values at Chebyshev nodes.

Recalling that $\mathcal{L}$ leaves both $\mathcal{X}^{N}_{\nu}$ and
$\mathcal{X}^{\infty}_{\nu}$ invariant, we have

$\displaystyle\Pi_{N}F(\bar{u})(t_{k})=e^{\tau(t_{k}+1)\mathcal{L}_{N}}\Pi_{N}f+\tau\int_{-1}^{t_{k}}e^{\tau(t_{k}-s)\mathcal{L}_{N}}\Pi_{N}\gamma(\bar{u}(s))\mbox{d}s-\bar{u}\left(t_{k}\right).$

Therefore, we need to be able to rigorously enclose integrals of the form

$\int_{-1}^{t_{k}}e^{\tau(t_{k}-s)\mathcal{L}_{N}}\varphi(s)\mbox{d}s,$

where $\varphi(s)\in\mathbb{C}^{2N-1}$ and each component of $\varphi$ is a
polynomial in $s$. Here we use the fact that

$\mathcal{L}_{N}=Q\Lambda_{N}Q^{-1},$

where $\Lambda_{N}=\text{diag}\left(\lambda_{-N},\ldots,\lambda_{N}\right)$,
to write

$\int_{-1}^{t_{k}}e^{\tau(t_{k}-s)\mathcal{L}_{N}}\varphi(s)\mbox{d}s=Q\int_{-1}^{t_{k}}e^{\tau(t_{k}-s)\Lambda_{N}}Q^{-1}\varphi(s)\mbox{d}s,$

where $Q^{-1}\varphi$ is another vector of polynomials, and thus the whole
estimate boils down to rigorously enclosing integrals of the form

$\int_{-1}^{t_{k}}e^{\tau(t_{k}-s)\lambda_{n}}\varphi(s)\mbox{d}s,$

where $\varphi$ is a scalar polynomial. We explain in detail in Appendix C how
this can be done.

######  Remark 5.2.

We expect to get $Y_{KN}$ small provided $\bar{u}$ is an accurate approximate
solution of the (truncated) problem. In practice, if the obtained value of
$Y_{KN}$ is not small enough for (17) to hold, we should try to increase $K$
(or to split the time interval, see Section 6).

####  5.1.2 Interpolation error estimates

We now turn our attention to the computation of the $Y_{\infty N}$ bound. We
need to estimate

$\displaystyle\left\|\Pi_{\infty
N}\left(T(\bar{u})-\bar{u}\right)\right\|_{\mathcal{X}^{\infty N}_{\nu}}$
$\displaystyle=\left\|\Pi_{\infty N}F(\bar{u})\right\|_{\mathcal{X}^{\infty
N}_{\nu}}$
$\displaystyle=\left\|\left\|\Pi_{N}F(\bar{u})-\Pi_{KN}F(\bar{u})\right\|_{C^{0}}\right\|_{\ell^{1}_{\nu}},$

i.e., to control the interpolation error for $\Pi_{N}F(\bar{u})$. Since
$\Pi_{N}F(\bar{u})$ is a quantity that we can compute explicitly, we apply the
idea presented in Remark 4.10, i.e. we consider $K_{0}\geq K$ and estimate

$\displaystyle\left\|\Pi_{N}F(\bar{u})-\Pi_{KN}F(\bar{u})\right\|_{\ell^{1}_{\nu}(C^{0})}$
$\displaystyle\qquad\qquad\leq\left\|\Pi_{N}F(\bar{u})-\Pi_{K_{0}N}F(\bar{u})\right\|_{\ell^{1}_{\nu}(C^{0})}+\left\|\Pi_{K_{0}N}F(\bar{u})-\Pi_{KN}F(\bar{u})\right\|_{\ell^{1}_{\nu}(C^{0})}.$

We then estimate
$\left\|\Pi_{K_{0}N}F(\bar{u})-\Pi_{KN}F(\bar{u})\right\|_{\ell^{1}_{\nu}(C^{0})}$
using (12), and
$\left\|\Pi_{N}F(\bar{u})-\Pi_{K_{0}N}F(\bar{u})\right\|_{\ell^{1}_{\nu}(C^{0})}$
by combining Lemma 4.6 and Lemma 4.9 to obtain the following statement.

######  Proposition 5.3.

Let $K_{0}\geq K$,

$\tilde{f}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,Q^{-1}\Pi_{N}f,\quad\tilde{c}^{Y}(s)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,Q^{-1}\Pi_{N}\gamma(\bar{u}(s)).$

For each $n\in\mathbb{Z}$ such that $|n|\leq N$, consider also $0\leq
q_{n}\leq K_{0}$, $\rho_{n},\check{\rho}_{n}>1$,

$\displaystyle\Omega_{n}=\frac{4\rho_{n}^{-K_{0}}}{\rho_{n}-1}\exp\left(\tau\left(\Re(\lambda_{n})+\frac{1}{2}\sqrt{\Re(\lambda_{n})^{2}(\rho_{n}+\rho_{n}^{-1})^{2}+\Im(\lambda_{n})^{2}(\rho_{n}-\rho_{n}^{-1})^{2}}\right)\right)|\tilde{f}_{n}|,$
$\tilde{C}_{n}=\sigma_{K_{0},q_{n}}\left(\left|\tau\lambda_{n}\right|^{q_{n}+1}\frac{e^{2\tau\Re(\lambda_{n})}-1}{\Re(\lambda_{n})}\left\|\tilde{c}^{Y}_{n}\right\|_{C^{0}}+\tau\sum_{i=0}^{q_{n}}\left|\tau\lambda_{n}\right|^{i}\left\|\left(\tilde{c}^{Y}_{n}\right)^{(q_{n}-i)}\right\|_{C^{0}}\right),$

and

$\displaystyle\check{C}_{n}=$
$\displaystyle\quad\frac{2\tau(\check{\rho}_{n}+\check{\rho}_{n}^{-1}+2)}{\check{\rho}_{n}^{K_{0}}(\check{\rho}_{n}-1)}\exp\left(\tau\left(\Re(\lambda_{n})+\frac{1}{2}\sqrt{\Re(\lambda_{n})^{2}(\check{\rho}_{n}+\check{\rho}_{n}^{-1})^{2}+\Im(\lambda_{n})^{2}(\check{\rho}_{n}-\check{\rho}_{n}^{-1})^{2}}\right)\right)\sup_{z\in\mathcal{E}_{\check{\rho}_{n}}}\left|\tilde{c}^{Y}_{n}\right|.$

Then

$\displaystyle\left\|\Pi_{N}F(\bar{u})-\Pi_{K_{0}N}F(\bar{u})\right\|_{\ell^{1}_{\nu}(C^{0})}\leq\left\||Q|\left(\Omega+\min\left(\tilde{C},\check{C}\right)\right)\right\|_{\ell^{1}_{\nu}},$

where $\Omega=\left(\Omega_{n}\right)_{-N\leq n\leq N}$,
$\tilde{C}=\left(\tilde{C}_{n}\right)_{-N\leq n\leq N}$,
$\check{C}=\left(\check{C}_{n}\right)_{-N\leq n\leq N}$, and
$\min\left(\tilde{C},\check{C}\right)$ has to be understood component-wise.

###### Proof.

First note that

$\displaystyle\Pi_{N}F(\bar{u})-\Pi_{K_{0}N}F(\bar{u})$
$\displaystyle=\left(I-P_{K_{0}}\right)\left(t\mapsto
e^{\tau(t+1)\mathcal{L}_{N}}\Pi_{N}f+\tau\int_{-1}^{t}e^{\tau(t-s)\mathcal{L}_{N}}\Pi_{N}\gamma(\bar{u}(s))\mbox{d}s\right)$
$\displaystyle=Q\left(I-P_{K_{0}}\right)\left(t\mapsto
e^{\tau(t+1)\Lambda_{N}}\right)\tilde{f}+Q\left(I-P_{K_{0}}\right)\left(t\mapsto\tau\int_{-1}^{t}e^{\tau(t-s)\Lambda_{N}}\tilde{c}^{Y}(s)\mbox{d}s\right).$

Therefore,

$\displaystyle\left|\Pi_{N}F(\bar{u})-\Pi_{K_{0}N}F(\bar{u})\right|$
$\displaystyle\leq|Q|\left(\left|\left(I-P_{K_{0}}\right)\left(t\mapsto
e^{\tau(t+1)\Lambda_{N}}\right)\right|\left|\tilde{f}\right|+\left|\left(I-P_{K_{0}}\right)\left(t\mapsto\tau\int_{-1}^{t}e^{\tau(t-s)\Lambda_{N}}\tilde{c}^{Y}(s)\mbox{d}s\right)\right|\right),$
(32)

where the absolute values have to be understood component-wise. We then apply,
component-wise, Lemma 4.6 and Lemma 4.9 to the first and the second term
respectively. ∎

######  Remark 5.4.

For each component, it is not clear a priori which choice of $q_{n}$ leads to
the smallest $\tilde{C}_{n}$, so in practice we just compute $\tilde{C}_{n}$
for all $q_{n}$ between $1$ and $K_{0}$ and then keep the smallest result.
Similarly, we approximately optimize for each $\rho_{n}$ and
$\check{\rho}_{n}$. Finally, in practice we replace the $C^{0}$ norms and the
supremum over Bernstein ellipses by the easy-to-evaluate upper-bounds provided
in Lemma 2.10.

####  5.1.3 Truncation in phase space

Next, we compute a bound for the residual associated to the truncation in
phase space. First note that

$\Pi_{\infty}\left(T(\bar{u})-\bar{u}\right)=\Pi_{\infty}F(\bar{u}).$

Therefore, for each $|n|>N$ we have

$\displaystyle
F_{n}(\bar{u})(t)=e^{\tau\lambda_{n}(t+1)}f_{n}+\tau\displaystyle\int_{-1}^{t}e^{\lambda_{n}\tau\left(t-s\right)}\gamma_{n}(\bar{u}(s))\mbox{d}s,$

which we estimate by

$\displaystyle\left\|F_{n}\left(\bar{u}\right)\right\|_{C^{0}}$
$\displaystyle\leq
e^{2\tau\Re(\lambda_{n})^{+}}|f_{n}|+\left\|\gamma_{n}(\bar{u})\right\|_{C^{0}}\frac{e^{2\tau\Re(\lambda_{n})}-1}{\Re(\lambda_{n})},$

where
$\Re(\lambda_{n})^{+}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\max(\Re(\lambda_{n}),0)$.
We then get the $Y_{\infty}$ bound by taking the $\ell^{1}_{\nu}$ norm of the
right-hand side. Notice that $\gamma(\bar{u})$ only has finitely many non-zero
coefficients, hence this $\ell^{1}_{\nu}$ norm amounts to a finite
computation. Assuming we have a bound on the $\ell^{1}_{\nu}$ norm of the tail
of $f$, this contributes to the $\mathcal{X}^{\infty}_{\nu}$ bound on the tail
of $F$ after estimating $e^{2\tau\Re(\lambda_{n})^{+}}$ uniformly for $|n|>N$.

###  5.2 Z-bounds

In this subsection, we now focus on the bound $Z$ satisfying assumption (15)
of Theorem 2.12. In order to estimate the norm of $DT(\bar{u})$, we take an
arbitrary $h$ in $B_{1,\epsilon}\left(0\right)$ (see (26)), and we again
derive three separate estimates, namely $Z_{KN}$, $Z_{\infty N}$ and
$Z_{\infty}$ such that

$\displaystyle\left\|\Pi_{KN}DT(\bar{u})h\right\|_{\mathcal{X}^{KN}_{\nu}}\leq
Z_{KN},\quad\left\|\Pi_{\infty N}DT(\bar{u})h\right\|_{\mathcal{X}^{\infty
N}_{\nu}}\leq Z_{\infty
N},\quad\left\|\Pi_{\infty}DT(\bar{u})h\right\|_{\mathcal{X}^{\infty}_{\nu}}\leq
Z_{\infty},$

so that

$\displaystyle
Z\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,Z_{KN}+\epsilon_{\infty
N}^{-1}Z_{\infty N}+\epsilon_{\infty}^{-1}Z_{\infty},$ (33)

satisfies (15). For each of these bounds, it will be helpful, recalling the
Definition 3.9 of $\widehat{DF}$, to rewrite $DT(\bar{u})$ as

$\displaystyle DT\left(\bar{u}\right)h$
$\displaystyle=\left(I-A\widehat{DF}\right)-A\left(DF\left(\bar{u}\right)-\widehat{DF}\right).$

######  Remark 5.5.

From (17) (with $M=1$), we see that we must have $Z<1$ in order to apply
Theorem 2.12. When deriving the estimates $Z_{KN}$, $Z_{\infty N}$ and
$Z_{\infty}$, we are going to see how the choices we made when defining
$\mathcal{L}$ are instrumental in achieving that goal. A careful choice of the
weights $\epsilon_{\infty N}$ and $\epsilon_{\infty}$ is also helpful in order
to make $Z$ small. However, unlike what (33) might suggest, we cannot simply
take these weights arbitrarily large, as they will also influence the bounds
$Z_{KN}$, $Z_{\infty N}$ and $Z_{\infty}$ themselves (as well as the
forthcoming bound $W$). Since this dependency is somewhat intricate and
difficult to study a priori, in practice we have to experiment with different
values of these weights in order to select suitable ones.

####  5.2.1 Finite dimensional projection

By definition of $A$ and $\widehat{DF}$,

$\displaystyle\left\|\Pi_{KN}\left(I-A\widehat{DF}\right)h\right\|_{\mathcal{X}^{KN}_{\nu}}$
$\displaystyle\leq\left\|I_{KN}-A_{KN}DF_{KN}\left(\bar{u}\right)\right\|_{\mathcal{B}\left(\mathcal{X}^{KN}_{\nu},\mathcal{X}^{KN}_{\nu}\right)},$
(34)

which is just a finite computation. In order to obtain the bound $Z_{KN}$, the
main task is therefore to enclose

$\Pi_{KN}\left(DF(\bar{u})-\widehat{DF}\right)h=\Pi_{KN}DF(\bar{u})\left(\Pi_{\infty
N}+\Pi_{\infty}\right)(h),$

for all $h\in B_{1,\epsilon}(0)$. We then have to multiply the result by
$|A_{KN}|$ in order to get a bound on

$\left\|\Pi_{KN}A\left(DF\left(\bar{u}\right)-\widehat{DF}\right)h\right\|_{\mathcal{X}^{KN}_{\nu}}.$

We compute

$\displaystyle\Pi_{N}\left[DF(\bar{u})\left(\Pi_{\infty
N}+\Pi_{\infty}\right)(h)\right](t_{k})$
$\displaystyle=\tau\int_{-1}^{t_{k}}e^{\tau(t_{k}-s)\mathcal{L}_{N}}\Pi_{N}D\gamma(\bar{u}(s))\left(\Pi_{\infty
N}+\Pi_{\infty}\right)(h)(s)\mbox{d}s$ $\displaystyle=\tau
Q\int_{-1}^{t_{k}}e^{\tau(t_{k}-s)\Lambda_{N}}Q^{-1}\Pi_{N}D\gamma(\bar{u}(s))\left(\Pi_{\infty
N}+\Pi_{\infty}\right)(h)(s)\mbox{d}s,$

where, according to (29),

$\displaystyle\Pi_{N}D\gamma(\bar{u}(s))\left(\Pi_{\infty
N}+\Pi_{\infty}\right)(h)(s)=$
$\displaystyle\qquad\sum_{j=0}^{2R-1}\mathcal{D}^{j}\Pi_{N}\left[\left(\left(g^{(j)}\right)^{\prime}(\bar{u}(s))-\bar{v}^{(j)}\right)\ast\Pi_{\infty
N}h(s)+\left(g^{(j)}\right)^{\prime}(\bar{u}(s))\ast\Pi_{\infty}h(s)\right]-\mathcal{R}_{N}\Pi_{\infty
N}h(s).$

By Lemma 4.3 we have that

$\displaystyle\left\|\Pi_{N}\left[\left(\left(g^{(j)}\right)^{\prime}(\bar{u}(s))-\bar{v}^{(j)}\right)\ast\Pi_{\infty
N}h(s)+\left(g^{(j)}\right)^{\prime}(\bar{u}(s))\ast\Pi_{\infty}h(s)\right]\right\|_{C^{0}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq\Pi_{N}\Phi^{\epsilon}\left(\left(g^{(j)}\right)^{\prime}(\bar{u})-\bar{v}^{(j)},\left(g^{(j)}\right)^{\prime}(\bar{u})\right),$

and therefore, recalling that $D_{N}(t)$ is defined in (30), and using Lemma
4.1, we end up with

$\displaystyle\left|\Pi_{N}\left[DF(\bar{u})\left(\Pi_{\infty
N}+\Pi_{\infty}\right)(h)\right](t_{k})\right|$
$\displaystyle\qquad\leq\tau|Q|\int_{-1}^{t_{k}}e^{\tau(t_{k}-s)\Re(\Lambda_{N})}\mbox{d}s$
$\displaystyle\qquad\qquad\times\left(|Q^{-1}|\sum_{j=0}^{2R-1}|\mathcal{D}^{j}|\Pi_{N}\Phi^{\epsilon}\left(\left(g^{(j)}\right)^{\prime}(\bar{u})-\bar{v}^{(j)},\left(g^{(j)}\right)^{\prime}(\bar{u})\right)+\left|Q^{-1}\mathcal{R}_{N}\right|\left\|\Pi_{\infty
N}h\right\|_{C^{0}}\right)$
$\displaystyle\qquad\leq|Q|\,D_{N}(t_{k})\left(|Q^{-1}|\sum_{j=0}^{2R-1}|\mathcal{D}^{j}|\Pi_{N}\Phi^{\epsilon}\left(\left(g^{(j)}\right)^{\prime}(\bar{u})-\bar{v}^{(j)},\left(g^{(j)}\right)^{\prime}(\bar{u})\right)+\epsilon_{\infty
N}\Upsilon\left(Q^{-1}\mathcal{R}_{N}\right)\right).$ (35)

######  Remark 5.6.

A careful inspection of this bound reveals why we can hope it to be small, so
that $Z_{KN}<1$. First, the last term $\epsilon_{\infty
N}\Upsilon\left(Q^{-1}\mathcal{R}_{N}\right)$ should be negligible, as
$\mathcal{R}_{N}$ should be very small: it simply accounts for the fact that
we do not necessarily diagonalize $\tilde{\mathcal{L}}_{N}$ exactly when
defining $\mathcal{L}$, see Section 3.4. More importantly, the term
$\Pi_{N}\Phi^{\epsilon}\left(\left(g^{(j)}\right)^{\prime}(\bar{u})-\bar{v}^{(j)},\left(g^{(j)}\right)^{\prime}(\bar{u})\right)$
is expected to be small because we take $\bar{v}^{(j)}$ as a constant-in-time
approximation of $\left(g^{(j)}\right)^{\prime}(\bar{u})$. Notice that, if we
had taken $\mathcal{L}$ to be simply $(-1)^{R+1}\mathcal{D}^{2R}$, then there
would be no $-\bar{v}$ terms, and the only way of making
$\Pi_{N}\Phi^{\epsilon}\left(\left(g^{(j)}\right)^{\prime}(\bar{u}),\left(g^{(j)}\right)^{\prime}(\bar{u})\right)$
small would have been to take $\epsilon_{\infty N}$ small, which would then be
detrimental for the $\epsilon_{\infty N}^{-1}$ factor in $Z$, see 33. We also
point out that this estimate will benefit from the domain decomposition
procedure (presented in details in Section 6): by splitting the time interval
into smaller subintervals, and by choosing the $\bar{v}^{(j)}$ piece-wise
constant, we will be able to get a more accurate approximation of
$\left(g^{(j)}\right)^{\prime}(\bar{u})$.

####  5.2.2 Interpolation error estimates

Next, regarding the projection onto $\mathcal{X}^{\infty N}_{\nu}$ we first
notice that

$\displaystyle\Pi_{\infty N}\left(I-A\widehat{DF}\right)=0.$

Therefore, we have to bound, for any $h\in B_{1,\epsilon}(0)$,

$\displaystyle\left\|\Pi_{\infty
N}A\left(DF\left(\bar{u}\right)-\widehat{DF}\right)h\right\|_{\mathcal{X}^{\infty
N}_{\nu}}$ $\displaystyle\qquad\qquad\qquad=\left\|\Pi_{\infty
N}\left(DF\left(\bar{u}\right)h+h\right)\right\|_{\mathcal{X}^{\infty
N}_{\nu}}$
$\displaystyle\qquad\qquad\qquad=\left\|\left\|\left(I-P_{K}\right)\left(t\mapsto\tau
Q\int_{-1}^{t}e^{\tau(t-s)\Lambda_{N}}Q^{-1}\Pi_{N}D\gamma(\bar{u}(s))h(s)\mbox{d}s\right)\right\|_{C^{0}}\right\|_{\ell^{1}_{\nu}},$

which we do by using Lemma 4.8, and then Lemma 4.3, namely

$\displaystyle\left\|\Pi_{\infty
N}A\left(DF\left(\bar{u}\right)-\widehat{DF}\right)h\right\|_{\mathcal{X}^{\infty
N}_{\nu}}$
$\displaystyle\leq\tau\sigma_{K,0}\Bigg{(}\sum_{j=0}^{2R-1}\left\||\mathcal{D}^{j}|+|Q||\Lambda_{N}|\,D_{N}(1)\,|Q^{-1}||\mathcal{D}^{j}|\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}$
$\displaystyle\qquad\qquad\quad\times\left\|\Pi_{N}\left[\left(\left(g^{(j)}\right)^{\prime}(\bar{u})-\bar{v}^{(j)}\right)\ast\Pi_{N}h+\left(g^{(j)}\right)^{\prime}(\bar{u})\ast\Pi_{\infty}h\right]\right\|_{\ell^{1}_{\nu}(C^{0})}$
$\displaystyle\qquad\qquad\quad+\left\|I_{N}+|Q||\Lambda_{N}|\,D_{N}(1)\,|Q^{-1}|\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}\left\|\mathcal{R}_{N}\Pi_{N}h\right\|_{\ell^{1}_{\nu}(C^{0})}\Bigg{)}$
$\displaystyle\leq\tau\sigma_{K,0}\Bigg{(}\sum_{j=0}^{2R-1}\left\||\mathcal{D}^{j}|+|Q||\Lambda_{N}|\,D_{N}(1)\,|Q^{-1}||\mathcal{D}^{j}|\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}\tilde{\Phi}^{\epsilon}\left(\left(g^{(j)}\right)^{\prime}(\bar{u})-\bar{v}^{(j)},\left(g^{(j)}\right)^{\prime}(\bar{u})\right)$
$\displaystyle\qquad\qquad\quad+\max(1,\epsilon_{\infty
N})\left\|I_{N}+|Q||\Lambda_{N}|\,D_{N}(1)\,|Q^{-1}|\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}\left\|\mathcal{R}_{N}\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}\Bigg{)},$
(36)

which gives us the $Z_{\infty N}$ estimate.

######  Remark 5.7.

As was the case for $Z_{KN}$, we see here that the definition of $\mathcal{L}$
and the corresponding choice of $\bar{v}$ helps to make $Z_{\infty N}$
smaller. Also, we see here the influence of $K$: the larger we take $K$, the
smaller the constant $\sigma_{K,0}$ becomes. In Section 6, we will introduce
another option to make $Z_{\infty N}$ smaller, namely to subdivide the time
interval.

####  5.2.3 Truncation in phase space

Again, we first notice that

$\displaystyle\Pi_{\infty}\left(I-A\widehat{DF}\right)=0.$

Therefore, our last remaining task regarding the $Z$ bound is to estimate,
still for any $h\in B_{1,\epsilon}(0)$,

$\displaystyle\left\|\Pi_{\infty}A\left(DF\left(\bar{u}\right)-\widehat{DF}\right)h\right\|_{\mathcal{X}^{\infty}_{\nu}}=\left\|\Pi_{\infty}\left(DF\left(\bar{u}\right)h+h\right)\right\|_{\mathcal{X}^{\infty}_{\nu}}.$

Introducing $\tilde{g}^{(j)}\in\mathcal{X}_{\nu}$, $j=0,\ldots,2R-1$, defined
by

$\tilde{g}^{(j)}_{n}(s)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\left\\{\begin{aligned}
&\left(\left(g^{(j)}\right)^{\prime}(\bar{u}(s))-\bar{v}^{(j)}\right)_{0}\quad&n=0\\\
&\left(\left(g^{(j)}\right)^{\prime}(\bar{u}(s))\right)_{n}\quad&n\neq 0,\\\
\end{aligned}\right.$

we have for all $|n|>N$ and $t\in[-1,1]$,

$\displaystyle(DF\left(\bar{u}\right)h+h)_{n}(t)$
$\displaystyle=\tau\int_{-1}^{t}e^{\tau(t-s)\lambda_{n}}\left(\sum_{j=0}^{2R-1}(in)^{j}\left(\tilde{g}^{(j)}(s)\ast
h(s)\right)_{n}\right)\mbox{d}s.$

By Lemmas 4.11 and 3.6 and the Banach algebra property, we infer that

$\displaystyle\left\|\Pi_{\infty}(DF\left(\bar{u}\right)h+h)\right\|_{\mathcal{X}^{\infty}_{\nu}}$
$\displaystyle\leq\sum_{j=0}^{2R-1}\chi_{N}^{(j)}\left\|\tilde{g}^{(j)}\ast
h\right\|_{\ell^{1}_{\nu}(C^{0})}\leq\vartheta_{\epsilon}\sum_{j=0}^{2R-1}\chi_{N}^{(j)}\left\|\left\|\tilde{g}^{(j)}\right\|_{C^{0}}\right\|_{\ell^{1}_{\nu}},$

which gives us the $Z_{\infty}$ bound.

######  Remark 5.8.

Looking at the constants $\chi_{N}^{(j)}$ (see Lemma 4.11), we see that
$Z_{\infty}$ can be made as small as we want by taking $N$ large enough.

###  5.3 W-bound

In this subsection, we finally derive a computable bound $W$ satisfying
assumption (16) of Theorem 2.12. In order to estimate the norm of
$DT(u)-DT(\bar{u})$, for any $u$ in $B_{r_{*},\epsilon}(\bar{u})$, we take
arbitrary $h$ and $v$ in $B_{1,\epsilon}\left(0\right)$, and we again derive
three separate estimates, namely $W_{KN}$, $W_{\infty N}$ and $W_{\infty}$
such that

$\displaystyle\left\|\Pi_{KN}\left(DT(\bar{u}+r_{*}v)-DT(\bar{u})\right)h\right\|_{\mathcal{X}^{KN}_{\nu}}$
$\displaystyle\leq W_{KN}\,r_{*}\left\|v\right\|_{\mathcal{X}_{\nu}},$
$\displaystyle\left\|\Pi_{\infty
N}\left(DT(\bar{u}+r_{*}v)-DT(\bar{u})\right)h\right\|_{\mathcal{X}^{\infty
N}_{\nu}}$ $\displaystyle\leq W_{\infty
N}\,r_{*}\left\|v\right\|_{\mathcal{X}_{\nu}},$
$\displaystyle\left\|\Pi_{\infty}\left(DT(\bar{u}+r_{*}v)-DT(\bar{u})\right)h\right\|_{\mathcal{X}^{\infty}_{\nu}}$
$\displaystyle\leq W_{\infty}\,r_{*}\left\|v\right\|_{\mathcal{X}_{\nu}},$

so that

$\displaystyle
W\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,W_{KN}+\epsilon_{\infty
N}^{-1}W_{\infty N}+\epsilon_{\infty}^{-1}W_{\infty}$

satisfies (16).

We start from the fact that
$DT(\bar{u}+r_{*}v)-DT(\bar{u})=-A\left(DF(\bar{u}+r_{*}v)-DF(\bar{u})\right)$,
and compute

$\displaystyle\left[\left(DF\left(\bar{u}+r_{*}v\right)-DF\left(\bar{u}\right)\right)h\right](t)$
$\displaystyle\qquad=\tau\int_{-1}^{t}e^{\tau\left(t-s\right)\mathcal{L}}\biggl{(}D\gamma\left(\bar{u}\left(s\right)+r_{*}v\left(s\right)\right)-D\gamma\left(\bar{u}\left(s\right)\right)\biggr{)}h(s)\
\mbox{d}s$ $\displaystyle\qquad=\tau
r_{*}\int_{-1}^{t}e^{\tau\left(t-s\right)\mathcal{L}}\int_{0}^{1}D^{2}\gamma\left(\bar{u}\left(s\right)+\tilde{s}r_{*}v\left(s\right)\right)\left[h\left(s\right),v\left(s\right)\right]\
\mbox{d}\tilde{s}\ \mbox{d}s$ $\displaystyle\qquad=\tau
r_{*}\int_{-1}^{t}e^{\tau\left(t-s\right)\mathcal{L}}\sum_{j=0}^{2R-1}\mathcal{D}^{j}\int_{0}^{1}\left(g^{(j)}\right)^{\prime\prime}\\!\left(\bar{u}\left(s\right)+\tilde{s}r_{*}v\left(s\right)\right)\ast
v(s)\ast h(s)\ \mbox{d}\tilde{s}\ \mbox{d}s.$ (37)

####  5.3.1 Finite dimensional projection

Let us start with estimating the finite dimensional projection of (37):

$\displaystyle\left|\Pi_{N}\tau
r_{*}\left(\int_{-1}^{t_{k}}e^{\tau\left(t_{k}-s\right)\mathcal{L}}\sum_{j=0}^{2R-1}\mathcal{D}^{j}\int_{0}^{1}\left(g^{(j)}\right)^{\prime\prime}\left(\bar{u}\left(s\right)+\tilde{s}r_{*}v\left(s\right)\right)\ast
v(s)\ast h(s)\ \mbox{d}\tilde{s}\ \mbox{d}s\right)\right|$
$\displaystyle\quad\leq
r_{*}\sum_{j=0}^{2R-1}\tau\left|Q\int_{-1}^{t_{k}}e^{\tau\left(t_{k}-s\right)\Lambda_{N}}Q^{-1}\mathcal{D}^{j}\int_{0}^{1}\Pi_{N}\left(\left(g^{(j)}\right)^{\prime\prime}\left(\bar{u}\left(s\right)+\tilde{s}r_{*}v\left(s\right)\right)\ast
v(s)\ast h(s)\right)\ \mbox{d}\tilde{s}\ \mbox{d}s\right|$
$\displaystyle\quad\leq r_{*}\sum_{j=0}^{2R-1}|Q|\,D_{N}(t_{k})\,|Q^{-1}|\
|\mathcal{D}^{j}|\,\Pi_{N}\left(\left|\left(g^{(j)}\right)^{\prime\prime}\right|\left(\|\bar{u}\|_{C^{0}}+r_{*}\|v\|_{C^{0}}\right)\ast\|v\|_{C^{0}}\ast\|h\|_{C^{0}}\right),$
(38)

where $\left|\left(g^{(j)}\right)^{\prime\prime}\right|$ is a polynomial
defined as in Lemma 4.5. Considering the operators
$\Xi^{(j)}:\mathbb{C}^{2N+1}\to\mathcal{X}^{KN}_{\nu}$ defined by

$\Xi^{(j)}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\begin{pmatrix}|Q|\,D_{N}(t_{0})\,|Q^{-1}|\
|\mathcal{D}^{j}|\\\ \vdots\\\ |Q|\,D_{N}(t_{K})\,|Q^{-1}|\
|\mathcal{D}^{j}|\end{pmatrix},$ (39)

and introducing

$B^{(j)}=\begin{pmatrix}B^{(j)}_{0}\\\ \vdots\\\
B^{(j)}_{K}\end{pmatrix}=|A_{KN}|\,\Xi^{(j)}$

with $B^{(j)}_{k}:\mathbb{C}^{2N+1}\to\mathbb{C}^{2N+1}$, we then use Lemma
3.6 and Lemma 4.5 to obtain

$\displaystyle\left\|\Pi_{KN}\left(A\left[\left(DF\left(\bar{u}+r_{*}v\right)-DF\left(\bar{u}\right)\right)h\right]\right)\right\|_{\mathcal{X}^{KN}_{\nu}}$
$\displaystyle\leq
r_{*}\sum_{j=0}^{2R-1}\left\|B^{(j)}\Pi_{N}\left(\left|\left(g^{(j)}\right)^{\prime\prime}\right|\left(\|\bar{u}\|_{C^{0}}+r_{*}\|v\|_{C^{0}}\right)\ast\|v\|_{C^{0}}\ast\|h\|_{C^{0}}\right)\right\|_{\mathcal{X}^{KN}_{\nu}}$
$\displaystyle\leq
r_{*}\sum_{j=0}^{2R-1}\left(\left\|B^{(j)}_{0}\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}+2\sum_{k=1}^{K}\left\|B^{(j)}_{k}\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}\right)$
$\displaystyle\hskip
85.35826pt\times\left\|\Pi_{N}\left(\left|\left(g^{(j)}\right)^{\prime\prime}\right|\left(\|\bar{u}\|_{C^{0}}+r_{*}\|v\|_{C^{0}}\right)\ast\|v\|_{C^{0}}\ast\|h\|_{C^{0}}\right)\right\|_{\ell^{1}_{\nu}}$
$\displaystyle\leq
r_{*}\left\|v\right\|_{\mathcal{X}_{\nu}}\vartheta_{\epsilon}^{2}\sum_{j=0}^{2R-1}\left(\left\|B^{(j)}_{0}\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}+2\sum_{k=1}^{K}\left\|B^{(j)}_{k}\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}\right)\left|\left(g^{(j)}\right)^{\prime\prime}\right|\left(\left\|\bar{u}\right\|_{\ell^{1}_{\nu}(C^{0})}+\vartheta_{\epsilon}r_{*}\right),$

which, after removing the $r_{*}\|v\|_{\mathcal{X}_{\nu}}$ factor, gives the
$W_{KN}$ bound.

####  5.3.2 Interpolation error estimates

Next, we estimate

$\left\|\Pi_{\infty
N}A\left(DF\left(\bar{u}+r_{*}v\right)-DF\left(\bar{u}\right)\right)h\right\|_{\mathcal{X}^{\infty
N}_{\nu}}=\left\|\Pi_{\infty
N}\left(DF\left(\bar{u}+r_{*}v\right)-DF\left(\bar{u}\right)\right)h\right\|_{\mathcal{X}^{\infty
N}_{\nu}},$

by using Lemma 4.8 on (37), and then once again Lemma 3.6 and Lemma 4.5. This
yields

$\displaystyle\left\|\Pi_{\infty
N}A\left(DF\left(\bar{u}+r_{*}v\right)-DF\left(\bar{u}\right)\right)h\right\|_{\mathcal{X}^{\infty
N}_{\nu}}$ $\displaystyle\leq
r_{*}\tau\sigma_{K,0}\sum_{j=0}^{2R-1}\left\||\mathcal{D}^{j}|+|Q||\Lambda_{N}|D_{N}(1)|Q^{-1}||\mathcal{D}^{j}|\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}$
$\displaystyle\qquad\qquad\qquad\quad\times\left\|\left\|\Pi_{N}\int_{0}^{1}\left(g^{(j)}\right)^{\prime\prime}\left(\bar{u}\left(t\right)+\tilde{s}rv\left(t\right)\right)\ast
v(t)\ast h(t)\ \mbox{d}\tilde{s}\right\|_{C^{0}}\right\|_{\ell^{1}_{\nu}}$
$\displaystyle\leq
r_{*}\left\|v\right\|_{\mathcal{X}_{\nu}}\vartheta_{\epsilon}^{2}\tau\sigma_{K,0}\sum_{j=0}^{2R-1}\left\||\mathcal{D}^{j}|+|Q||\Lambda_{N}|D_{N}(1)|Q^{-1}||\mathcal{D}^{j}|\right\|_{B(\ell^{1}_{\nu},\ell^{1}_{\nu})}\left|\left(g^{(j)}\right)^{\prime\prime}\right|\left(\left\|\bar{u}\right\|_{\ell^{1}_{\nu}(C^{0})}+\vartheta_{\epsilon}r_{*}\right),$

which, after removing the $r_{*}\|v\|_{\mathcal{X}_{\nu}}$ factor, gives the
$W_{\infty N}$ bound.

####  5.3.3 Truncation in phase space

Finally, for the tail part, we use once again Lemmas 4.11, 3.6 and 4.5, to get

$\displaystyle\left\|\Pi_{\infty}A\left(DF\left(\bar{u}+r_{*}v\right)-DF\left(\bar{u}\right)\right)h\right\|_{\mathcal{X}^{\infty}_{\nu}}$
$\displaystyle=\left\|\Pi_{\infty}\left(DF\left(\bar{u}+r_{*}v\right)-DF\left(\bar{u}\right)\right)h\right\|_{\mathcal{X}^{\infty}_{\nu}}$
$\displaystyle\leq
r_{*}\left\|v\right\|_{\mathcal{X}_{\nu}}\vartheta_{\epsilon}^{2}\sum_{j=0}^{2R-1}\chi^{(j)}_{N}\left|\left(g^{(j)}\right)^{\prime\prime}\right|\left(\left\|\bar{u}\right\|_{\ell^{1}_{\nu}(C^{0})}+\vartheta_{\epsilon}r_{*}\right),$
(40)

which, after removing the $r_{*}\|v\|_{\mathcal{X}_{\nu}}$ factor, gives the
$W_{\infty}$ bound.

##  6 Domain decomposition

In this section, we introduce a generalization of the setup introduced in
Section 3, which is more efficient for handling longer integration times. In
that case, rather than increasing $K$, we can split the time domain into
multiple smaller subdomains, and use several “copies” of the zero finding
problem introduced previously, which are only weakly coupled via “matching
conditions” at the boundary between two successive subintervals.

The key aspect that we wish to emphasize is that we still validate the entire
solution at once, which means that this setup can then be used to solve
boundary value problems in time, for instance between local unstable and
stable manifolds of objects of interest. This will be the subject of a future
work.

An alternative approach for long time integration is time stepping: we also
decompose the time domain, but then validate the solution on each subdomain
successively, while rigorously propagating the errors/enclosures obtained at
each step. While this direction is not the main focus of our work, the
approach we developed can also handle time-stepping, which we describe in more
detail in Appendix A.

In this section, we describe the modifications required for the domain
decomposition approach. One key aspect is that the approximate inverse $A$
should be modified judiciously. Once this is done, most of the computations
presented in Section 5 can be re-used, and we therefore mainly describe the
new estimates that have to be incorporated.

###  6.1 How to adapt the setup

We start again from (20), for $t\in[0,2\tau]$, but instead of dealing with the
entire time interval at once, we consider $M\in\mathbb{N}_{\geq 2}$, and
$\tau_{1},\tau_{2},\ldots,\tau_{M}>0$ such that $\sum_{m=1}^{M}\tau_{m}=\tau$.
Splitting $t\in[0,2\tau]$ into $M$ subintervals (each of length $2\tau_{m}$),
and then rescaling to $[-1,1]$ on each subinterval, the initial value problem
(21) is then equivalent to

$\displaystyle\begin{cases}\dfrac{du^{(m)}}{dt}(t)-\tau_{m}\mathcal{L}^{(m)}u^{(m)}(t)=\tau_{m}\left((-1)^{R+1}\mathcal{D}^{2R}u^{(m)}(t)+\displaystyle\sum_{j=0}^{2R-1}\mathcal{D}^{j}g^{(j)}(u^{(m)})(t)-\mathcal{L}^{(m)}u^{(m)}(t)\right),\\\\[8.61108pt]
\hfill t\in[-1,1],\\\\[8.61108pt]
u^{(m)}\left(-1\right)=u^{(m-1)}(1),\end{cases}$ (41)

for all $m=1,\ldots,M$, where
$u^{(0)}(1)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,f$.
A natural space in which to study this domain decomposition problem is
$\mathcal{X}_{\nu}^{M}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\left(\mathcal{X}_{\nu}\right)^{M}$,
with the max-norm

$\displaystyle\left\|u\right\|_{\mathcal{X}_{\nu}^{M}}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\max_{1\leq
m\leq M}\left\|u^{(m)}\right\|_{\mathcal{X}_{\nu}}.$

######  Definition 6.1.

The zero finding map $F:\mathcal{X}_{\nu}^{M}\rightarrow\mathcal{X}_{\nu}^{M}$
for (1) is defined by

$\displaystyle F^{(m)}\left(u\right)\left(t\right)$
$\displaystyle\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,e^{\tau_{m}(t+1)\mathcal{L}^{(m)}}u^{(m-1)}(1)+\tau_{m}\int_{-1}^{t}e^{\tau_{m}(t-s)\mathcal{L}^{(m)}}\gamma^{(m)}(u^{(m)}(s))\mbox{d}s-u^{(m)}\left(t\right),$

for all $m=1,\ldots,M$, where

$\gamma^{(m)}(u^{(m)})=(-1)^{R+1}\mathcal{D}^{2R}u^{(m)}+\displaystyle\sum_{j=0}^{2R-1}\mathcal{D}^{j}g^{(j)}(u^{(m)})-\mathcal{L}^{(m)}u^{(m)}.$

Each $\mathcal{L}^{(m)}$ is defined, on its corresponding subinterval, as
$\mathcal{L}$ was defined in Section 3.4.

One can now define a finite dimensional reduction of $F$ and a Newton-like
operator $T$ in almost exactly the same way as in Section 3.3. For the sake of
completeness, we present the essential details. To keep the coupling between
the successive subdomains more easily tractable, we work with a single choice
of $K$ and $N$ for all subdomains.

Before introducing the approximate inverse, let us also consider the linear
operator
$G:\left(\mathcal{X}_{\nu}\right)^{M}\rightarrow\left(\mathcal{X}_{\nu}\right)^{M}$
defined by

$G^{(m)}\left(u\right)\left(t\right)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\left\\{\begin{aligned}
&-u^{(m)}\left(t\right)&\qquad m=1,\\\
&e^{(t+1)\tau_{m}\mathcal{L}^{(m)}}u^{(m-1)}(1)-u^{(m)}\left(t\right)&\qquad
m=2,\ldots,M.\end{aligned}\right.$

Notice that $G$ is invertible, and that

$\displaystyle\left(G^{-1}(u)\right)^{(m)}(t)\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,$
$\displaystyle\qquad\qquad\left\\{\begin{aligned}
&-u^{(m)}\left(t\right)&\qquad m=1,\\\
&-u^{(m)}(t)-e^{(t+1)\tau_{m}\mathcal{L}^{(m)}}\left(\sum_{l=1}^{m-1}\exp\left(2\textstyle\sum_{j=l+1}^{m-1}\tau_{j}\mathcal{L}^{(j)}\right)u^{(l)}(1)\right)&\qquad
m=2,\ldots,M.\end{aligned}\right.$

where $\sum_{j=m}^{m-1}\tau_{j}\mathcal{L}^{(j)}=0$.

We extend the projections $\Pi_{KN},\Pi_{\infty N}$ and $\Pi_{\infty}$ from
the single domain to the multiple domains setting in the natural way. This
gives rise to the subspaces

$\displaystyle(\mathcal{X}^{KN}_{\nu})^{M}=\Pi_{KN}\left(\mathcal{X}_{\nu}^{M}\right),\quad(\mathcal{X}^{\infty
N}_{\nu})^{M}=\Pi_{\infty
N}\left(\mathcal{X}_{\nu}^{M}\right),\quad(\mathcal{X}^{\infty}_{\nu})^{M}=\Pi_{\infty}\left(\mathcal{X}_{\nu}^{M}\right)$

and to the decomposition

$\displaystyle\mathcal{X}^{M}_{\nu}=(\mathcal{X}^{KN}_{\nu})^{M}\oplus(\mathcal{X}^{\infty
N}_{\nu})^{M}\oplus(\mathcal{X}^{\infty}_{\nu})^{M}.$

In order to introduce an appropriate approximate inverse, it will be
convenient to use a block-decomposition of linear operators acting on
$\mathcal{X}_{\nu}^{M}$, based on the above decomposition.

######  Definition 6.2 (Approximation of $DF\left(\bar{u}\right)$).

Given $\bar{u}$ in $\left(\mathcal{X}^{KN}_{\nu}\right)^{M}$, we introduce an
approximate derivative
$\widehat{DF}:\mathcal{X}^{M}_{\nu}\rightarrow\mathcal{X}^{M}_{\nu}$ of $F$ at
$\bar{u}$ defined by

$\displaystyle\widehat{DF}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\left(\begin{array}[]{c|c|c}DF^{M}_{KN}\left(\bar{u}\right)&0&0\\\
\hline\cr\Pi_{\infty N}G\Pi_{KN}&-I^{M}_{\infty N}&0\\\ \hline\cr
0&0&\Pi_{\infty}G\Pi_{\infty}\end{array}\right).$

######  Remark 6.3.

Note that $\Pi_{KN}G\Pi_{\infty N}=0$ and $\Pi_{\infty N}G\Pi_{\infty
N}=-I^{M}_{\infty N}$, so that this approximate derivative $\widehat{DF}$ is
simply obtained by always keeping the “important” part of $DF$, namely $G$,
and by neglecting the others terms except on the finite dimensional subspace
$\left(\mathcal{X}^{KN}_{\nu}\right)^{M}$.

It is tempting be somewhat lazy, go one step further and also neglect the
$\Pi_{\infty N}G\Pi_{KN}$ term, so that $\widehat{DF}$ and then its inverse
$A$ become block-diagonal, which makes subsequent estimates easier. However,
in practice we want the approximate derivative, and especially the approximate
inverse of the derivative, to become more and more accurate when $N$ and $K$
increase, so that, at least in principle, we always get a contraction for $T$
by taking the finite dimensional projection large enough. If the $\Pi_{\infty
N}G\Pi_{KN}$ term is not included, the consequence is that quantities like

$\left\|\left(I-P_{K}\right)\left(t\mapsto
e^{(t+1)\tau_{m}\lambda_{n}^{(m)}}\right)\right\|_{C^{0}}$

for all $|n|\leq N$ would have to be estimated in the new $Z^{1}_{\infty N}$
bound, and would have to be small enough for us to prove that $T$ is a
contraction. Because these exponentials get stiffer and stiffer when $N$
increases, this would introduce a very undesirable dependency between $K$ and
$N$, somewhat reminiscent of the CFL condition, where $K$ (the time
discretization parameter) has to be large enough with respect to $N$ (the
space discretization parameter).

Including the $\Pi_{\infty N}G\Pi_{KN}$ term in $\widehat{DF}$, and adapting
$A$ accordingly, allows us to alleviate this limitation.

######  Definition 6.4 (Approximate inverse of $DF\left(\bar{u}\right)$).

Given $\bar{u}$ in $\left(\mathcal{X}^{KN}_{\nu}\right)^{M}$, we introduce an
approximate inverse $A:\mathcal{X}^{M}_{\nu}\rightarrow\mathcal{X}^{M}_{\nu}$
of $DF\left(\bar{u}\right)$, defined by

$\displaystyle
A\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\left(\begin{array}[]{c|c|c}A_{KN}&0&0\\\
\hline\cr\Pi_{\infty N}G\Pi_{KN}A_{KN}&-I^{M}_{\infty N}&0\\\ \hline\cr
0&0&\Pi_{\infty}G^{-1}\Pi_{\infty}\end{array}\right),$

where $A_{KN}$ is a numerically computed approximate inverse of
$DF^{M}_{KN}(\bar{u})$.

Up to the fact that $A_{KN}$ is not the exact inverse of
$DF^{M}_{KN}(\bar{u})$, $A$ is the inverse of $\widehat{DF}$. We can now again
consider a Newton-like operator $T=I-AF$, this time mapping the product space
$\mathcal{X}^{M}_{\nu}$ into itself, and try to validate a posteriori a
numerically computed approximate solution $\bar{u}$ by using Theorem 2.12.

On each copy of $\mathcal{X}_{\nu}$ in $\mathcal{X}_{\nu}^{M}$, we use the
norm defined in (25), possibly with different weights $\epsilon_{\infty N}$
and $\epsilon_{\infty}$, which are thus denoted by $\epsilon_{\infty N}^{(m)}$
and $\epsilon_{\infty}^{(m)}$. For any $u\in\mathcal{X}^{M}_{\nu}$ and
$r=(r^{(1)},\ldots,r^{(m)})\in\mathbb{R}^{M}_{>0}$, we define

$\displaystyle
B^{M}_{r,\epsilon}(u)=\prod_{m=1}^{M}B_{r^{(m)},\epsilon^{(m)}}\left(u^{(m)}\right),$

which will play the role of $\operatorname{Box}(u,r)$ when applying Theorem
2.12.

###  6.2 Estimates on $G$ and $G^{-1}$

We collect here basic estimates on $G$ and $G^{-1}$ that will be used several
times in the next subsection, where we describe the extra terms appearing in
the bounds $Y$, $Z$ and $W$ when domain decomposition is used.

######  Lemma 6.5.

Let $\bar{w}\in\left(\mathcal{X}_{\nu}^{KN}\right)^{M}$. Then,

$\displaystyle\left\|\Pi_{\infty
N}G^{(1)}\left(\bar{w}\right)\right\|_{\ell^{1}_{\nu}(C^{0})}=0$

and, for $m=2,\ldots,M$,

$\displaystyle\left\|\Pi_{\infty
N}G^{(m)}\left(\bar{w}\right)\right\|_{\ell^{1}_{\nu}(C^{0})}\leq\left\||Q|\left\|\left(I-P_{K}\right)\left(t\mapsto
e^{(t+1)\tau_{m}\Lambda_{N}^{(m)}}\right)\right\|_{C^{0}}|Q^{-1}|\
\left|\bar{w}^{(m-1)}(1)\right|\right\|_{\ell^{1}_{\nu}}.$

###### Proof.

By definition of $G$, we have

$\Pi_{\infty N}G^{(m)}\left(\bar{w}\right)=\left\\{\begin{aligned} &0&\qquad
m=1,\\\ &\Pi_{\infty N}\left(t\mapsto
e^{(t+1)\tau_{m}\mathcal{L}^{(m)}}\bar{w}^{(m-1)}(1)\right)&\qquad
m=2,\ldots,M.\end{aligned}\right.$

For all $m=2,\ldots,M$, we then estimate

$\displaystyle\left\|\left\|\left(I-P_{K}\right)\left(t\mapsto
e^{(t+1)\tau_{m}\mathcal{L}_{N}^{(m)}}\bar{w}^{(m-1)}(1)\right)\right\|_{C^{0}}\right\|_{\ell^{1}_{\nu}}$
$\displaystyle\qquad\leq\left\|\left\|\left(I-P_{K}\right)\left(t\mapsto
e^{(t+1)\tau_{m}\mathcal{L}_{N}^{(m)}}\right)\right\|_{C^{0}}\left|\bar{w}^{(m-1)}(1)\right|\right\|_{\ell^{1}_{\nu}}$
$\displaystyle\qquad\leq\left\||Q|\left\|\left(I-P_{K}\right)\left(t\mapsto
e^{(t+1)\tau_{m}\Lambda_{N}^{(m)}}\right)\right\|_{C^{0}}|Q^{-1}|\
\left|\bar{w}^{(m-1)}(1)\right|\right\|_{\ell^{1}_{\nu}}.$

∎

While these interpolation errors for stiff exponentials were exactly the terms
we were worried about when we discussed the definition of $\widehat{DF}$ and
$A$ in Remark 6.3, they are less worrisome here because they get multiplied by
$\left|\bar{w}^{(m-1)}(1)\right|$, which are not specified here but which will
always be small when we apply Lemma 6.5 in Section 6.3 repeatedly.

######  Remark 6.6.

In order to make this estimate computable, we have to get estimates on

$\left\|\left(I-P_{K}\right)\left(t\mapsto
e^{(t+1)\tau_{m}\lambda_{n}^{(m)}}\right)\right\|_{C^{0}}$

for all $|n|\leq N$. This is accomplished using Lemma 4.6 together with Remark
4.10. Similarly to what we did in Section 5.1.2, we estimate this error by
splitting it into an almost exact part, using an interpolation polynomial of
high degree $K_{0}$, and an interpolation error estimate:

$\displaystyle\left\|\left(I-P_{K}\right)\left(t\mapsto
e^{(t+1)\tau_{m}\lambda_{n}^{(m)}}\right)\right\|_{C^{0}}\leq$
$\displaystyle\qquad\qquad\left\|\left(P_{K_{0}}-P_{K}\right)\left(t\mapsto
e^{(t+1)\tau_{m}\lambda_{n}^{(m)}}\right)\right\|_{C^{0}}+\left\|\left(I-P_{K_{0}}\right)\left(t\mapsto
e^{(t+1)\tau_{m}\lambda_{n}^{(m)}}\right)\right\|_{C^{0}},$

where the first term can be easily estimated via (12), and for the second one
we use again Lemma 4.6, for some well chosen $\rho>1$ (in practice a different
$\rho$ is taken for each $m$ and $n$).

######  Lemma 6.7.

Let $u\in\mathcal{X}_{\nu}^{M}$. For $m=1,\ldots,M$ and $|n|>N$,

$\displaystyle\left\|\left(G^{-1}(u)\right)^{(m)}_{n}\right\|_{C^{0}}\leq\left\|u_{n}^{(m)}\right\|_{C^{0}}+\sum_{l=1}^{m-1}\exp\left(2\tau_{m}\Re\left(\lambda^{(m)}_{n}\right)^{+}+2\sum_{j=l+1}^{m-1}\tau_{j}\Re\left(\lambda_{n}^{(j)}\right)\right)\left|u^{(l)}_{n}(1)\right|,$

where we again use the convention that empty sums are equal to $0$. Moreover,

$\displaystyle\left\|\Pi_{\infty}\left(G^{-1}(u)\right)^{(m)}\right\|_{\ell^{1}_{\nu}(C^{0})}\leq\left\|\Pi_{\infty}u^{(m)}\right\|_{\ell^{1}_{\nu}(C^{0})}+\sum_{l=1}^{m-1}e^{\mu_{m,l}}\left\|\Pi_{\infty}u^{(l)}\right\|_{\ell^{1}_{\nu}(C^{0})},$

where

$\mu_{m,l}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\sup_{n>N}\left(2\tau_{m}\Re\left(\lambda_{n}^{(m)}\right)^{+}+2\sum_{j=l+1}^{m-1}\tau_{j}\Re\left(\lambda_{n}^{(j)}\right)\right).$

###### Proof.

The proof of Lemma 6.7 follows directly from the definition of $G^{-1}$ and
the triangle inequality. ∎

###  6.3 How to adapt the bounds

In this subsection, we derive computable bounds $Y$, $Z$ and $W$ satisfying
assumptions (14)-(16) of Theorem 2.12, for the operator $T$, the space
$X=\mathcal{X}_{\nu}^{M}$ and an approximate solution $\bar{u}$ in
$\left(\mathcal{X}_{\nu}^{KN}\right)^{M}$. The dependencies in $F$ on the
various subdomains are mostly uncoupled, except for the matching conditions at
the boundaries, which means many of the estimations derived in Section 5 can
be re-used. Throughout this section we will comment sparsely on the
bookkeeping involved to obtain the “component” bounds $Y^{(m)}$, $Z_{i}^{(m)}$
and $W^{(m)}_{ij}$, which are needed for using Theorem 2.12 with $M>1$.

As was done in Section 5, each bound $Y^{(m)}$, $Z_{i}^{(m)}$ and
$W^{(m)}_{ij}$ will be split into three parts corresponding to the
decomposition (23) of $\mathcal{X}_{\nu}$, i.e.

$\displaystyle Y^{(m)}:=Y_{KN}^{(m)}+\left(\epsilon^{(m)}_{\infty
N}\right)^{-1}Y^{(m)}_{\infty
N}+\left(\epsilon^{(m)}_{\infty}\right)^{-1}Y^{(m)}_{\infty},\quad\text{for
all }1\leq m\leq M,$

and similarly for $Z^{(m)}_{i}$ and $W^{(m)}_{ij}$.

######  Remark 6.8.

Similarly to what we did in Section 5, it will be convenient to consider
arbitrary $h$ and $v$ in $B^{M}_{1,\epsilon}(0)$ in order to derive the bounds
$Z^{(m)}_{i}$ and $W^{(m)}_{ij}$. That is we are going to look for
$Z^{(m)}_{i}$ such that

$\displaystyle\left\|\pi^{(m)}DT(\bar{u})h\right\|_{\mathcal{X}_{\nu}}\leq\sum_{i=1}^{M}Z^{(m)}_{i}\left\|h^{(i)}\right\|_{\mathcal{X}_{\nu}},$

and for $W^{(m)}_{ij}$ such that, writing an arbitrary $u\in
B^{M}_{r_{*},\epsilon}(\bar{u})$ as $\bar{u}+r_{*}v$,

$\displaystyle\left\|\pi^{(m)}\left(DT(\bar{u}+r_{*}v)-DT(\bar{u})\right)h\right\|_{\mathcal{X}_{\nu}}\leq
r_{*}\sum_{i=1}^{M}\sum_{j=1}^{M}W^{(m)}_{ij}\left\|h^{(i)}\right\|_{\mathcal{X}_{\nu}}\left\|v^{(j)}\right\|_{\mathcal{X}_{\nu}}.$

####  6.3.1 Modifications for $Y_{KN}$

Each entry of $F_{KN}(\bar{u})$ can still be computed (or more precisely
enclosed), and so the generalization is straightforward: we simply consider

$\displaystyle
Y^{(m)}_{KN}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\left\|\pi^{(m)}\left(A_{KN}F_{KN}(\bar{u})\right)\right\|_{\mathcal{X}^{KN}_{\nu}},\quad
m=1,\ldots,M,$

with the $F$ and $A_{KN}$ of Section 6.1.

####  6.3.2 Modifications for $Y_{\infty N}$

On each subdomain $1\leq m\leq M$, we have to estimate $\left\|\Pi_{\infty
N}F^{(m)}(\bar{u})\right\|_{\mathcal{X}_{\nu}}$, for which we can reproduce
verbatim the analysis of Section 5.1.2. However, we also get an additional
term

$\Pi_{\infty N}G\Pi_{KN}A_{KN}F_{KN}(\bar{u})$

to estimate. We therefore consider
$\bar{w}=A_{KN}F_{KN}(\bar{u})\in\left(\mathcal{X}_{\nu}^{KN}\right)^{M}$, and
use Lemma 6.5 together with Remark 6.6 to get an upper bound for this
quantity.

####  6.3.3 Modifications for $Y_{\infty}$

We have to bound

$\pi^{(m)}\Pi_{\infty}\left(T(\bar{u})-\bar{u}\right)=\pi^{(m)}G^{-1}\left(\Pi_{\infty}F(\bar{u})\right),$

and to that aim we use the first part of Lemma 6.7, together with

$\displaystyle\left|F^{(m)}_{n}\left(\bar{u}\right)(1)\right|=\left|e^{2\tau_{m}\lambda_{n}^{(m)}}\bar{u}^{(m-1)}_{n}(1)+\tau_{m}\int_{-1}^{1}e^{(1-s)\tau_{m}\lambda_{n}^{(m)}}\gamma_{n}^{(m)}(\bar{u}(s))\mbox{d}s\right|,$
(42)

and

$\displaystyle\left\|F^{(m)}_{n}\left(\bar{u}\right)\right\|_{C^{0}}$
$\displaystyle\leq
e^{2\tau_{m}\Re\left(\lambda_{n}^{(m)}\right)^{+}}\left|\bar{u}^{(m-1)}_{n}(1)\right|+\left\|\gamma_{n}^{(m)}\left(\bar{u}\right)\right\|_{C^{0}}\frac{e^{2\tau_{m}\Re(\lambda_{n}^{(m)})}-1}{\Re(\lambda_{n}^{(m)})},$
(43)

for each $m=1,\ldots,M$ and $n>N$. Notice that, except maybe for $m=1$,
$\bar{u}^{(m-1)}_{n}=0$ because we only consider $|n|>N$, and that there are
only finitely many $n$ for which $\gamma_{n}^{(m)}\left(\bar{u}\right)$, and
hence $F^{(m)}_{n}\left(\bar{u}\right)$ , are non-zero. Therefore we can
compute the associated $\ell^{1}_{\nu}$ norm, which gives us the bounds
$Y^{(m)}_{\infty}$ for $1\leq m\leq M$.

####  6.3.4 Modifications for $Z_{KN}$

Up to using the new definition of $A$ and $\widehat{DF}$ from Section 6.1,
there is no significant change to the $\Pi_{KN}(I-A\widehat{DF})$ part of that
estimate, which remains a finite computation. In view of (15) we just need to
extract the different components $(Z_{KN})^{(m)}_{i}$, which is essentially a
bookkeeping task. Let $A_{KN}^{(m,m^{\prime})}$ be the block of the matrix
$A_{KN}$ linking the $m^{\prime}$-th to the $m$-th domain, and similarly for
$DF^{M}_{KN}(\bar{u})^{(m,m^{\prime})}$, then we need to compute
$\|I_{KN}-\sum_{m^{\prime}=1}^{M}A_{KN}^{(m,m^{\prime})}DF^{M}_{KN}(\bar{u})^{(m^{\prime},i)}\|_{B(\mathcal{X}_{\nu}^{KN},\mathcal{X}_{\nu}^{KN})}$
as the first part of the bound $(Z_{KN})^{(m)}_{i}$.

Regarding the $\Pi_{KN}\left(DF(\bar{u})-\widehat{DF}\right)h$ part, we can
reproduce verbatim the analysis of Section 5.2.1 on each subdomain separately.
There are no new terms coming from the matching conditions, since $\Pi_{\infty
N}h^{(m)}(1)=0$ for $1\leq m\leq M$. When determining the contribution of this
term to the bound $(Z_{KN})^{(m)}_{i}$, the only coupling between the domains
is through left-multiplication by $A_{KN}^{M}$.

####  6.3.5 Modifications for $Z_{\infty N}$

Regarding the $\Pi_{\infty N}\left(I-A\widehat{DF}\right)$ part of that
estimate, we again get an extra term of the form $\Pi_{\infty
N}G\left(\bar{w}\right)$, this time with

$\displaystyle\bar{w}=\left(I^{M}_{KN}-A_{KN}^{M}DF^{M}_{KN}(\bar{u})\right)h_{KN},$

for $h\in B^{M}_{1,\epsilon}(0)$. We can then easily estimate each
$|\bar{w}^{(m)}(1)|$, and subsequently use Lemma 6.5. The decomposition into
components $(Z_{\infty N})^{(m)}_{i}$ is a bookkeeping exercise as already
outlined in Section 6.3.4.

Regarding the $\Pi_{\infty
N}A\left(DF\left(\bar{u}\right)-\widehat{DF}\right)$ part of that estimate, we
also pick up another term of the form $\Pi_{\infty N}G\left(\bar{w}\right)$,
this time with

$\bar{w}=A_{KN}\Pi_{KN}\left(DF(\bar{u})-\widehat{DF}\right)h,$

for $h\in B^{M}_{1,\epsilon}(0)$. The
$\Pi_{KN}\left(DF(\bar{u})-\widehat{DF}\right)h$ part was already estimated in
(5.2.1), since all the extra terms related to the coupling appearing in
$DF(\bar{u})$ are also included in $\widehat{DF}$ and therefore cancel out. We
then simply have to multiply this by $|A_{KN}|$, evaluate the value at 1, use
Lemma 6.5, and do diligent bookkeeping to decompose the contributions to
$(Z_{\infty N})^{(m)}_{i}$.

####  6.3.6 Modifications for $Z_{\infty}$

We have to estimate, for all $m=1,\ldots,M$,

$\left\|\pi^{(m)}\Pi_{\infty}A\left(DF\left(\bar{u}\right)-\widehat{DF}\right)h\right\|_{\mathcal{X}^{\infty}_{\nu}}=\left\|\pi^{(m)}G^{-1}\left(\Pi_{\infty}w\right)\right\|_{\mathcal{X}^{\infty}_{\nu}}$

where

$\displaystyle w=DF(\bar{u})(h)-G(h),$

for $h\in B^{M}_{1,\epsilon}(0)$. For all $|n|>N$, and $1\leq m\leq M$, with
the notation $\tilde{g}^{(m,j)}$ inherited from Section 5.2.3,

$\displaystyle w^{(m)}_{n}(t)$
$\displaystyle=\tau_{m}\int_{-1}^{t}e^{\tau_{m}(t-s)\lambda^{(m)}_{n}}\left(\sum_{j=0}^{2R-1}(in)^{j}\left(\tilde{g}^{(m,j)}(s)\ast
h^{(m)}(s)\right)_{n}\right)\mbox{d}s,$

hence

$\displaystyle\left\|\Pi_{\infty}w^{(m)}\right\|_{\ell^{1}_{\nu}(C^{0})}\leq\vartheta_{\epsilon}\left\|h^{(m)}\right\|_{\mathcal{X}_{\nu}}\sum_{j=0}^{2R-1}\chi_{N}^{(m,j)}\left\|\tilde{g}^{(m,j)}\right\|_{\ell^{1}_{\nu}(C^{0})},\qquad\text{for
}m=1,\ldots,M,$

which can be combined with the second part of Lemma 6.7 to obtain the
$(Z_{\infty})^{(m)}_{i}$ estimates.

####  6.3.7 Modifications for $W_{KN}$

As was the case for the $Y$ and the $Z$ bounds, there is nothing to change in
the finite part of the $W$ bound, we can just re-use, component-wise, the
bounds from Section 5.3.1.

####  6.3.8 Modifications for $W_{\infty N}$

We get a final extra term of the form $\Pi_{\infty N}G\left(\bar{w}\right)$,
this time with

$\bar{w}=A_{KN}\Pi_{KN}\left(DF(\bar{u}+r_{*}v)-DF(\bar{u})\right)h,$

for $h,v\in B^{M}_{1,\epsilon}(0)$. The
$\Pi_{KN}\left(DF(\bar{u}+r_{*}v)-DF(\bar{u})\right)h$ part was already
estimated in (5.3.1). Starting back from there, and also generalizing the
notation $\Xi^{(j)}$ from (39) to multiple domains:

$\Xi^{(m,j)}\,\stackrel{{\scriptstyle\mbox{\tiny{\raisebox{0.0pt}[0.0pt][0.0pt]{def}}}}}{{=}}\,\begin{pmatrix}|Q^{(m)}|\,D^{(m)}_{N}(t_{0})\,|(Q^{(m)})^{-1}|\
|\mathcal{D}^{j}|\\\ \vdots\\\
|Q^{(m)}|\,D^{(m)}_{N}(t_{K})\,|(Q^{(m)})^{-1}|\
|\mathcal{D}^{j}|\end{pmatrix},$

we get, for all $m=1,\ldots,M$,

$\displaystyle\left|\Pi_{KN}\left(DF^{(m)}(\bar{u}+r_{*}v)-DF^{(m)}(\bar{u})\right)h\right|\leq$
$\displaystyle\qquad\qquad
r_{*}\sum_{j=0}^{2R-1}\Xi^{(m,j)}\,\Pi_{N}\left(\left|\left(g^{(j)}\right)^{\prime\prime}\right|\left(\|\bar{u}^{(m)}\|_{C^{0}}+r_{*}\|v^{(m)}\|_{C^{0}}\right)\ast\|v^{(m)}\|_{C^{0}}\ast\|h^{(m)}\|_{C^{0}}\right).$

We then define the “evaluation at $t=1$” operator
$E_{1}:\mathcal{X}^{KN}_{\nu}\to\mathbb{C}^{2N+1}$ by
$(E_{1}u)_{n}=u_{n0}+2\sum_{k=1}^{K}u_{nk}$, and use Lemmas 3.6, 4.1 and 4.5
to obtain the $(W_{\infty N})^{(m)}_{ij}$ estimates.

$\displaystyle|\bar{w}^{(m)}(1)|\leq$ $\displaystyle\quad
r_{*}\vartheta_{\epsilon}^{2}\sum_{m^{\prime}=1}^{M}\|v^{(m^{\prime})}\|_{\mathcal{X}_{\nu}}\|h^{(m^{\prime})}\|_{\mathcal{X}_{\nu}}\sum_{j=0}^{2R-1}\Upsilon\left(|E_{1}A^{(m,m^{\prime})}_{KN}|\,\Xi^{(m^{\prime},j)}\right)\,\left|\left(g^{(j)}\right)^{\prime\prime}\right|\left(\left\|\bar{u}^{(m^{\prime})}\right\|_{\ell^{1}_{\nu}(C^{0})}+\vartheta_{\epsilon}r^{*}\right),$
(44)

where $A^{(m,m^{\prime})}_{KN}$ is the block of the matrix $A_{KN}$ linking
the $m^{\prime}$-th to the $m$-th domain. Finally, we apply Lemma 6.5 to
obtain the $(W_{\infty N})^{(m)}_{ij}$ estimates.

####  6.3.9 Modifications for $W_{\infty}$

We have to estimate, for all $m=1,\ldots,M$,

$\left\|\pi^{(m)}\Pi_{\infty}A\left(DF\left(\bar{u}+r_{*}v\right)-DF\left(\bar{u}\right)\right)h\right\|_{\mathcal{X}^{\infty}_{\nu}}=\left\|\pi^{(m)}G^{-1}\left(\Pi_{\infty}z\right)\right\|_{\mathcal{X}^{\infty}_{\nu}},$

where

$z=\left(DF\left(\bar{u}+r_{*}v\right)-DF\left(\bar{u}\right)\right)h.$

Such a term was already estimated in (5.3.3), and we obtain

$\left\|\Pi_{\infty}z^{(m)}\right\|_{\ell^{1}_{\nu}(C^{0}}\leq\vartheta_{\epsilon}^{2}r_{*}\left\|v^{(m)}\right\|_{\mathcal{X}_{\nu}}\left\|h^{(m)}\right\|_{\mathcal{X}_{\nu}}\sum_{j=0}^{2R-1}\chi^{(m,j)}_{N}\left|\left(g^{(j)}\right)^{\prime\prime}\right|\left(\left\|\bar{u}^{(m)}\right\|_{\ell^{1}_{\nu}(C^{0})}+\vartheta_{\epsilon}r^{*}\right),$

for all $1\leq m\leq M$, which can be combined with the second part of Lemma
6.7 to obtain the $(W_{\infty})^{(m)}_{ij}$ estimates.

##  7 Applications

In this section, we present some applications of the validation procedure
introduced in this paper, and discuss the obtained results. First, using the
domain decomposition approach, we showcase the broad applicability of our
method in Section 7.1, by using it to rigorously integrate several PDEs of the
form (1) having different properties and different dynamics. In Section 7.2,
we highlight the influence of one of the most significant aspects of this
work, namely the solution-adapted choice of the operator $\mathcal{L}$ made in
Section 3.4, by reproducing some of the proofs from Section 7.1 but now with a
trivial choice of $\mathcal{L}$, and by comparing the results. In Section 7.3,
we discuss the merits of time stepping as an alternative to domain
decomposition.

###  7.1 Different examples of equations

We start with what is arguably the simplest nonlinear parabolic equation.

######  Theorem 7.1.

Consider the Fisher-KPP equation

$\displaystyle\begin{cases}\dfrac{\partial u}{\partial
t}=\dfrac{\partial^{2}u}{\partial
x^{2}}+u(1-u),&(t,x)\in(0,t_{\text{{end}}}]\times[0,L],\\\\[8.61108pt]