# A Posteriori Error Estimates for Energy-Based Quasicontinuum Approximations
of a Periodic Chain

Hao Wang  Hao Wang, Oxford University Mathematical Institute, 24-29 St Giles’,
Oxford, OX1 3LB, UK<EMAIL_ADDRESS>
###### Abstract.

We present a posteriori error estimates for a recently developed
atomistic/continuum coupling method, the Consistent Energy-Based QC Coupling
method. The error estimate of the deformation gradient combines a residual
estimate and an a posteriori stability analysis. The residual is decomposed
into the residual due to the approximation of the stored energy and that due
to the approximation of the external force, and are bounded in negative
Sobolev norms. In addition, the error estimate of the total energy using the
error estimate of the deformation gradient is also presented. Finally,
numerical experiments are provided to illustrate our analysis.

##  1\. Introduction

Quasicontinuum (QC) methods, or in general atomistic/continuum coupling
methods, are a class of multiscale methods for coupling an atomistic model of
a solid with a continuum model. These methods have been widely employed in
computational nano-technology, where a fully atomistic model will result in a
prohibitive computational cost but an exact configuration is required in a
certain region of the material. In this situation, atomistic model is applied
in the region which contains the defect core to retain certain accuracy, while
continuum model is applied in the far field to reduce the computational cost.

A number of QC methods have been developed in the past decades and are
classified in two groups: energy-based coupling methods and force-based
coupling methods. Despite the fact that the force-based methods are easy to
implement and extend to higher dimensional cases, energy-based methods have
certain advantages. For example, the forces derived from an energy potential
are conservative which could leads to a faster convergence rate in
computation, and the energy of an atomistic system can also be a quantity of
interest in real application. However, consistent energy-based coupling
methods can be tedious and restrictive on the shape of the coupling interface
in more than one dimension (see [11, 4]) and it was not until recent that a
practical consistent energy-based coupling method was created by Shapeev [10],
which is the Consistent Energy-Based QC Coupling method that we analyze in the
present paper.

A number of literature on the rigorous analysis of different QC methods have
been proposed since the first one by Lin [5]. However, most of the analysis
are on the a priori error analysis, and only a few are on the a posteriori
error analysis. Arndt and Luskin give a posteriori error estimates for the QC
approximation of a Frenkel-Kontorova model [2, 3, 1]. A goal-oriented approach
is used and error estimates on different quantity of interests, each of which
is essentially the difference between the values of a linear functional at the
atomistic solution and the QC solution, are proposed. The estimates are
decomposed into two parts, one is used to correctly chose the atomistic region
and another is used to optimally choose the mesh in the continuum region.
Serge et al. [9] give error estimates, also through a goal-oriented approach,
of the original energy-based QC approximation, whose consistency is not
guaranteed. Both of the above works employ the technique of deriving and
solving dual problems as a result of the goal-oriented approach. Ortner and
Süli [7] derive an a posteriori error indicator for a global norm through a
similar approach as ours. However, the QC method analyzed there does not
contain an approximation of the stored energy which is essentially different
from the QC method we are interested.

The present paper provides the a posteriori error analysis for the Consistent
Energy-Based QC Coupling method [10] for a one dimensional periodic chain with
nearest and next nearest neighbour interactions. The formulation of the QC
approximation has the feature that the finite element nodes in the continuum
region are not restricted to reside at the atomistic positions, which creates
the situation that interaction bonds often cross the element boundaries, which
is common in two dimensional formulation. We then derive the residual in
negative Sobolev norms and then the a posteriori stability constant as a
function of the QC solution. The error estimator of the deformation gradient
in $L_{2}$-norm is then obtained by combining these two analysis. In addition,
we derive an error estimator for the total energy difference by using that of
the deformation gradient. It should be remarked that though both of the error
estimators are global quantities, they consist of contributions from element.
As a result, an adaptive mesh refinement algorithm is developed and applied to
a problem that mimics the vacancy in the two dimensional case, and the
numerical results are presented.

###  1.1. Outline

In Section 2, we first formulate the atomistic model through both a continuous
approach, i.e., the deformation and the displacement are considered as
continuous functions on the reference lattice, and a discrete approach, which
is always taken in previous literature. We then formulate the Consistent
Energy-Based QC Coupling method in one dimensional setting.

In Section 3, we derive the residual estimates for the Consistent Energy-Based
QC Coupling method in a negative Sobolev norm. The residual is split into two
part, one is due to the approximation of the stored energy and the other is
due to the approximation of the external force.

In Section 4, we give the a posteriori stability analysis.

In Section 5, we combine the residual estimate and the stability analysis to
give the a posteriori error estimate of the deformation gradient in
$L_{2}$-norm and that of the total energy.

In Section 6, we present a numerical example to complement our analysis.

##  2\. Model Problem and QC Approximation

###  2.1. Atomistic Model

As opposed to taking only a discrete point of view in many QC researches, we
use both continuous functions and discretized vectors to denote the
displacement and the deformation. The reason for doing this is that the
Consistent Energy-Based QC coupling method, which we analyze in this paper, is
easily formulated through the continuous approach, while discrete formulations
could make the residual analysis of the external forces much easier.

For an infinite reference lattice with atomistic spacing $\varepsilon$, we
make the partition
$\mathcal{T}^{\varepsilon}=\\{T^{\varepsilon}_{\ell}\\}_{\ell=-\infty}^{\infty}$
of the domain $\mathbb{R}$ such that
$\mathbb{R}=\cup_{\ell=-\infty}^{\infty}T^{\varepsilon}_{\ell}$ and
$T^{\varepsilon}_{\ell}=[(\ell-1)\varepsilon,\ell\varepsilon]$. We then define
the displacement and deformation of this infinite lattice to be continuous
piecewise linear functions $u$,
$y\in\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})\cap C^{0}(\mathbb{R})$. We use
$\boldsymbol{u}$ and $\boldsymbol{y}$ to denote the vectorizations of $u$ and
$y$ such that $u_{\ell}=u(\ell\varepsilon)$ and $y_{\ell}=y(\ell\varepsilon)$.
We know that $u_{\ell}$ and $y_{\ell}$ are the physical displacement and
deformation of atom $\ell$ respectively.

To avoid technical difficulties with boundaries, we apply periodic boundary
conditions. We rescale the problem so that there are $N\in\mathbb{N}$ atoms in
each period and $\varepsilon=1/N$, which implies that $u$ and $y$ are
1-periodic functions and $\boldsymbol{u}$ and $\boldsymbol{y}$ are
$N$-periodic vectors. We also impose a zero-mean condition to the admissible
space of displacements, which is defined to be

$\mathcal{U}=\big{\\{}u\in\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})\cap
C^{0}(\mathbb{R}):u(x+1)=u(x)\text{ and }\int_{0}^{1}u(x)\,{\rm
d}x=0\big{\\}}.$ (2.1)

The set of admisible deformations is given by

$\mathcal{Y}=\big{\\{}y\in\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})\cap
C^{0}(\mathbb{R}):y(x)=Fx+u(x),u\in\mathcal{U}\big{\\}},$ (2.2)

where $F>0$ is a given macroscopic deformation gradient.

As we mentioned above, it is necessary in the analysis of the external forces
to employ the discretization of the displacement and the deformation.
Therefore, by the relationship between $u,y$ and their vectorizations
$\boldsymbol{u},\boldsymbol{y}$, the discrete space of displacement and the
admissible set of deformation are defined by

$\mathcal{U}^{\varepsilon}=\\{\boldsymbol{u}\in\mathbb{R}^{\mathbb{Z}}:u_{\ell+N}=u_{\ell},\varepsilon\sum_{\ell=1}^{N}u_{\ell}=0\\},$
(2.3)

and

$\mathcal{Y}^{\varepsilon}=\\{\boldsymbol{y}\in\mathbb{R}^{\mathbb{Z}}:y_{\ell+N}=F\ell\varepsilon+u_{\ell},\boldsymbol{u}\in\mathcal{U}^{\varepsilon}\\},$
(2.4)

where the zero-mean condition on the displacements, i.e.,
$\varepsilon\sum_{\ell=1}^{N}u_{\ell}=0$ is obatined by applying the
trapezoidal rule to evaluate the integration $\int_{0}^{1}u(x)\,{\rm d}x$ with
respect to the partition $\mathcal{T}^{\varepsilon}$ and using the periodicity
of $u$.

For simplicity of analysis, we adopt a pair interaction model and assume that
only nearest neighbours and the next-nearest neighbours interact. With a
slight abuse of notation, the stored atomistic energy (per period) of an
admissible deformation is then given by

$\displaystyle\mathcal{E}_{\rm a}(y)$
$\displaystyle:=\varepsilon\sum_{\ell=1}^{N}\phi\Big{(}\frac{y(\ell\varepsilon)-y((\ell-1)\varepsilon)}{\varepsilon}\Big{)}+\varepsilon\sum_{\ell=1}^{N}\phi\Big{(}\frac{y(\ell\varepsilon)-y((\ell-2)\varepsilon)}{\varepsilon}\Big{)}$
$\displaystyle=\varepsilon\sum_{\ell=1}^{N}\phi\Big{(}\frac{y_{\ell}-y_{\ell-1}}{\varepsilon}\Big{)}+\varepsilon\sum_{\ell=1}^{N}\phi\Big{(}\frac{y_{\ell}-y_{\ell-2}}{\varepsilon}\Big{)}=:\mathcal{E}_{\rm
a}(\boldsymbol{y}),$ (2.5)

where $\phi\in C^{3}((0,+\infty))$ is a Lennard-Jones type interaction
potential. We assume that there exists $r_{\ast}>0$ such that $\phi$ is convex
in $(0,r_{\ast})$ and concave in $(r_{\ast},+\infty)$.

For the formulation of the external energy, we first define the linear nodal
interpolation operator
$I_{\varepsilon}:C^{0}(\mathbb{R})\rightarrow\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})\cap
C^{0}(\mathbb{R})$ such that

$I_{\varepsilon}g(\ell\varepsilon)=g(\ell\varepsilon)\quad\forall g\in
C^{0}(\mathbb{R}).$ (2.6)

Then given a dead load $f\in\mathcal{U}$, we define the external energy (per
period) caused by $f$ to be

$\langle
f,u\rangle_{\varepsilon}:=\int_{0}^{1}I_{\varepsilon}(fu)dx=\sum_{\ell=1}^{N}\varepsilon
f_{\ell}u_{\ell}=:\langle\boldsymbol{f},\boldsymbol{u}\rangle_{\varepsilon},$
(2.7)

where $\boldsymbol{f}$ and $\boldsymbol{u}$ are the vectorizations of the
external force $f$ and the displacement $u$ according to
$\mathcal{T}^{\varepsilon}$.

Thus, the total energy (per period) under a deformation $y\in\mathcal{Y}$ is
given by

$E_{\rm a}(y;F)=\mathcal{E}_{\rm a}(y)-\langle f,u\rangle_{\varepsilon},$

as $u$ is determined by $y$ and $F$. However, in our analysis, we always
assume that $F$ is given and as a result, we simply write $E_{\rm a}(y;F)$ as
$E_{\rm a}(y)$.

The problem we wish to solve is to find

$y_{\rm a}\in{\rm argmin}E_{\rm a}(\mathcal{Y}),$ (2.8)

where ${\rm argmin}$ denotes the set of local minimizers.

###  2.2. Notation of Partitions, Norms and Discrete Derivatives

Though it is natural to introduce the QC approximation after the atomistic
model, we decide to pause here and introduce some important notation that are
used throughout the paper in order to make the flow of the paper more smooth
and save some space.

In Section 2.1, we have introduced the partition $\mathcal{T}^{\varepsilon}$
of the domain $\mathbb{R}$. We now fix the notation for a generalized
partition.

Let $\mathcal{T}^{m}=\\{T_{k}^{m}\\}_{k=-\infty}^{\infty}$ be a given
partition such that $T_{k}^{m}=[x^{m}_{k-1},x^{m}_{k}]$, where
$x^{m}_{k}>x^{m}_{k-1}$ are the nodes of the partition. We denote the size (or
the length) of the $k$’th element by
$\varepsilon^{m}_{k}:=|T_{k}^{m}|=x^{m}_{k}-x^{m}_{k-1}$. We also define the
mesh size vector $\boldsymbol{\varepsilon}^{m}$ such that
$\boldsymbol{\varepsilon}^{m}:=(\varepsilon^{m}_{k})_{k=-\infty}^{\infty}\in(\mathbb{R}^{+})^{\mathbb{Z}}$.

Given a partition $\mathcal{T}^{m}$ and a function $g\in C^{0}(R)$, we define
the $\mathcal{P}_{1}$ direct interpolation
$I_{m}:C^{0}(\mathbb{R})\rightarrow\mathcal{P}_{1}(\mathcal{T}^{m})\cap
C^{0}(\mathbb{R})$ by

$(I_{m}g)(x^{m}_{i})=g(x^{m}_{i})\quad\forall g\in C^{0}(\mathbb{R}),$ (2.9)

and $I_{m}g$ is often denoted by $g_{m}$. We also denote the vectorization of
$g\in C^{0}(\mathbb{R})$ with respect to $\mathcal{T}^{m}$ by
$\boldsymbol{g}^{m}$ such that

$g^{m}_{j}=g(x^{m}_{j}).$ (2.10)

Let $\mathcal{D}$ be a subset of $\mathbb{Z}$. For a vector
$\boldsymbol{v}\in\mathbb{R}^{\mathbb{Z}}$ and a partition $\mathcal{T}^{m}$,
we define the (semi-)norms

$\displaystyle\|\boldsymbol{v}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{m}}(\mathcal{D})}=\left\\{\begin{array}[]{l
l}\Big{(}\sum_{\ell\in\mathcal{D}}\varepsilon^{m}_{\ell}|v_{\ell}|^{p}\Big{)}^{1/p},&1\leq
p<\infty,\\\ \max_{\ell\in\mathcal{D}}|v_{\ell}|,&p=\infty.\end{array}\right.$

In particular, if $n_{m}$ is the number of the nodes of $\mathcal{T}^{m}$ that
are in $[0,1]$, we simply define

$\displaystyle\|\boldsymbol{v}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{m}}}=\left\\{\begin{array}[]{l
l}\Big{(}\sum_{\ell=1}^{n_{m}}\varepsilon^{m}_{\ell}|v_{\ell}|^{p}\Big{)}^{1/p},&1\leq
p<\infty,\\\
\max_{\ell=1,\ldots,n_{m}}|v_{\ell}|,&p=\infty.\end{array}\right.$

We now define discrete derivatives. Suppose $v\in C^{0}(\mathbb{R})$ and
$\boldsymbol{v}^{m}$ is its vectorization according to $\mathcal{T}^{m}$. We
define the first and second order discrete derivative
${\boldsymbol{v}^{m}}^{\prime}$ by

${v^{m}}^{\prime}_{k}=\frac{v^{m}_{j}-v^{m}_{j-1}}{\varepsilon^{m}_{j}},\text{
and
},{v^{m}}^{\prime\prime}_{k}=\frac{{v^{m}}^{\prime}_{j+1}-{v^{m}}^{\prime}_{j}}{\bar{\varepsilon}^{m}_{j}},$
(2.11)

where
$\bar{\varepsilon}^{m}_{j}:={\textstyle\frac{1}{2}}(\varepsilon^{h}_{j}+\varepsilon^{h}_{j+1})$.

It can be proved that for $v^{m}\in
C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{m})$ and
$\boldsymbol{v}^{m}$ being its vectorization, we have the identity

$\|{v^{m}}^{\prime}\|_{L^{p}[0,1]}=\|{\boldsymbol{v}^{m}}^{\prime}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{m}}}.$
(2.12)

Since $\mathcal{T}^{\varepsilon}$ is special and uniform, we simply use
$\boldsymbol{\varepsilon}$ and $\varepsilon$ to denote its mesh size vector
and mesh size without superscripts and subscripts.

In addition to these, we denote the left and the right limit of an open
interval $\omega$ by $L_{\omega}$ and $R_{\omega}$, which are also used later
in our analysis.

###  2.3. QC Approximation

The QC approximation we analyze in this paper is essentially the Consistent
Atomistic/Continuum Coupling method developed in [10]. We briefly redevelop
this method in 1D so that it is easily understood and enough for us to carry
out the analysis.

We first decompose the reference lattice, which occupies $\mathbb{R}$, into an
atomistic region $\Omega_{\rm a}$, which should contain any ’defects’, and a
continuum region $\Omega_{c}$, where the solution is expected to be smooth.
Moreover, we assume $\Omega_{\rm a}$ to be a union of open intervals and
$\Omega_{c}$ to be a union of closed intervals, and $\Omega_{\rm
a}\cup\Omega_{c}=\mathbb{R}$. Since we impose periodic boundary conditions on
the displacement and notationally it is easier to assume the atomistic region
is away from the boundary of the period we analyze, we make the following
assumptions on $\Omega_{\rm a}$ and $\Omega_{c}$:

  * •

$\Omega_{\rm a}$ and $\Omega_{c}$ appear periodically with exactly period of
$1$, i.e, if $x\in\Omega_{\rm a}$ then $x+1\in\Omega_{\rm a}$ and the same for
$\Omega_{c}$.

  * •

$\exists\delta>2\varepsilon$ such that
$\big{(}(0,\delta)\cup(1-\delta,1)\big{)}\subset\Omega_{c}$, i.e., the
atomistic region is contained in the ’middle’ of the chain.

Note that, there is no such a restriction, that the interfaces where different
regions meet should lie on the positions of the atoms, i.e., it is not
necessary that $\Omega_{c}\cap\Omega_{\rm a}\in\varepsilon\mathbb{Z}$, which
was always assumed in previous modeling and analysis of 1D QC method.

Then, in order to reduce the number of degrees of freedom, we make the
partition $\mathcal{T}^{h}=\\{T^{h}_{k}\\}_{k=-\infty}^{\infty}$ of the domain
$\mathbb{R}$ according to the above region decomposition of $\mathbb{R}$ as
follows:

  * •

$T_{k}^{h}=[x^{h}_{k},x^{h}_{k-1}]$ and
$T_{k+K}=[x^{h}_{k}+1,x^{h}_{k-1}+1]=[x^{h}_{k+K},x^{h}_{k-1+K}]$, which
implies that the partition is $K$-periodic with $|\cup_{k=1}^{K}T_{k}|=1$, and
there are $K$ elements in each $[0,1]$. We also assume that $x^{h}_{1}$ is the
left most node and $x^{h}_{K}$ is the right most node in $[0,1]$.

  * •

If $\ell\varepsilon\in\Omega_{\rm a}$, then $\exists i\in\mathbb{Z}$ such that
$x_{i}=\ell\varepsilon$, i.e., every position of an atom in the atomistic
region is a node of this partition.

  * •

$\partial\Omega_{c}$ is a node in this partition which means that each element
is contained in only one of the two regions.

  * •

$|T_{k}^{h}|=\varepsilon^{h}_{k}\geq 2\varepsilon$ if
$T^{h}_{k}\subset\Omega_{c}$, i.e., the size of each element in the continnum
region is larger than or equal to $2\varepsilon$.

We emphasize two definitions

$\ell_{k}:=\max_{\ell}\\{\ell:\ell\varepsilon\leq x^{h}_{k}\\}\text{ and
}\theta_{k}:=\frac{x^{h}_{k}-\ell_{k}\varepsilon}{\varepsilon},$ (2.13)

which are extensively used in the analysis and significantly simply the
notation. Note that $0\leq\theta_{k}\leq 1$.

Based on this partition of the domain, the QC space of displacement and the QC
set of admissible deformation are defined by

$\mathcal{U}_{{\rm qc}}=\big{\\{}u\in\mathcal{P}_{1}(\mathcal{T}^{h})\cap
C^{0}(\mathbb{R}):u(x+1)=u(x)\text{ and }\int_{0}^{1}u(x)\,{\rm
d}x=0\big{\\}},$ (2.14)

and

$\mathcal{Y}_{{\rm qc}}=\big{\\{}y\in\mathcal{P}_{1}(\mathcal{T}^{h})\cap
C^{0}(\mathbb{R}):y(x)=Fx+u(x),u\in\mathcal{U}_{{\rm qc}}\big{\\}}.$ (2.15)

The discrete QC space of displacement and the QC set of admissible deformation
are defined by

$\mathcal{U}^{h}_{{\rm
qc}}=\big{\\{}\boldsymbol{u}^{h}\in\mathbb{R}^{\mathbb{Z}}:u^{h}_{k}=u^{h}_{k+K},\forall
k\in\mathbb{Z},\text{ and
}\sum_{k=1}^{K}\frac{1}{2}(x^{h}_{k+1}-x^{h}_{k-1})u^{h}_{k}=0\big{\\}},$
(2.16)

and

$\mathcal{Y}^{h}_{{\rm
qc}}=\big{\\{}\boldsymbol{y}^{h}\in\mathbb{R}^{\mathbb{Z}}:y^{h}_{k}=Fx_{k}+u^{h}_{k},\boldsymbol{u}\in\mathcal{U}^{h}_{{\rm
qc}}\big{\\}}.$ (2.17)

Note that unlike $\mathcal{U}^{\varepsilon}$ and $\mathcal{Y}^{\varepsilon}$,
in which every vector has the physical displacements and deformations of the
atoms as its components, $\mathcal{U}^{h}$ and $\mathcal{Y}^{h}$ only contain
vectors whose components are the values of displacements and deformations at
the nodes of $\mathcal{T}^{h}$.

The approach to couple the atomistic and continuum energy is to associate the
energy with interaction bonds. The term bond between atoms $i\in\mathbb{Z}$
and $i+r\in\mathbb{Z}$ refer to the open interval
$b=(i\varepsilon,(i+r)\varepsilon)$. In our case, since only nearest neighbour
and next nearest neighbour bonds are taken into account, $r=1,2$ only.

To develop the coupling method, we define the operator $D_{\omega}y$ for an
open interval $\omega=(L_{\omega},R_{\omega})\subset\mathbb{R}$ and $y\in
C^{0}(\mathbb{R})$ such that

$D_{\omega}y:=\frac{1}{|\omega|}\big{(}y(R_{\omega})-y(L_{\omega})\big{)}.$
(2.18)

If we take any $y\in C^{0}(\mathbb{R})\cap W^{1,\infty}(\mathbb{R})$ as a
deformation (note that this ’deformation’ might be non-physical) and a bond
$b=(i\varepsilon,(i+r_{b})\varepsilon)$, we can define the atomistic energy
contribution of bond $b$ to the stored energy to be

$a_{b}(y)=\frac{|b\cap\Omega_{\rm
a}|}{r_{b}}\phi\big{(}r_{b}D_{b\cap\Omega_{\rm a}}y\big{)},$ (2.19)

and its continuum energy contribution to the stored energy to be

$c_{b}(y)=\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x,$ (2.20)

where $\nabla_{r_{b}}y=r_{b}y^{\prime}(x)$.

Since we are only interested in the situation in $[0,1]$, which is extended
periodically to the whole domain, the set of bonds that we will consider is

$\mathcal{B}=\big{\\{}(i\varepsilon,(i+r)\varepsilon):r=1,2,i=0,1,\ldots,N-1\big{\\}}.$
(2.21)

Therefore, coupling the two energy contributions together, the stored QC
energy (per period) of a deformation $y\in C^{0}(\mathbb{R})\cap
W^{1,\infty}(\mathbb{R})$ is then given by

$\mathcal{E}_{{\rm qc}}(y)=\sum_{b\in B}\big{[}a_{b}(y)+c_{b}(y)\big{]},$
(2.22)

which was shown in [10] to be a consistent coupling method, where the
definition of consistency is as follows:

$\mathcal{E}^{\prime}_{\rm a}(Fx)[v]=\mathcal{E}^{\prime}_{{\rm
qc}}(Fx)[v]=0\quad\forall v\in C^{0}(\mathbb{R})\cap
W^{1,\infty}(\mathbb{R}).$ (2.23)

Given a dead load $f\in\mathcal{U}$ the QC approximation of the external
energy (per period) caused by $f$ is given by

$\langle f,u_{h}\rangle_{h}:=\int_{0}^{1}I_{h}(fu_{h})\,{\rm
d}x=\sum_{k=1}^{K}\frac{1}{2}(x^{h}_{k+1}-x^{h}_{k-1})f^{h}_{k}u^{h}_{k}=:\langle\boldsymbol{f}^{h},\boldsymbol{u}^{h}\rangle_{h},$
(2.24)

where $I_{h}$ is the linear nodal interpolation with respect to
$\mathcal{T}^{h}$ , and $\boldsymbol{f}^{h}$ and $\boldsymbol{u}^{h}$ are the
vectorizations of $f$ and $u_{h}$.

Thus, the total energy (per period) of a deformation
$y_{h}\in\mathcal{Y}_{{\rm qc}}$ is given by

$E_{{\rm qc}}(y_{h};F)=\mathcal{E}_{{\rm qc}}(y_{h})-\langle
f,u_{h}\rangle_{h}.$

For the same reason that $F$ is given, we write $E_{{\rm qc}}(y_{h};F)$ as
$E_{{\rm qc}}(y_{h})$. The problem we wish to solve is to find

$y_{{\rm qc}}\in{\rm argmin}E_{{\rm qc}}(\mathcal{Y}_{{\rm qc}}).$ (2.25)

##  3\. Residual Analysis

In this section, we bound the residual in a negative Sobolev norms. We equip
the space $\mathcal{U}$ with the Sobolev norm

$\|v\|_{\mathcal{U}^{1,2}}=\|v^{\prime}\|_{L^{2}[0,1]},\quad\text{ for
}v\in\mathcal{U},$

and denote it by $\mathcal{U}^{1,2}$. The norm on the dual
$\mathcal{U}^{-1,2}:=(\mathcal{U}^{1,2})^{\ast}$ is defined by

$\|T\|_{\mathcal{U}^{-1,2}}:=\sup_{\begin{subarray}{c}v\in\mathcal{U}\\\
\|v\|_{\mathcal{U}^{1,2}}=1\end{subarray}}T[v],\quad\text{ for
}T\in\mathcal{U}^{-1,2}.$

In the following sections, we formulate the problems in variational forms and
then analyze the residual.

###  3.1. Variational Formulation and Residual

Let $y_{\rm a}$ be a solution of the atomistic problem (2.8). If ${y_{\rm
a}}^{\prime}(x)>0$ on $[0,1]$, $\mathcal{E}_{\rm a}(y)$ has the variational
derivative at $y_{\rm a}$ and therefore, the first order optimality condition
for (2.8) in variational form is

$\mathcal{E}^{\prime}_{\rm a}(y_{\rm a})[v]=\langle
f,v\rangle_{\varepsilon}\qquad\forall v\in\mathcal{U},$ (3.1)

where

$\mathcal{E}^{\prime}_{\rm a}(y_{\rm
a})[v]=\varepsilon\sum_{b\in\mathcal{B}}\phi^{\prime}(r_{b}D_{b}y_{\rm
a})r_{b}D_{b}v.$ (3.2)

Let $y_{{\rm qc}}$ be a solution of the QC problem (2.25). If ${y_{{\rm
qc}}}^{\prime}(x)>0$ on $[0,1]$, then $\mathcal{E}_{{\rm qc}}(y)$ has the
variational derivative at $y_{{\rm qc}}$ and the first order optimality
condition for (2.25) in variational form is

$\mathcal{E}^{\prime}_{{\rm qc}}(y_{{\rm qc}})[v_{h}]=\langle
f,v_{h}\rangle_{h}\quad\forall v_{h}\in\mathcal{U}_{{\rm qc}},$ (3.3)

where

$\displaystyle\mathcal{E}^{\prime}_{{\rm qc}}(y_{h})[v_{h}]$
$\displaystyle=\sum_{b\in
B}\big{[}a^{\prime}_{b}(y_{h})[v_{h}]+c_{b}(y_{h})[v_{h}]\big{]}$
$\displaystyle=|b\cap\Omega_{\rm a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm
a}}y_{h})D_{b\cap\Omega_{\rm
a}}v_{h}+\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v_{h}\,{\rm
d}x\quad\forall v_{h}\in\mathcal{U}_{{\rm qc}}.$ (3.4)

In conforming finite element analysis, where the finite element solution space
is a subspace of the original solution space, the residual is defined as the
quantity we obtain by inserting the computed solution to the equation which
the real solution satisfies. However, in our case, $\mathcal{Y}_{{\rm qc}}$ is
in general not a subspace of $\mathcal{Y}_{\rm a}$, and hence the functional
$\mathcal{E}_{\rm a}(\cdot)$ is not defined on $\mathcal{Y}_{{\rm qc}}$ in
general and $\mathcal{E}_{{\rm qc}}(\cdot)$ is not defined on
$\mathcal{Y}_{\rm a}$ either. The way through which we circumvent this
difficulty is to define mappings between the solution spaces so that the
residual could be well defined. In concrete, we define
$J_{\mathcal{U}}:\mathcal{U}\rightarrow\mathcal{U}_{{\rm qc}}$ and
$J_{\mathcal{U}_{{\rm qc}}}:\mathcal{U}_{{\rm qc}}\rightarrow\mathcal{U}$ such
that

$J_{\mathcal{U}}u=I_{h}u-\frac{1}{2}\sum_{\ell=1}^{K}(x^{h}_{k+1}-x^{h}_{k-1})u(x^{h}_{k})\quad\forall
u\in\mathcal{U},$ (3.5)

and

$J_{\mathcal{U}_{{\rm
qc}}}u_{h}=I_{\varepsilon}u_{h}-\varepsilon\sum_{\ell=1}^{N}u_{h}(\ell\varepsilon)\quad\forall
u_{h}\in\mathcal{U}_{{\rm qc}}.$ (3.6)

It is easy to check that $J_{\mathcal{U}}u$ and $J_{\mathcal{U}_{{\rm
qc}}}u_{h}$ satisfy the corresponding mean zero condition of $\mathcal{U}$ and
$\mathcal{U}_{{\rm qc}}$, which implies that
$J_{\mathcal{U}}u\in\mathcal{U}_{{\rm qc}}$ and $J_{\mathcal{U}_{{\rm
qc}}}u_{h}\in\mathcal{U}$. With a slight abuse of notation, we define

$J_{\mathcal{U}}y=Fx+J_{\mathcal{U}}u=Fx+I_{h}u-\frac{1}{2}\sum_{\ell=1}^{K}(x^{h}_{k+1}-x^{h}_{k-1})u(x^{h}_{k})=I_{h}y-\frac{1}{2}\sum_{\ell=1}^{K}(x^{h}_{k+1}-x^{h}_{k-1})u(x^{h}_{k})\quad\forall
y\in\mathcal{U},$ (3.7)

and

$J_{\mathcal{U}_{{\rm qc}}}y_{h}=Fx+J_{\mathcal{U}_{{\rm
qc}}}u_{h}=Fx+I_{\varepsilon}u_{h}-\frac{1}{2}\varepsilon\sum_{\ell=1}^{N}u_{h}(\ell\varepsilon)=I_{\varepsilon}y-\frac{1}{2}\varepsilon\sum_{\ell=1}^{N}u_{h}(\ell\varepsilon)\quad\forall
y_{h}\in\mathcal{U}_{{\rm qc}}.$ (3.8)

We then define the residual (at the solution $y_{{\rm qc}}$) to be

$\displaystyle R[v]$ $\displaystyle=E^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{{\rm qc}})[v]$ $\displaystyle=E^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{{\rm qc}})[v]-E^{\prime}_{{\rm qc}}(y_{{\rm qc}})[v]$
$\displaystyle=\big{[}\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{{\rm qc}})[v]-\langle
f,v\rangle_{\varepsilon}\big{]}-\big{[}\mathcal{E}^{\prime}_{{\rm qc}}(y_{{\rm
qc}})[J_{\mathcal{U}}v]-\langle f,J_{\mathcal{U}}v\rangle_{h}\big{]}$
$\displaystyle=\big{[}\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{{\rm qc}})[v]-\mathcal{E}^{\prime}_{{\rm qc}}(y_{{\rm
qc}})[J_{\mathcal{U}}v]\big{]}+\big{[}\langle
f,J_{\mathcal{U}}v\rangle_{h}-\langle f,v\rangle_{\varepsilon}\big{]}.$ (3.9)

and understand $R$ as a functional in $\mathcal{U}^{-1,2}$. By this
formulation, we essentially split the residual into two parts: the first part
is the residual of the stored energy and the second part is the residual of
the external force. We will bound these two parts in the following sections.

###  3.2. Estimate of the Residual of the Stored Energy

In this section, we analyze the first part of (3.1), which is the residual of
the stored energy.

Before we give the theorem, we make several definitions that simplify our
notation.

First, we define the set $\mathcal{K}_{c}$ to be

$\mathcal{K}_{c}:=\big{\\{}k:k\in\\{1,\ldots,K\\}\text{ such that
}T_{k}\cap[0,1]\neq\emptyset\text{ but }T_{k}\cap(1,+\infty)=\emptyset\text{
and }T_{k}\subset\Omega_{c}\big{\\}},$ (3.10)

which is essentailly the set of indices of the elements in the continuum
region in $[0,1]$.

Second, suppose the atomistic region consists of $M$ disjoint subregions in
$[0,1]$, i.e., $\Omega_{\rm a}\cap[0,1]=\cup_{i=1}^{M}\Omega^{i}_{\rm a}$
among which $\Omega^{i}_{\rm a}\cap\Omega^{j}_{\rm a}=\emptyset$ if $i\neq j$,
we define the nodes lie on the atomistic-continuum interface of the atomistic
regions be $x^{h}_{La_{i}}$, $i=1,\ldots,M$ and those lie on the right
interface be $x^{h}_{Rc_{i}}$, $i=1,\ldots,M$.

Third, we define $\mathcal{K}^{\prime}_{c}\subset\mathcal{K}_{c}$ to be the
set of indices of the elements in the continuum region but not adjacent to an
atomistic region, i.e., $\forall k\in\mathcal{K}^{\prime}_{c}$, $k\neq
L_{a_{i}}$ and $k-1\neq R_{a_{i}}$, $\forall i\in\\{1,2,\ldots,M\\}$.

Using these definitions, we have the following theorem.

Theorem 1.  For $y_{h}\in\mathcal{Y}$ with $y_{h}^{\prime}(x)>0$, we have

$\big{\|}\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})[\cdot]-\mathcal{E}^{\prime}_{{\rm
qc}}(y_{h})[J_{\mathcal{U}}\cdot]\big{\|}_{\mathcal{U}^{-1,2}}\leq\big{\\{}\sum_{k\in\mathcal{K}_{c}}{\eta_{k}^{e}}^{2}\big{\\}}^{\frac{1}{2}}=:\mathscr{E}_{\rm
store}(y_{h}),$ (3.11)

where

$\eta_{k}^{e}=\big{(}\frac{1}{2}\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{k-1}+j}^{2}+\frac{1}{2}\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{k}+j}^{2}\big{)}^{\frac{1}{2}},$
(3.12)

if $k\in\mathcal{K}^{\prime}_{c}$,

$\eta_{k}^{e}=\big{(}\varepsilon\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{La_{i}}+j}^{2}+\frac{1}{2}\varepsilon\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{k-1}+j}^{2}\big{)}^{\frac{1}{2}},$
(3.13)

if $k=L_{a_{i}}$ for some $i\in\\{1,2,\ldots,M\\}$, i.e., $T_{k}$ is adjacent
to and to the left of an atomistic region, and

$\eta_{k}^{e}=\big{(}\varepsilon\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{Ra_{i}}+j}^{2}+\frac{1}{2}\varepsilon\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{k}+j}^{2}\big{)}^{\frac{1}{2}},$
(3.14)

if $k-1=R_{a_{i}}$ for some $i\in\\{1,2,\ldots,M\\}$, i.e., $T_{k}$ is
adjacent to and to the right of an atomistic region.
$[[\phi^{\prime}]]_{\ell}$’s will be defined in the proof.

###### Proof.

By (3.2) and (3.1), we have

$\displaystyle\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})[v]-\mathcal{E}^{\prime}_{{\rm qc}}(y_{h})[J_{\mathcal{U}}v]=$
$\displaystyle\varepsilon\sum_{b\in\mathcal{B}}\phi^{\prime}(r_{b}D_{b}J_{\mathcal{U}_{{\rm
qc}}}y_{h})r_{b}D_{b}v-\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}J_{\mathcal{U}}v$
$\displaystyle-\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}J_{\mathcal{U}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\sum_{b\in\mathcal{B}}\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}I_{h}v$
$\displaystyle-\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}I_{h}v\,{\rm
d}x,$ (3.15)

since $D_{b}J_{\mathcal{U}_{{\rm
qc}}}y_{h}=D_{b}I_{\varepsilon}y_{h}=D_{b}y_{h}$,
$D_{\omega}J_{\mathcal{U}}v=D_{\omega}I_{h}v$ for any $\omega$ being an open
interval, and $(J_{\mathcal{U}}v)^{\prime}=(I_{h}v)^{\prime}$, which can be
easily verified by noting that $J_{\mathcal{U}_{{\rm qc}}}y_{h}$ and
$J_{\mathcal{U}}v$ are $I_{\varepsilon}y_{h}$ and $I_{h}v$ shifted by some
constants.

To make further analysis of (3.2), we subtract and add the same terms

$\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v\text{ and
}\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$

to get

$\displaystyle\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})[v]-\mathcal{E}^{\prime}_{{\rm qc}}(y_{h})[J_{\mathcal{U}}v]=$
$\displaystyle\sum_{b\in\mathcal{B}}\bigg{\\{}\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm a}}v$
$\displaystyle-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x\bigg{\\}}$
$\displaystyle-\bigg{\\{}\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})\big{[}D_{b\cap\Omega_{\rm
a}}I_{h}v\,{\rm d}x-D_{b\cap\Omega_{\rm a}}v\big{]}\bigg{\\}}$
$\displaystyle-\bigg{\\{}\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\big{[}\nabla_{r_{b}}I_{h}v-\nabla_{r_{b}}v\big{]}\,{\rm
d}x\bigg{\\}}.$ (3.16)

We first analyze the second and third groups, which turn out to be $0$ as we
will see immediately.

For the second group, we have,

$\displaystyle\quad\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})\big{[}D_{b\cap\Omega_{\rm
a}}I_{h}v\,{\rm d}x-D_{b\cap\Omega_{\rm a}}v\big{]}$
$\displaystyle=\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm
a}}y_{h})\bigg{[}\frac{I_{h}v(R_{b\cap\Omega_{\rm
a}})-I_{h}v(L_{b\cap\Omega_{\rm a}})}{R_{b\cap\Omega_{\rm
a}}-L_{b\cap\Omega_{\rm a}}}-\frac{v(R_{b\cap\Omega_{\rm
a}})-v(L_{b\cap\Omega_{\rm a}})}{R_{b\cap\Omega_{\rm a}}-L_{b\cap\Omega_{\rm
a}}}\bigg{]}.$ (3.17)

We define the above to be $0$ if $b\cap\Omega_{\rm a}=\emptyset$. If
$b\cap\Omega_{\rm a}\neq\emptyset$, since both $R_{b\cap\Omega_{\rm a}}$ and
$L_{b\cap\Omega_{c}}$ are either at atomistic postions in $\Omega_{\rm a}$ or
on $\partial\Omega_{c}$, they must be nodes in $\mathcal{T}^{h}$. Therefore,
by the definition of $I_{h}v$, the following holds

$I_{h}v(L_{b\cap\Omega_{\rm a}})=v(L_{b\cap\Omega_{\rm a}})\text{ and
}I_{h}v(R_{b\cap\Omega_{\rm a}})=v(R_{b\cap\Omega_{\rm a}}),$

which implies that (3.2) is $0$.

For the third group, upon defining $\chi_{\mathcal{S}}$ to be the
characteristic function of a set $\mathcal{S}$, we can rewrite it as

$\displaystyle\quad\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\big{[}\nabla_{r_{b}}I_{h}v-\nabla_{r_{b}}v\big{]}\,{\rm
d}x$
$\displaystyle=\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{\Omega_{c}}\chi_{b}\phi^{\prime}(\nabla_{r_{b}}y_{h})\big{[}\nabla_{r_{b}}I_{h}v-\nabla_{r_{b}}v\big{]}\,{\rm
d}x$
$\displaystyle=\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\sum_{k\in\mathcal{K}_{c}}\int_{T_{k}}\chi_{b}\phi^{\prime}(\nabla_{r_{b}}y_{h})\big{[}\nabla_{r_{b}}I_{h}v-\nabla_{r_{b}}v\big{]}\,{\rm
d}x$
$\displaystyle=\sum_{r=1}^{2}\sum_{b\in\mathcal{B},r_{b}=r}\sum_{k\in\mathcal{K}_{c}}\frac{1}{r_{b}}\int_{T_{k}}\chi_{b}\phi^{\prime}(\nabla_{r_{b}}y_{h})\big{[}\nabla_{r_{b}}I_{h}v-\nabla_{r_{b}}v\big{]}\,{\rm
d}x$
$\displaystyle=\sum_{r=1}^{2}\sum_{k\in\mathcal{K}_{c}}\sum_{b\in\mathcal{B},r_{b}=r}\frac{1}{r_{b}}\int_{T_{k}}\chi_{b}\phi^{\prime}(\nabla_{r}y_{h})\big{[}\nabla_{r}I_{h}v-\nabla_{r}v\big{]}\,{\rm
d}x$
$\displaystyle=\sum_{r=1}^{2}\sum_{k\in\mathcal{K}_{c}}\phi^{\prime}(\nabla_{r}y_{h}|_{T_{k}})\int_{T_{k}}\big{[}\sum_{b\in\mathcal{B},r_{b}=r}\frac{1}{r_{b}}\chi_{b}\big{]}\big{[}\nabla_{r}I_{h}v-\nabla_{r}v\big{]},$
(3.18)

since $\nabla_{r}y_{h}|_{T_{k}}$ is a constant on each element. By the 1D bond
density lemma[10, Lemma 3.4],

$\sum_{b\in\mathcal{B},r_{b}=r}\frac{1}{r}\chi_{b}(x)=_{a.e.}1,$

we have

$\displaystyle\sum_{r=1}^{2}\sum_{k\in\mathcal{K}_{c}}\phi^{\prime}(\nabla_{r}y_{h}|_{T_{k}})\int_{T_{k}}\big{[}\sum_{b\in\mathcal{B},r_{b}=r}\frac{1}{r_{b}}\chi_{b}\big{]}\big{[}\nabla_{r}I_{h}v-\nabla_{r}v\big{]}$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{2}\sum_{k\in\mathcal{K}_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h}|_{T_{k}})\bigg{[}r\big{(}I_{h}v(x_{k})-I_{h}v(x_{k-1})\big{)}-r\big{(}v(x_{k})-v(x_{k-1})\big{)}\bigg{]}.$
(3.19)

Again by the definition of $I_{h}v$,

$I_{h}v(x^{h}_{k})=v(x^{h}_{k})\text{ and
}I_{h}v(x^{h}_{k-1})=v(x^{h}_{k-1}),$

and thus (3.19) is $0$.

Now we turn to the analysis of the first group and analyze

$\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ (3.20)

for each interaction bond $b$.

If $b\subset\Omega_{\rm a}$, we have $|b\cap\Omega_{c}|=r_{b}\varepsilon$,
$|b\cap\Omega_{\rm a}|=0$ and the equivalence of the operators
$D_{b}=D_{b\cap\Omega_{\rm a}}$. We know that (3.20) is $0$ by substituting
these equivalences.

If $b\subset\Omega_{c}\cap T_{k}$ for some $k\in\mathcal{K}_{c}$, then
$|b\cap\Omega_{c}|=r_{b}\varepsilon$ and $|b\cap\Omega_{\rm a}|=0$. We also
note that $\nabla_{r_{b}}y_{h}(x)=r_{b}D_{b}y_{h}$, as $y_{h}$ is affine on
$T_{k}$, and $\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\nabla_{r_{b}}v=\varepsilon
r_{b}D_{b}v$. Using these equivalences, we know that (3.20) is again $0$.

Therefore, we only need to analyze the bonds crossing the atomistic-continuum
interface or the boundaries of two adjacent elements in $\Omega_{c}$. Because
of its tediousness, we leave the detailed analysis to the Appendix but just
present the result here. Employing the notation often adopted by a posteriori
error analysis for elliptic equations, we have the following result

$\displaystyle\quad\varepsilon\sum_{b\in\mathcal{B}}\phi^{\prime}(r_{b}D_{b}I_{\varepsilon}y_{h})r_{b}D_{b}v-\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v+\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$
$\displaystyle=\sum_{i=1}^{M}\varepsilon\bigg{\\{}[[\phi^{\prime}]]_{\ell_{La_{i}}}v^{\prime}_{\ell_{La_{i}}}+[[\phi^{\prime}]]_{\ell_{La_{i}}+1}v^{\prime}_{\ell_{La_{i}}+1}+[[\phi^{\prime}]]_{\ell_{La_{i}}+2}v^{\prime}_{\ell_{La_{i}}+2}\bigg{\\}}$
$\displaystyle\quad+\sum_{i=1}^{M}\varepsilon\bigg{\\{}[[\phi^{\prime}]]_{\ell_{Ra_{i}}}v^{\prime}_{\ell_{Ra_{i}}}+[[\phi^{\prime}]]_{\ell_{Ra_{i}}+1}v^{\prime}_{\ell_{Ra_{i}}+1}+[[\phi^{\prime}]]_{\ell_{Ra_{i}}+2}v^{\prime}_{\ell_{Ra_{i}}+2}\bigg{\\}}$
$\displaystyle\quad+\sum_{k\in\mathcal{K}^{\prime}_{c}}\varepsilon\bigg{\\{}[[\phi^{\prime}]]_{\ell_{k}}v^{\prime}_{\ell_{k}}+[[\phi^{\prime}]]_{\ell_{k}+1}v^{\prime}_{\ell_{k}+1}+[[\phi^{\prime}]]_{\ell_{k}+2}v^{\prime}_{\ell_{k}+2}\bigg{\\}},$
(3.21)

where for $k=La_{i}$,

$\displaystyle[[\phi^{\prime}]]_{\ell}$
$\displaystyle=\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)},$
(3.22) $\displaystyle[[\phi^{\prime}]]_{\ell+1}$
$\displaystyle=\bigg{[}\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k+1}}\big{)}-\theta_{k}\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
$\displaystyle\quad+\bigg{[}(1-\theta_{k})\phi^{\prime}\big{(}\frac{2}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k+2}}+\frac{2(1-\theta_{k})}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k}+1}\big{)}$
$\displaystyle\quad\quad+\theta_{k}\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k+2}}+(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
$\displaystyle\quad+\bigg{[}(1-\theta_{k})\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}+\theta_{k}\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
(3.23) $\displaystyle[[\phi^{\prime}]]_{\ell+2}$
$\displaystyle=\phi^{\prime}(\frac{2(1-\theta_{k})}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k+1}}+\frac{2}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k+2}}\big{)}-\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k+2}}+(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)},$
(3.24)

for $k=Ra_{i}$

$\displaystyle[[\phi^{\prime}]]_{\ell}$
$\displaystyle=\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}+y_{h}^{\prime}|_{T_{k-1}}\big{)}-\phi^{\prime}\bigg{(}\frac{2\theta_{k}}{(1+\theta_{k})}y_{h}^{\prime}|_{T_{k}}+\frac{2}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k-1}}\bigg{)},$
(3.25) $\displaystyle[[\phi^{\prime}]]_{\ell+1}$
$\displaystyle=\bigg{[}\theta_{k}\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k}}\big{)}+(1-\theta_{k})\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k+1}}\big{)}-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
$\displaystyle\quad+\bigg{[}\theta_{k}\phi^{\prime}\big{(}\frac{2\theta_{k}}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k}}+\frac{2}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k-1}}\big{)}+(1-\theta_{k})\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}$
$\displaystyle\quad-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}+y_{h}^{\prime}|_{T_{k-1}}\big{)}\bigg{]}$
$\displaystyle\quad+\bigg{[}(1-\theta_{k})\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}+\theta_{k}\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
(3.26) $\displaystyle[[\phi^{\prime}]]_{\ell+2}$
$\displaystyle=\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}-\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)},$
(3.27)

and for $k\in\mathcal{K}^{\prime}_{c}$

$\displaystyle[[\phi^{\prime}]]_{\ell}$
$\displaystyle=\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y^{\prime}_{h}|_{T_{k}}\big{)},$
(3.28) $\displaystyle[[\phi^{\prime}]]_{\ell+1}$
$\displaystyle=\bigg{[}(1-\theta_{k})\phi^{\prime}\big{(}y^{\prime}_{h}|_{T_{k+1}}\big{)}+\theta_{k}\phi^{\prime}\big{(}y^{\prime}_{h}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
$\displaystyle+\bigg{[}2(1-\theta_{k})\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}+2\theta_{k}\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}$
$\displaystyle-\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]},$
(3.29) $\displaystyle[[\phi^{\prime}]]_{\ell+2}$
$\displaystyle=\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}-\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}.$
(3.30)

Distributing the contribution of (3.2) to each element and applying Cauchy-
Schwarz inequality, we obtain the estimate stated in the theorem. ∎

###  3.3. Estimate of the Residual of the External Force

We now turn to the estimate of the residual of the external energy. Upon
defining $J_{\mathcal{U}}v:=v_{h}$, the residual of the external force is
given by

$\langle f,v_{h}\rangle_{h}-\langle f,v\rangle_{\varepsilon},$ (3.31)

where $f,v\in\mathcal{U}$.

To further analyze (3.31), we introduce a new partition
$\mathcal{T}^{r}=\\{T^{r}_{j}\\}_{j=-\infty}^{+\infty}$ of the domain
$\mathbb{R}$, such that all the nodes in partition $\mathcal{T}^{\varepsilon}$
and partition $\mathcal{T}^{h}$ are included in this partition. The indexing
of the nodes in $\mathcal{T}^{r}$ follow the rule that the node $x^{h}_{k}$ in
$\mathcal{T}^{h}$ is labeled as $x_{j_{k}}^{r}$ in $\mathcal{T}^{r}$. We also
assume there are $n$ nodes in $\mathcal{T}^{r}$ in $[0,1]$, i.e.,

$n=\big{|}\\{\varepsilon,2\varepsilon,\ldots,N\varepsilon\\}\cup\\{x_{1},x_{2},\ldots,x_{K}\\}\big{|},$

where $|\mathcal{A}|$ denote the cardinality of a finite set $\mathcal{A}$.

The inner product associated with $\mathcal{T}^{r}$ partition is then defined
by

$\langle f,g\rangle_{r}:=\int_{0}^{1}I_{r}(fg)\,{\rm
d}x=\sum_{j=1}^{n}\frac{1}{2}(x^{r}_{j+1}-x^{r}_{j-1})f^{r}_{j}g^{r}_{j}=:\langle\boldsymbol{f}^{r},\boldsymbol{g}^{r}\rangle_{r}\quad\forall
f,g\in C^{0}(\mathbb{R}),$ (3.32)

where $I_{r}$ is the linear nodal interpolation operator with respect to
$\mathcal{T}^{r}$, and $\boldsymbol{f}^{r}$ and $\boldsymbol{g}^{r}$ are the
vectorizations of $f$ and $g$ with respect to $\mathcal{T}^{r}$.

Now we decompose the residual of the external force into three parts by adding
and subtracting the same terms,

$\displaystyle\langle f,v_{h}\rangle_{h}-\langle
f,v\rangle_{\varepsilon}=\big{[}\langle f,v\rangle_{r}-\langle
f,v\rangle_{\varepsilon}\big{]}+\big{[}\langle f,v_{h}\rangle_{r}-\langle
f,v\rangle_{r}\big{]}+\big{[}\langle f,v_{h}\rangle_{h}-\langle
f,v_{h}\rangle_{r}\big{]}.$ (3.33)

The following three lemma are derived to give the estimates of the three
parts.

Lemma 2.  Let
$\boldsymbol{f},\boldsymbol{v},\boldsymbol{f}^{r},\boldsymbol{v}^{r}$ be the
vectorizations of $f,v\in C^{0}(\mathbb{R})$ according to
$\mathcal{T}^{\varepsilon}$ and $\mathcal{T}^{r}$. Then the following
inequality holds

$\big{|}\langle f,v\rangle_{r}-\langle
f,v\rangle_{\varepsilon}\big{|}=\big{|}\langle\boldsymbol{f}^{r},\boldsymbol{v}^{r}\rangle_{r}-\langle\boldsymbol{f},\boldsymbol{v}\rangle_{\varepsilon}\big{|}\leq\frac{1}{8}\varepsilon^{2}\|\boldsymbol{f}^{\prime}\|_{\ell_{\varepsilon}^{2}(\mathcal{K}_{U})}\|\boldsymbol{v}^{\prime}\|_{\ell_{\varepsilon}^{2}},$
(3.34)

where
$\mathcal{K}_{U}=\big{\\{}k\in\\{1,\ldots,K\\}:x_{k}\neq\ell_{k}\varepsilon\big{\\}}$,
in other words, $\mathcal{K}_{U}$ is the set of indices of the nodes
$x^{h}_{k}$ in $\mathcal{T}^{h}$ such that $x^{h}_{k}$ does not coincide with
any of the nodes in $\mathcal{T}^{\varepsilon}$.

###### Proof.

We first write out the two inner products and eliminate the terms that are the
same

$\displaystyle\langle\boldsymbol{f}^{r},\boldsymbol{v}^{r}\rangle_{r}-\langle\boldsymbol{f},\boldsymbol{v}\rangle_{\varepsilon}$
$\displaystyle=\sum_{\ell=1}^{n}\varepsilon^{r}_{\ell}\frac{1}{2}(f^{r}_{\ell}v^{r}_{\ell}+f^{r}_{\ell+1}v^{r}_{\ell+1})-\sum_{\ell=1}^{N}\varepsilon\frac{1}{2}(f_{\ell}v_{\ell}+f_{\ell+1}v_{\ell+1})$
$\displaystyle=\sum_{k\in\mathcal{K}_{U}}\big{(}\varepsilon_{j_{k}}\frac{1}{2}(f^{r}_{j_{k}-1}v^{r}_{j_{k}-1}+f^{r}_{j_{k}}v^{r}_{j_{k}})+\varepsilon_{j_{k}+1}\frac{1}{2}(f^{r}_{j_{k}}v^{r}_{j_{k}}+f^{r}_{j_{k}+1}v^{r}_{j_{k}+1})\big{)}$
$\displaystyle\quad-\sum_{k\in\mathcal{K}_{U}}\varepsilon\big{(}f_{\ell_{k}}v_{\ell_{k}}+f_{\ell_{k}+1}v_{\ell_{k}+1}\big{)},$
(3.35)

as
$\varepsilon_{j_{k}+2}=\varepsilon_{j_{k}+3}=\ldots=\varepsilon_{j_{k+1}-1}=\varepsilon$
and $f_{\ell_{k}+i}v_{\ell_{k}+i}=f^{r}_{j_{k}+i}v^{r}_{j_{k}+i},\quad
i=1,2,\ldots,\ell_{k+1}-\ell_{k}$, if $\ell_{k}\varepsilon\neq x_{k}$ and
$\ell_{k+1}\varepsilon\neq x_{k+1}$.

For $k$ such that $\ell_{k}\varepsilon\neq x_{k}$, by the definition of
$\boldsymbol{f}$, $\boldsymbol{v}$, $\boldsymbol{f}^{r}$ and
$\boldsymbol{v}^{r}$, we have $f_{\ell_{k}}=f^{r}_{j_{k}-1}$,
$v_{\ell_{k}}=v^{r}_{j_{k}-1}$, $f_{\ell_{k}+1}=f^{r}_{j_{k}+1}$ and
$v_{\ell_{k}+1}=v^{r}_{j_{k}+1}$. We also have
$f^{r}_{j_{k}}=(1-\theta_{k})f_{\ell_{k}}+\theta_{k}f_{\ell_{k}+1}$ and
$v^{r}_{j_{k}}=(1-\theta_{k})v_{\ell_{k}}+\theta_{k}v_{\ell_{k}+1}$. Inserting
these equalities, (3.35) can be estimated as

$\displaystyle\big{|}\langle\boldsymbol{f}^{r},\boldsymbol{v}^{r}\rangle_{r}-\langle\boldsymbol{f},\boldsymbol{v}\rangle_{\varepsilon}\big{|}$
$\displaystyle=$
$\displaystyle\bigg{|}\sum_{k\in\mathcal{K}_{U}}\bigg{\\{}\frac{1}{2}\theta_{k}\varepsilon
f_{\ell_{k}}v_{\ell_{k}}+\frac{1}{2}\theta_{k}\varepsilon\big{[}(1-\theta_{k})f_{\ell_{k}}+\theta_{k}f_{\ell_{k+1}}\big{]}\big{[}(1-\theta_{k})v_{\ell_{k}}+\theta_{k}v_{\ell_{k}+1}\big{]}$
$\displaystyle+\frac{1}{2}(1-\theta_{k})\varepsilon
f_{\ell_{k}+1}v_{\ell_{k}+1}+\frac{1}{2}(1-\theta_{k})\varepsilon\big{[}(1-\theta_{k})f_{\ell_{k}}+\theta_{k}f_{\ell_{k}+1}\big{]}\big{[}(1-\theta_{k})v_{\ell_{k}}+\theta_{k}v_{\ell_{k}+1}\big{]}$
$\displaystyle-\frac{1}{2}\varepsilon
f_{\ell_{k}}v_{\ell_{k}}-\frac{1}{2}\varepsilon
f_{\ell_{k}+1}v_{\ell_{k}+1}\bigg{\\}}\bigg{|}$ $\displaystyle=$
$\displaystyle\big{|}\sum_{k\in\mathcal{K}_{U}}\frac{1}{2}\varepsilon\bigg{\\{}\big{[}\theta_{k}(\theta_{k}-1)(f_{\ell_{k}+1}-f_{\ell_{k}})v_{\ell_{k}+1}\big{]}-\big{[}\theta_{k}(\theta_{k}-1)(f_{\ell_{k}+1}-f_{\ell_{k}})v_{\ell_{k}}\big{]}\bigg{\\}}\big{|}$
$\displaystyle=$
$\displaystyle\sum_{k\in\mathcal{K}_{U}}\frac{1}{2}\varepsilon^{3}\big{|}\theta_{k}(1-\theta_{k})f^{\prime}_{\ell_{k}+1}v^{\prime}_{\ell_{k}+1}\big{|}$
$\displaystyle\leq\frac{1}{8}\varepsilon^{2}\bigg{(}\sum_{k\in\mathcal{K}_{U}}\varepsilon|f^{\prime}_{\ell_{k}+1}|^{2}\bigg{)}^{\frac{1}{2}}\bigg{(}\sum_{k\in\mathcal{K}_{U}}\varepsilon|v^{\prime}_{\ell_{k}+1}|^{2}\bigg{)}^{\frac{1}{2}}\leq\frac{1}{8}\varepsilon^{2}\|\boldsymbol{f}^{\prime}\|_{\ell_{\varepsilon}^{2}(\mathcal{K}_{U})}\|\boldsymbol{v}^{\prime}\|_{\ell_{\varepsilon}^{2}},$
(3.36)

which concludes the proof. ∎

Remark 1. If $\mathcal{K}=\emptyset$, i.e., every node in $T^{h}$ is also in
$T^{\varepsilon}$, then this part of the residual is $0$. ∎

Lemma 3.  Let $f,v\in
C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})$ and
$v_{h}=I_{h}v\in C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{h})$ be the
$\mathcal{P}_{1}$ interpolation of $v$ according to $\mathcal{T}^{h}$
partition. Let $\boldsymbol{f}^{r},\boldsymbol{v}^{r}$ and
$\boldsymbol{v}_{h}^{r}$ be the vectorizations of $f,v,v_{h}$ respectively
according to $\mathcal{T}^{r}$, and $\mathcal{K}_{c}$ is defined in (3.10).
Then we have the following estimate

$\langle f,v_{h}\rangle_{r}-\langle
f,v\rangle_{r}=\langle\boldsymbol{f}^{r},\boldsymbol{v}_{h}^{r}\rangle_{r}-\langle\boldsymbol{f}^{r},\boldsymbol{v}^{r}\rangle_{r}\leq\bigg{[}\sum_{k=\mathcal{K}_{c}}\tilde{h}_{k}^{2}\|\boldsymbol{f}^{r}\|^{2}_{\ell^{2}_{\bar{\varepsilon}^{r}}(\mathcal{D}^{2}_{k})}\bigg{]}^{\frac{1}{2}}\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}},$
(3.37)

$\bar{\varepsilon}^{r}_{j}=\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})$,
$\tilde{h}_{k}=\frac{1}{2}(j_{k+1}-j_{k})\varepsilon$ and
$\mathcal{D}^{2}_{k}=\\{j_{k}+1,\ldots,j_{k+1}-1\\}$.

###### Proof.

Using the fact that $(v_{h}^{r})_{j_{k}}=v^{r}_{j_{k}}$ and by Cauchy-Schwarz
inequality, we have

$\displaystyle\langle\boldsymbol{f}^{r},\boldsymbol{v}_{h}^{r}\rangle_{r}-\langle\boldsymbol{f}^{r},\boldsymbol{v}^{r}\rangle_{r}=$
$\displaystyle\sum_{j=1}^{n}\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})(f^{r}_{j}(v_{h}^{r})_{j}-f^{r}_{j}v^{r}_{j})$
$\displaystyle=$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\sum_{j=j_{k-1}}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}f^{r}_{j}\big{[}(v_{h}^{r})_{j}-v^{r}_{j}\big{]}$
$\displaystyle\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\bigg{[}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}(f^{r}_{j})^{2}\bigg{]}^{\frac{1}{2}}\bigg{\\{}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}\big{[}(v_{h}^{r})_{j}-v^{r}_{j}\big{]}^{2}\bigg{\\}}^{\frac{1}{2}},$
(3.38)

where
$\bar{\varepsilon}^{r}_{j}=\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})$.
Upon defining $\boldsymbol{g}$ such that $g_{j}=(v_{h}^{r})_{j}-v^{r}_{j}$
(note $g_{j_{k}}=g_{j_{k+1}}=0$) and by Lemma C in Appendix C (Discrete
Friedrich’s Inequality) and Rieze-Thorin Theorem,

$\displaystyle\bigg{\\{}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}\big{[}(v_{h}^{r})_{j}-v^{r}_{j}\big{]}^{2}\bigg{\\}}^{\frac{1}{2}}=$
$\displaystyle\bigg{\\{}\sum_{j=j_{k-1}}^{j_{k}}\bar{\varepsilon}^{r}_{j}g_{j}^{2}\bigg{\\}}^{\frac{1}{2}}\leq\frac{1}{2}(j_{k}-j_{k-1})\varepsilon\bigg{\\{}\sum_{j=j_{k-1}+1}^{j_{k}}\varepsilon^{r}_{j}{g^{\prime}_{j}}^{2}\bigg{\\}}^{\frac{1}{2}},$
(3.39)

where
$g_{j}^{\prime}={\textstyle\frac{g_{j}-g_{j-1}}{\varepsilon^{r}_{j}}}=(v^{r})^{\prime}_{j}-(v_{h}^{r})^{\prime}_{j}$.
$\varepsilon$ appears in the last inequality since
$\max_{j}\bar{\varepsilon}^{r}_{j}\leq\varepsilon$. Since $v_{h}^{r}$ and
$v^{r}$ are both piecewise linear on $\mathcal{T}^{r}$, we have
$(v_{h}^{r})^{\prime}_{j}-(v^{r})^{\prime}_{j}=(v^{\prime}-v_{h}^{\prime})(x)\
\forall x\in(x^{r}_{j-1},x^{r}_{j})$, and as a result,

$\bigg{\\{}\sum_{j=j_{k-1}+1}^{j_{k}}\varepsilon^{r}_{j}\big{[}(v_{h}^{r})^{\prime}_{j}-(v^{r})^{\prime}_{j}\big{]}^{2}\bigg{\\}}^{\frac{1}{2}}=\int_{x^{r}_{j_{k-1}}}^{x^{r}_{j_{k}}}|(v^{\prime}-v_{h}^{\prime})(x)|^{2}\,{\rm
d}x=\|v^{\prime}-v_{h}^{\prime}\|_{L^{2}[x^{r}_{j_{k-1}},x^{r}_{j_{k}}]}.$

By Lemma B in Appendix B,

$\|v^{\prime}-v_{h}^{\prime}\|_{L^{2}[x^{r}_{j_{k-1}},x^{r}_{j_{k}}]}\leq\|v^{\prime}\|_{L^{2}[x^{r}_{j_{k-1}},x^{r}_{j_{k}}]}.$

Put all the results above together and apply Cauchy-Schwarz inequality, we
obtain

$\displaystyle\sum_{k\in\mathcal{K}_{c}}\bigg{[}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}(f^{r}_{j})^{2}\bigg{]}^{\frac{1}{2}}\bigg{\\{}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}\big{[}(v_{h}^{r})_{j}-v^{r}_{j}\big{]}^{2}\bigg{\\}}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\Bigg{\\{}\tilde{h}_{k}\Big{(}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}{f^{r}_{j}}^{2}\Big{)}^{\frac{1}{2}}\|v^{\prime}\|_{L^{2}(x^{r}_{j_{k-1}},x^{r}_{j_{k}})}\Bigg{\\}}\leq\bigg{[}\sum_{k=\mathcal{K}_{c}}\tilde{h}_{k}^{2}\Big{(}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}{f^{r}_{j}}^{2}\Big{)}\bigg{]}^{\frac{1}{2}}\|v^{\prime}\|_{L^{2}{[0,1]}}.$
(3.40)

The eatimate in the theorem holds as
$\|v^{\prime}\|_{L^{2}{[0,1]}}=\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}}$
for $v\in C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})$. ∎

Lemma 4.  Let $f,v\in
C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})$ and
$v_{h}=I_{h}v\in C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{h})$ be the
$\mathcal{P}_{1}$ interpolation of $v$ according to the $\mathcal{T}^{h}$. Let
$\boldsymbol{f}^{r},\boldsymbol{v}^{r}$ and $\boldsymbol{v}_{h}^{r}$ be the
vectorizations $f,v$ and $v_{h}$ according to $\mathcal{T}^{r}$. If
$\int_{0}^{1}v_{h}=0$, then we have the following estimate

$\langle f,v_{h}\rangle_{h}-\langle
f,v_{h}\rangle_{r}\leq\Bigg{\\{}\frac{1}{8}\bigg{[}(n\varepsilon)^{4}\sum_{k=\mathcal{K}_{c}}\hat{h}_{k+1}^{4}\|{\boldsymbol{f}^{r}}^{\prime\prime}\|^{2}_{\ell^{2}_{\bar{\varepsilon}^{r}}(\mathcal{D}^{2}_{k})}\bigg{]}^{\frac{1}{2}}+\bigg{[}\sum_{k=\mathcal{K}_{c}}\hat{h}_{k+1}^{4}\|{\boldsymbol{f}^{r}}^{\prime}\|^{2}_{\ell^{2}_{\varepsilon^{r}(\mathcal{D}^{1}_{k})}}\bigg{]}^{\frac{1}{2}}\Bigg{\\}}\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}}.$
(3.41)

where $\mathcal{K}_{c}$ is defined in (3.10),
$\mathcal{D}_{k}^{1}=\\{j_{k}+1,\ldots,j_{k+1}\\}$ and $\hat{h}_{k}$ will be
defined in the proof.

###### Proof.

Since $I_{h}(fv_{h})$ is also piecewise linear with respect to the
$\mathcal{T}^{r}$ partition, we apply the trapezoidal rule here to evaluate
$\langle f,v_{h}\rangle_{h}=\int_{0}^{1}I_{h}(fv_{h})\,{\rm d}x$ to obtain

$\displaystyle\langle f,v_{h}\rangle_{h}-\langle f,v_{h}\rangle_{r}$
$\displaystyle=$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\Bigg{\\{}\bigg{[}\frac{1}{2}\varepsilon^{r}_{j_{k-1}+1}I_{h}(fv_{h})(x^{r}_{k})+\sum_{j=j_{k-1}+1}^{j_{k}-1}\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})I_{h}(fv_{h})(x^{r}_{j})+\frac{1}{2}\varepsilon^{r}_{j_{k}}I_{h}(fv_{h})(x^{r}_{j_{k}})\bigg{]}$
$\displaystyle-\bigg{[}\frac{1}{2}\varepsilon^{r}_{j_{k-1}+1}(fv_{h})(x^{r}_{j_{k-1}+1})+\sum_{j=j_{k-1}+1}^{j_{k}-1}\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})(fv_{h})(x^{r}_{j})+\frac{1}{2}\varepsilon^{r}_{j_{k}}(fv_{h})(x^{r}_{j_{k}})\bigg{]}\Bigg{\\}}.$
(3.42)

We define $\boldsymbol{g}$ and $\boldsymbol{G}$ such that
$g_{j}=(fv_{h})(x^{r}_{j})$ and $G_{j}=(I_{h}(fv_{h}))(x^{r}_{j})$. It is easy
to check that $g_{j_{k}}=G_{j_{k}}$ and

$G_{j_{k-1}+i}=g_{j_{k-1}}+\frac{\sum_{\ell=1}^{i}\varepsilon^{r}_{j_{k-1}+\ell}}{\varepsilon^{h}_{k}}(g_{j_{k}}-g_{j_{k-1}})\quad\forall
k\in\mathcal{K}_{c}\quad and\quad i=1,\ldots,j_{k}-j_{k-1},$ (3.43)

where
$\varepsilon^{h}_{k}=\sum_{j=j_{k-1}+1}^{j_{k}}\varepsilon^{r}_{j}=x_{k}-x_{k-1}$.
Therefore, by Theorem C, we obatin the following estimate

$\displaystyle|\langle
f,v_{h}\rangle_{h}-\langle\boldsymbol{f}^{r},I_{r}\boldsymbol{v}^{h}\rangle_{r}|$
$\displaystyle=$
$\displaystyle\Bigg{|}\sum_{k\in\mathcal{K}_{c}}\bigg{[}\frac{1}{2}\varepsilon^{r}_{j_{k-1}}(g_{j_{k-1}}-G_{j_{k-1}})+\sum_{j=j_{k-1}+1}^{j_{k}-1}\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})(g_{j}-G_{j})+\frac{1}{2}\varepsilon^{r}_{j_{k}}(g_{j_{k}}-G_{j_{k}})\bigg{]}\Bigg{|}$
$\displaystyle\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\frac{1}{4}\frac{\big{(}(j_{k}-j_{k-1})\varepsilon\big{)}\big{(}(j_{k}-j_{k-1}+1)\varepsilon\big{)}^{2}}{\varepsilon^{h}_{k}}\|\boldsymbol{g}^{\prime\prime}\|_{\ell_{\bar{\boldsymbol{\varepsilon}}^{r}}^{1}(\mathcal{D}^{2}_{k})},$
(3.44)

where $\boldsymbol{g}^{\prime\prime}$ is the second finite difference
derivative with respect to the $\mathcal{T}^{r}$.

By the definition of $\boldsymbol{f}^{r}$ and $\boldsymbol{v}_{h}^{r}$,
$g_{j}=(fv_{h})(x^{r}_{j})=f^{r}_{j}(v_{h}^{r})_{j}$. Using
$(v_{h}^{r})^{\prime\prime}_{j}=0\ \forall j\in\mathcal{D}^{2}_{k}$,
$g_{j}^{\prime\prime}$ can be written as

$g_{j}^{\prime\prime}=(fv_{h}^{r})^{\prime\prime}_{j}=(f^{r})^{\prime\prime}_{j}(v_{h}^{r})_{j}+\frac{\varepsilon^{r}_{j}}{\bar{\varepsilon}^{r}_{j}}(f^{r})^{\prime}_{j}(v_{h}^{r})^{\prime}_{j}+\frac{\varepsilon^{r}_{j+1}}{\bar{\varepsilon}^{r}_{j}}(f^{r})^{\prime}_{j+1}(v_{h}^{r})^{\prime}_{j+1}.$
(3.45)

Noting that $\frac{\varepsilon^{r}}{\bar{\varepsilon}_{j}}\leq 2$ and
$\frac{\varepsilon^{r}_{j+1}}{\bar{\varepsilon}_{j}}\leq 2$ and defining
$\hat{h}_{k}:=\bigg{[}\frac{((j_{k}-j_{k-1})\varepsilon)(j_{k}-j_{k-1}+1)\varepsilon\big{)}^{2}}{\varepsilon^{h}_{k}}\bigg{]}^{\frac{1}{2}}$,
we have the following estimate

$\displaystyle\langle f,v_{h}\rangle_{h}-\langle f,v_{h}\rangle_{r}\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\frac{1}{4}\hat{h}_{k}^{2}\bigg{[}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}\big{|}(fv_{h}^{r})^{\prime\prime}_{j}\big{|}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\frac{1}{4}\hat{h}_{k}^{2}\bigg{[}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}\big{|}(f^{r})^{\prime\prime}_{j}\big{|}\big{|}(v_{h}^{r})_{j}\big{|}+4\sum_{j=j_{k-1}+1}^{j_{k}}\varepsilon^{r}_{j}\big{|}(f^{r})^{\prime}_{j}\big{|}\big{|}(v_{h}^{r})^{\prime}_{j}\big{|}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\frac{1}{4}\hat{h}_{k}^{2}\bigg{[}\|{\boldsymbol{f}^{r}}^{\prime\prime}\|_{\ell^{2}_{\bar{\varepsilon}^{r}}(\mathcal{D}_{k}^{2})}\|\boldsymbol{v}_{h}^{r}\|_{\ell^{2}_{\bar{\varepsilon}^{r}}(\mathcal{D}_{k}^{2})}+4\|{\boldsymbol{f}^{r}}^{\prime}\|_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}(\mathcal{D}_{k}^{1})}}\|(\boldsymbol{v}_{h}^{r})^{\prime}\|_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}(\mathcal{D}_{k}^{1})}}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\frac{1}{4}\bigg{[}\sum_{k\in\mathcal{K}_{c}}\hat{h}_{k}^{4}\|{\boldsymbol{f}^{r}}^{\prime\prime}\|^{2}_{\ell^{2}_{\bar{\boldsymbol{\varepsilon}}^{r}}(\mathcal{D}_{k}^{2})}\bigg{]}^{\frac{1}{2}}\|\boldsymbol{v}_{h}^{r}\|_{\ell^{2}_{\bar{\boldsymbol{\varepsilon}}^{r}}}+\bigg{[}\sum_{k\in\mathcal{K}_{c}}\hat{h}_{k}^{4}\|{\boldsymbol{f}^{r}}^{\prime}\|^{2}_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}(\mathcal{D}_{k}^{1})}}\bigg{]}^{\frac{1}{2}}\|(\boldsymbol{v}_{h}^{r})^{\prime}\|_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}}}.$
(3.46)

For further estimate, we first bound
$\|\boldsymbol{v}_{h}^{r}\|_{\ell^{2}_{\bar{\boldsymbol{\varepsilon}}^{r}}}$
by
$\|(\boldsymbol{v}_{h}^{r})^{\prime}\|_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}}}$.
Since $v_{h}(x)$ is piecewise linear with respect to $\mathcal{T}^{r}$
partition, we can apply the trapezoidal rule to the integration on each
element to get

$\sum_{j=1}^{n}\bar{\varepsilon}^{r}_{j}(v_{h}^{r})_{j}=\sum_{j=1}^{n}\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j-1})(v^{r}_{h})_{j}=\sum_{j=1}^{n}\varepsilon^{r}_{j}\frac{1}{2}\big{[}(v^{r}_{h})_{j}+(v^{r}_{h})_{j+1}\big{]}=\int_{0}^{1}v_{h}(x)\,{\rm
d}x=0.$ (3.47)

The last equality holds by the periodic condition on $v_{h}$. Thus, we can
apply Lemma C in Appendix C and Riez-Thorin Theorem to obtain

$\|\boldsymbol{v}_{h}^{r}\|_{\ell^{2}_{\bar{\varepsilon}^{r}}}\leq\frac{1}{2}n\varepsilon\bigg{(}\sum_{j=1}^{n}\varepsilon^{r}_{j}{(v_{h}^{r})^{\prime}_{j}}^{2}\bigg{)}^{\frac{1}{2}}.$
(3.48)

Since $v_{h}^{\prime}(x)=(v_{h}^{r})^{\prime}_{j}$ on
$(x^{r}_{j-1},x^{r}_{j})$,

$\sum_{j=1}^{n}\varepsilon^{r}_{j}{(v_{h}^{r})^{\prime}_{j}}^{2}=\int_{0}^{1}(v_{h}^{\prime})^{2}\,{\rm
d}x=\|v_{h}^{\prime}\|_{L^{2}[0,1]}.$ (3.49)

By Lemma B in Appendix B,

$\|v_{h}^{\prime}\|_{L^{2}[0,1]}\leq\|v^{\prime}\|_{L^{2}[0,1]}=\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}}.$
(3.50)

Combine these results, the estimate stated in the theorem is easy to
establish. ∎

Having the three lemma and distribute the contribution to each element, we now
give the theorem which essentially gives the estimate of the residual due to
the external force.

Theorem 5.  For $f,v\in\mathcal{U}$ and $J_{\mathcal{U}}$ defined in (3.5), we
have

$\|\langle f,J_{\mathcal{U}}\cdot\rangle_{h}-\langle
f,\cdot\rangle_{\varepsilon}\|_{\mathcal{U}^{-1,2}}\leq\big{\\{}\sum_{k\in\mathcal{K}_{c}}{\eta_{k}^{f}}^{2}\big{\\}}^{\frac{1}{2}}=:\mathscr{E}_{\rm
ext}(f),$ (3.51)

where

$\displaystyle\eta_{k}^{f}=$
$\displaystyle\bigg{\\{}\frac{1}{128}\big{[}\varepsilon^{3}(f^{\prime}_{\ell_{k-1}+1})^{2}+\varepsilon^{3}(f^{\prime}_{\ell_{k}+1})^{2}\big{]}^{2}+\tilde{h}_{k}^{2}\|\boldsymbol{f}^{r}\|^{2}_{\ell^{2}_{\bar{\boldsymbol{\varepsilon}}^{r}}(\mathcal{D}^{2}_{k})}$
$\displaystyle+\frac{1}{64}(n\varepsilon)^{4}\hat{h}_{k+1}^{4}\|{\boldsymbol{f}^{r}}^{\prime\prime}\|^{2}_{\ell^{2}_{\bar{\boldsymbol{\varepsilon}}^{r}}(\mathcal{D}^{2}_{k})}+\hat{h}_{k+1}^{4}\|{\boldsymbol{f}^{r}}^{\prime}\|^{2}_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}(\mathcal{D}^{1}_{k})}}\bigg{\\}}^{\frac{1}{2}},$
(3.52)

and $\mathcal{K}_{c}$ is defined in (3.10), $\mathcal{D}_{k}^{2}$ is defined
in Lemma 3.3, $\tilde{h}_{k}$, $\mathcal{D}_{k}^{1}$ is defined in Lemma 3.3,
and $\hat{h}_{k+1}$ is defined in Lemma 3.3.

###### Proof.

We can not directly apply the three lemma to estimate the three parts in
(3.33). The reason is that $J_{\mathcal{U}}v\neq v_{h}$, which is the direct
interpolation of $v$ according to $\mathcal{T}^{h}$. The way to circumvent
this difficulty is by defining

$w:=v-\sum_{k=1}^{K}\frac{1}{2}(x_{k+1}-x_{k-1})v(x_{k})\text{ and
}w_{h}:=I_{h}w=J_{\mathcal{U}}v,$ (3.53)

and noting that

$\langle f,J_{\mathcal{U}}v\rangle_{h}-\langle
f,v\rangle_{\varepsilon}=\langle f,w_{h}\rangle_{h}-\langle
f,w\rangle_{\varepsilon}-\langle f,C\rangle_{\varepsilon}=\langle
f,w_{h}\rangle_{h}-\langle f,w\rangle_{\varepsilon},$ (3.54)

and $w^{\prime}(x)=v^{\prime}(x)\ \forall x\in\mathbb{R}$. Then by the three
lemma, we have

$\big{|}\langle f,J_{\mathcal{U}}v\rangle_{h}-\langle
f,v\rangle_{\varepsilon}\big{|}=\big{|}\langle f,w_{h}\rangle_{h}-\langle
f,w\rangle_{\varepsilon}\big{|}\leq\big{\\{}\sum_{k\in\mathcal{K}_{c}}{\eta_{k}^{f}}^{2}\big{\\}}^{\frac{1}{2}}\|w^{\prime}\|_{L^{2}[0,1]}=\big{\\{}\sum_{k\in\mathcal{K}_{c}}{\eta_{k}^{f}}^{2}\big{\\}}^{\frac{1}{2}}\|v^{\prime}\|_{L^{2}[0,1]},$
(3.55)

which establishes the estimate in the theorem. ∎

##  4\. Stability

Stability of the QC approximation is the second key ingredient for deriving an
a posteriori error bounds. Since we would like to bound the error of the
deformation gradient in $L^{2}$-norm, we derive the $L^{2}$ stability estimate
in this section. The procedure of deriving the a posteriori stability
condition largely follows that of the a priori stability condition in [8].

For an a posteriori error analysis, the natural notion of stability for energy
minimization problem is the coercivity(or, positivity) of the atomistic
Hessian at the projected QC solution $J_{\mathcal{U}_{{\rm qc}}}y_{{\rm qc}}$:

$E^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{{\rm qc}})[v,v]\geq
c_{\rm a}(y_{{\rm qc}})\|v\|_{L^{2}[0,1]}^{2}\quad\forall v\in\mathcal{U},$
(4.1)

for some constant $c_{\rm a}(y_{{\rm qc}})>0$. To avoid notational difficulty,
we vectorize the above inequality and work on $\mathcal{U}^{\varepsilon}$
instead. Let $J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}}$ be the
vectorization of $J_{\mathcal{U}_{{\rm qc}}}y_{{\rm qc}}$, then (4.1) is
equivalent to

$E^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})[\boldsymbol{v},\boldsymbol{v}]\geq c_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm
qc}})\|\boldsymbol{v}\|_{\ell^{2}_{\boldsymbol{\varepsilon}}}^{2}\quad\forall\boldsymbol{v}\in\mathcal{U}.$
(4.2)

In the remainder of this section, we derive the explicit condition on
$\boldsymbol{y}^{{\rm qc}}$ such that (4.2) holds.

The Hessian operator of the atomistic model is given by

$E^{\prime\prime}_{\rm
a}(\boldsymbol{y})[\boldsymbol{v},\boldsymbol{v}]=\varepsilon\sum_{\ell=1}^{N}\phi^{\prime\prime}(y_{\ell}^{\prime})|v_{\ell}^{\prime}|^{2}+\varepsilon\sum_{\ell=1}^{N}\phi^{\prime\prime}(y_{\ell}^{\prime}+y_{\ell+1}^{\prime})|v_{\ell}^{\prime}+v^{\prime}_{\ell+1}|^{2}\quad\forall
y\in\mathcal{Y}.$

We note that the ’non-local’ Hessian terms
$|v^{\prime}_{\ell}+v^{\prime}_{\ell+1}|^{2}$ can be rewritten in terms of the
’local’ terms $|v^{\prime}_{\ell}|^{2}$ and $|v^{\prime}_{\ell+1}|^{2}$ and a
strain-gradient correction,

$|v^{\prime}_{\ell}+v^{\prime}_{\ell+1}|^{2}=2|v^{\prime}_{\ell}|^{2}+2|v^{\prime}_{\ell+1}|^{2}-\varepsilon^{2}|v^{\prime\prime}_{\ell}|^{2}.$

Using this formula, we can rewrite the Hessian in the form

$E_{\rm
a}^{\prime\prime}(\boldsymbol{y})[\boldsymbol{v},\boldsymbol{v}]=\varepsilon\sum_{\ell=1}^{N}A_{\ell}|v_{\ell}^{\prime}|^{2}+\varepsilon\sum_{\ell=1}^{N}B_{\ell}|v_{\ell}^{\prime\prime}|^{2},$

where

$\displaystyle
A_{\ell}(\boldsymbol{y})=\phi^{\prime\prime}(y_{\ell}^{\prime})+2\phi^{\prime\prime}(y_{\ell-1}^{\prime}+y_{\ell}^{\prime})+2\phi^{\prime\prime}(y_{\ell}^{\prime}+y_{\ell+1}^{\prime})$
(4.3) $\displaystyle
B_{\ell}(\boldsymbol{y})=-\phi^{\prime\prime}(y^{\prime}_{\ell}+y^{\prime}_{\ell+1}).$

Recall our assumption that $\phi$ is convex in $(0,{r_{\ast}})$ and concave in
$({r_{\ast}},+\infty)$. For typical pair interaction potentials,
$y^{\prime}_{\ell}<{r_{\ast}}/2$ can only be achieved under extreme
compressive forces. Since, under such extreme conditions a pair potential may
be an inappropriate model to employ anyhow, it is not too restrictive to
assume that the the projected QC solution $J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}}$ satisfies

$(J_{\mathcal{U}_{{\rm qc}}}y^{{\rm
qc}})^{\prime}_{\ell}\geq{r_{\ast}}/2\qquad\forall\ell\in\mathbb{Z}.$

As a result of this assumption, and the properties of $\phi$, we have
$-\phi^{\prime\prime}(y^{\prime}_{\ell}+y^{\prime}_{\ell+1})\geq 0\
\forall\ell\in\mathbb{Z}$ and thus $B_{\ell}\geq 0\ \forall\ell\in\mathbb{Z}$.

As an immediate consequence we obtain the following lemma, which gives
sufficient conditions under which the a posteriori stability of QC
approximation can be guaranteed.

Lemma 6.  Let $J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}}\in\mathcal{Y}^{\varepsilon}$ satisfies $\min_{\ell}(J_{\mathcal{U}_{{\rm
qc}}})^{\prime}_{\ell}\geq{r_{\ast}}/2$; then,

$E^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})[\boldsymbol{v},\boldsymbol{v}]\geq A_{\ast}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm
qc}})\|\boldsymbol{v}^{\prime}\|_{\ell_{\varepsilon}^{2}}^{2}\qquad\forall\boldsymbol{v}\in\mathcal{U},\quad\text{where}\quad
A_{\ast}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})=\min_{\ell=1,\dots,N}A_{\ell}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}}).$

The coefficients $A_{\ell}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})$ are defined in (4.3).

###### Proof.

If $\min_{\ell}(J_{\mathcal{U}_{{\rm qc}}}y^{{\rm
qc}})^{\prime}_{\ell}\geq{r_{\ast}}/2$, then

$\displaystyle\qquad E^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm
qc}})[\boldsymbol{v},\boldsymbol{v}]\geq\varepsilon\sum_{\ell=1}^{N}A_{\ell}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})|v^{\prime}_{\ell}|^{2}\geq
A_{\ast}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})\varepsilon\sum_{\ell=1}^{N}|v^{\prime}_{\ell}|^{2}=A_{\ast}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm
qc}})\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}}^{2}.\qquad\qquad\qquad\qed$

Before we present the main theorem and its proof in the next section, we state
a useful auxiliary result: a local Lipschitz bound on $E^{\prime\prime}_{\rm
a}$. The proof of this Lipschitz bound is straightforward and is therefore
omitted.

Lemma 7.  Let $\boldsymbol{y},\boldsymbol{z}\in\mathcal{Y}^{\varepsilon}$ such
that $\min_{\ell}y_{\ell}^{\prime}\geq\mu$ and
$\min_{\ell}z_{\ell}^{\prime}\geq\mu$ for some constant $\mu>0$, then

$\big{|}\\{\mathcal{E}_{\rm a}^{\prime\prime}(\boldsymbol{y})-\mathcal{E}_{\rm
a}^{\prime\prime}(\boldsymbol{z})\\}[\boldsymbol{v},\boldsymbol{w}]\big{|}\leq
C_{\rm
Lip}\|\boldsymbol{y}^{\prime}-\boldsymbol{z}^{\prime}\|_{\ell_{\varepsilon}^{\infty}}\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}}\|\boldsymbol{w}^{\prime}\|_{\ell^{2}_{\varepsilon}}\qquad\forall\boldsymbol{v},\boldsymbol{w}\in\mathcal{U},$

where $C_{\rm Lip}=M_{3}([\mu,+\infty))+{8M_{3}([2\mu,+\infty))}$ and
$M_{i}(S)=\max_{\xi\in\mathcal{S}}|\phi^{i}(\mathcal{S})|$.

##  5\. A Posteriori Error Estimates

###  5.1. The a posterior error estimates for the deformation gradient

The error we estimate is $e:=y_{\rm a}-J_{\mathcal{U}_{{\rm qc}}}y_{{\rm qc}}$
in the $\mathcal{U}^{1,2}$-norm, as for $y_{1},y_{2}\in\mathcal{Y}$,
$y_{1}-y_{2}\in\mathcal{U}$. To avoid technicalities associated with the
nonlinearity of our models, we make an a priori assumption: we assume the
existence of the atomistic and QC solutions and make a mild requirement on
their smoothness and closeness (cf. (5.1)).

Theorem 8.  Let $y_{{\rm qc}}$ be a solution of the QC problem (2.25) whose
gradients are such that $\min_{\ell}\big{(}J_{\mathcal{U}_{{\rm qc}}}y^{{\rm
qc}}\big{)}^{\prime}_{\ell}\geq r_{*}/2\ \forall\ell\in\mathbb{Z}$ and
$A_{*}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}})>0$, where $A_{*}$
is defined in the statement of Lemma 4. Suppose, further, that $y^{{\rm a}}$
is a solution of the atomistic model (2.8) such that, for some $\tau>0$,

$\|(y^{{\rm a}}-J_{\mathcal{U}_{{\rm qc}}}y_{{\rm
qc}})^{\prime}\|_{L^{\infty}[0,1]}=\|(\boldsymbol{y}^{{\rm
a}}-J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})^{\prime}\|_{\ell^{\infty}_{\varepsilon}}\leq\tau.$ (5.1)

Then, if $\tau$ is sufficiently small, we have the error estimate

$\|y^{\rm a}-J_{\mathcal{U}_{{\rm qc}}}y_{{\rm
qc}}\|_{L^{2}[0,1]}=\|(\boldsymbol{y}^{{\rm a}}-J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm
qc}})^{\prime}\|_{\ell^{2}_{\varepsilon}}=\leq{{\textstyle\frac{2}{A_{*}(\boldsymbol{y}^{{\rm
a}})}}}\big{(}\mathscr{E}_{\rm store}(y_{{\rm qc}})+\mathscr{E}_{\rm
ext}(f)\big{)},$ (5.2)

where the functional of the residual of the stored energy $\mathscr{E}_{\rm
store}(\cdot)$ is defined in (3.11) and the functional of the approximation
error for the external forces $\mathscr{E}_{\rm ext}(\cdot)$ is defined in
(3.51).

###### Proof.

From the mean value theorem we deduce that there exists
$\boldsymbol{\theta}\in{\rm conv}\\{\boldsymbol{y}^{{\rm
a}},J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}}\\}$ such that

$\displaystyle E^{\prime\prime}_{\rm
a}(\boldsymbol{\theta})[\boldsymbol{e},\boldsymbol{e}]=$ $\displaystyle
E^{\prime}_{\rm a}(\boldsymbol{y}^{{\rm a}})[\boldsymbol{v}]-E^{\prime}_{{\rm
qc}}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})[J_{\mathcal{U}}\boldsymbol{v}]$ $\displaystyle=$
$\displaystyle\big{(}\mathcal{E}^{\prime}_{\rm a}(\boldsymbol{y}^{{\rm
a}})[\boldsymbol{v}]-\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})[J_{\mathcal{U}}\boldsymbol{v}]\big{)}$
$\displaystyle-$
$\displaystyle\big{(}\langle\boldsymbol{f},\boldsymbol{v}\rangle_{\varepsilon}-\langle\boldsymbol{f},J_{\mathcal{U}}\boldsymbol{v}\rangle_{h}\big{)}.$

The first group was analyzed in section 3.2 Theorem 3.2 and the second group
was analyzed in section 3.3 Theorem 3.3. Inserting these estimates we arrive
at

$E^{\prime\prime}_{\rm
a}(\boldsymbol{\theta})[\boldsymbol{e},\boldsymbol{e}]\leq\big{(}\mathscr{E}_{\rm
store}(y^{{\rm qc}})+\mathscr{E}_{\rm
ext}(f)\big{)}\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}.$ (5.3)

It remains to prove a lower bound on $\mathcal{E}^{\prime\prime}_{\rm
a}(\boldsymbol{\theta})[\boldsymbol{e},\boldsymbol{e}]$. From our assumption
that $\min(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})^{\prime}_{\ell}\geq r_{*}/2$, and from (5.1) it follows that

$\min_{\ell}\theta^{\prime}_{\ell}\geq r_{*}/2-\tau.$

Assuming that $\tau$ is sufficiently small, e.g.,
$\tau\leq\tau_{1}:=\frac{1}{4}\min_{\ell}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})^{\prime}_{\ell}$, we can apply Lemma 4 to
deduce that

$\displaystyle\mathcal{E}^{\prime\prime}_{\rm
a}(\boldsymbol{\theta})[\boldsymbol{e},\boldsymbol{e}]\geq$
$\displaystyle\mathcal{E}^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})[\boldsymbol{e},\boldsymbol{e}]-C_{\rm
Lip}\|(\boldsymbol{\theta}-\boldsymbol{y}^{{\rm
a}})\|_{\ell^{\infty}_{\varepsilon}}\|\boldsymbol{e}^{\prime}\|^{2}_{\ell^{2}_{\varepsilon}}$
$\displaystyle\geq$ $\displaystyle\mathcal{E}^{\prime\prime}_{\rm
a}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})[\boldsymbol{e},\boldsymbol{e}]-C_{\rm
Lip}\tau\|\boldsymbol{e}^{\prime}\|^{2}_{\ell^{2}_{\varepsilon}},$ (5.4)

where $C_{\rm Lip}$ may depend on $\tau_{1}$.

We can now apply our stability analysis in Section 4. Since
$(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}})_{\ell}^{\prime}\geq
r_{*}/2$ for all $\ell$, Lemma 4 implies that

$\mathcal{E}^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})[\boldsymbol{e},\boldsymbol{e}]\geq
A_{*}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}^{2},$

which, combined with (5.3) and (5.1) , yields

$\big{(}A_{*}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}})-C_{\rm
Lip}\tau\big{)}\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}^{2}\leq\mathcal{E}^{\prime\prime}_{\rm
a}(\theta)[\boldsymbol{e},\boldsymbol{e}]\leq\big{(}\mathscr{E}_{\rm
store}(y_{{\rm qc}})+\mathscr{E}_{\rm
ext}(f)\big{)}\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}.$

Dividing through by $\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}$,
and assuming that $\tau\leq\min(\tau_{1},\tau_{2})$ where
$\tau_{2}=A_{*}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}})/(2C_{\rm
Lip})$, we deduce that

${\textstyle\frac{A_{*}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})}{2}}\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}\leq\big{(}\mathscr{E}_{\rm
store}(y_{{\rm qc}})+\mathscr{E}_{\rm ext}(f)\big{)},$

which concludes the proof of the a posteriori error estimate for the
deformation gradient. ∎

###  5.2. The a posterior error estimate for the energy

Besides the deformation gradient, the energy of the system is another quantity
of interest. In this section, we derive an a posteriori error estimator for
the energy difference between the atomistic model and the QC approximation,
namely,

$E_{\rm a}(y^{{\rm a}})-E_{{\rm qc}}(y_{{\rm qc}}).$ (5.5)

To analyze this difference, we decompose (5.5) as

$\big{|}E_{\rm a}(y^{{\rm a}})-E_{{\rm qc}}(y_{{\rm qc}})\big{|}=\big{|}E_{\rm
a}(y^{{\rm a}})-E_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{{\rm
qc}})\big{|}+\big{|}E_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{{\rm qc}})-E_{{\rm
qc}}(y_{{\rm qc}})\big{|}.$ (5.6)

We then analyze the two groups separately.

To analyze the first group, we have the following Lemma.

Lemma 9.  Let $y,z\in\mathcal{Y}$ and
$\boldsymbol{y},\boldsymbol{z}\in\mathcal{Y}^{\varepsilon}$ be their
vectorizations, such that $\min_{\ell}y_{\ell}^{\prime}\geq\mu$ and
$\min_{\ell}z_{\ell}^{\prime}\geq\mu$ for some constant $\mu>0$, and $y\in{\rm
argmin}E_{\rm a}(\mathcal{Y})$. Let
$\boldsymbol{e}=\boldsymbol{y}-\boldsymbol{z}$, then

$\big{|}E_{\rm a}(\boldsymbol{y})-E_{\rm a}(\boldsymbol{z})\big{|}\leq
C^{E}_{\rm Lip}\|\boldsymbol{e}^{\prime}\|^{2}_{\ell^{2}_{\varepsilon}},$
(5.7)

where $C^{E}_{\rm
Lip}=\frac{1}{2}M_{2}([\mu,+\infty))+{2M_{3}([2\mu,+\infty))}$, where
$M_{i}(S)=\max_{\xi\in\mathcal{S}}|\phi^{i}(\mathcal{S})|$.

###### Proof.

We first rewrite the difference of the total energy as the summation of the
differences of the stored energy and that of the external energy:

$E_{\rm a}(\boldsymbol{y})-E_{\rm a}(\boldsymbol{z})=\big{(}\mathcal{E}_{\rm
a}(\boldsymbol{y})-\mathcal{E}_{\rm
a}(\boldsymbol{z})\big{)}-\big{(}\langle\boldsymbol{f},\boldsymbol{z}\rangle_{\varepsilon}-\langle\boldsymbol{f},\boldsymbol{y}\rangle_{\varepsilon}\big{)}$

For the difference of the stored energy, we have

$\displaystyle\mathcal{E}_{\rm a}(\boldsymbol{y})-\mathcal{E}_{\rm
a}(\boldsymbol{z})=$
$\displaystyle\varepsilon\sum_{\ell=1}^{N}\big{[}\phi(y^{\prime}_{\ell})-\phi(z^{\prime}_{\ell})\big{]}+\varepsilon\sum_{\ell=1}^{N}\big{[}\phi(y^{\prime}_{\ell}+y^{\prime}_{\ell+1})-\phi(z^{\prime}_{\ell}+z^{\prime}_{\ell+1})\big{]}$
$\displaystyle=$
$\displaystyle\varepsilon\sum_{\ell=1}^{N}\big{[}\phi^{\prime}(y^{\prime}_{\ell})e^{\prime}_{\ell}-\frac{1}{2}\phi^{\prime\prime}(\xi^{1}_{\ell}){e^{\prime}_{\ell}}^{2}\big{]}$
$\displaystyle+$
$\displaystyle\varepsilon\sum_{\ell=1}^{N}\big{[}\phi^{\prime}(y^{\prime}_{\ell}+y^{\prime}_{\ell+1})(e^{\prime}_{\ell}+e^{\prime}_{\ell+1})-\frac{1}{2}\phi^{\prime\prime}(\xi^{2}_{\ell})(e^{\prime}_{\ell}+e^{\prime}_{\ell+1})^{2}\big{]}$
$\displaystyle=$ $\displaystyle\mathcal{E}_{\rm
a}^{\prime}(\boldsymbol{y})[\boldsymbol{e}]-\frac{1}{2}\varepsilon\sum_{\ell=1}^{N}\phi^{\prime\prime}(\xi^{1}_{\ell}){e^{\prime}_{\ell}}^{2}-\frac{1}{2}\varepsilon\sum_{\ell=1}^{N}\phi^{\prime\prime}(\xi^{2}_{\ell})(e^{\prime}_{\ell}+e^{\prime}_{\ell+1})^{2},$

where $\xi^{1}_{\ell}\in{\rm conv}\\{y^{\prime}_{\ell},z^{\prime}_{\ell}\\}$
and $\xi^{2}_{\ell}\in{\rm
conv}\\{y^{\prime}_{\ell}+y^{\prime}_{\ell+1},z^{\prime}_{\ell}+z^{\prime}_{\ell+1}\\}$.

For the difference of the energy caused by the external forces, we have

$\langle\boldsymbol{f},\boldsymbol{z}\rangle_{\varepsilon}-\langle\boldsymbol{f},\boldsymbol{y}\rangle_{\varepsilon}=-\langle\boldsymbol{f},\boldsymbol{e}\rangle_{\varepsilon}=-\mathcal{E}_{\rm
a}^{\prime}(\boldsymbol{y})[\boldsymbol{e}],$

by the first optimality condition of $y\in{\rm argmin}E_{\rm a}(\mathcal{Y})$.
It is then easy to obtain the estimate stated in the Lemma by using Cauchy-
Schwaz inequality to the non-local term. ∎

Lemma 10.  For $y_{h}\in\mathcal{Y}_{{\rm qc}}$ and $y^{\prime}_{h}(x)>0$, we
have

$\big{|}E_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{h})-E_{{\rm
qc}}(y_{h})\big{|}\leq\sum_{k\in\mathcal{K}_{c}}{\eta_{E}^{e}}_{k}+{\eta_{E}^{f}}_{k},$
(5.8)

where

$\eta_{k}^{e}=\frac{1}{2}\sum_{j=-1}^{1}[[\phi]]_{\ell_{k-1}+j}+\frac{1}{2}\sum_{j=-1}^{1}[[\phi]]_{\ell_{k}+j},$
(5.9)

if $k\in\mathcal{K}^{\prime}_{c}$,

$\eta_{k}^{e}=\sum_{j=-1}^{1}[[\phi]]_{\ell_{La_{i}}+j}+\frac{1}{2}\sum_{j=-1}^{1}[[\phi]]_{\ell_{k-1}+j},$
(5.10)

if $k=L_{a_{i}}$ for some $i\in\\{1,2,\ldots,M\\}$, and

$\eta_{k}^{e}=\sum_{j=-1}^{1}[[\phi]]_{\ell_{Ra_{i}}+j}+\frac{1}{2}\sum_{j=-1}^{1}[[\phi]]_{\ell_{k}+j},$
(5.11)

if $k=R_{a_{i}}$ for some $i\in\\{1,2,\ldots,M\\}$. $[[\phi]]_{\ell_{k}}$’s
and ${\eta_{E}^{f}}_{k}$’s will be defined in the proof.

###### Proof.

We first decompose the energy difference to two parts:

$E_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{h})-E_{{\rm
qc}}(y_{h})=\big{(}\mathcal{E}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})-\mathcal{E}_{{\rm qc}}(y_{h})\big{)}-\big{(}\langle
f,J_{\mathcal{U}_{{\rm qc}}}y_{h}\rangle_{\varepsilon}-\langle
f_{h},y_{h}\rangle_{h}\big{)}.$

We first analyze the energy difference of the stored energy. Since
$r_{b}D_{b}J_{\mathcal{U}_{{\rm qc}}}y_{h}=r_{b}D_{b}y_{h}$, we have

$\mathcal{E}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})=\sum_{b\in\mathcal{B}}a_{b}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})=\sum_{b\in\mathcal{B}}\varepsilon\phi(r_{b}D_{b}J_{\mathcal{U}_{{\rm
qc}}}y_{h})=\sum_{b\in\mathcal{B}}\varepsilon\phi(r_{b}D_{b}y_{h}),$ (5.12)

and

$\mathcal{E}_{{\rm
qc}}(y_{h})=\sum_{b\in\mathcal{B}}\big{[}a_{b}(y_{h})+c_{b}(y_{h})\big{]}=\sum_{b\in\mathcal{B}}\bigg{[}\frac{|b\cap\Omega_{\rm
a}|}{r_{b}}\phi\big{(}r_{b}D_{b\cap\Omega_{\rm
a}}y\big{)}+\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x\bigg{]}.$ (5.13)

We analyze the energy difference bond by bond,

If $b\subset\Omega_{\rm a}$, then

$a_{b}(y_{h})+c_{b}(y_{h})=a_{b}(y_{h})=\varepsilon\phi(r_{b}D_{b}y_{h})=a_{b}(J_{\mathcal{U}_{{\rm
qc}}}y_{h}),$

and the energy difference in this bond is thus $0$.

If $b\subset\Omega_{c}\cap T_{k}$ for some $k\in\mathcal{K}_{c}$, then
$|b\cap\Omega_{\rm a}|=0$ and

$\displaystyle a_{b}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})-\big{[}a_{b}(y_{h})+c_{b}(y_{h})\big{]}=$ $\displaystyle
a_{b}(J_{\mathcal{U}_{{\rm qc}}}y_{h})-c_{b}(y_{h})$ $\displaystyle=$
$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y_{h}(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\frac{1}{r_{b}}\int_{b}\big{[}\phi(r_{b}D_{b}y_{h})-\phi(\nabla_{r_{b}}y_{h}(x))\big{]}\,{\rm
d}x.$ (5.14)

Since $y_{h}$ is affine on $T_{k}$, $\nabla_{r_{b}}y(x))=r_{b}D_{b}y_{h}$ and
subsequently, (5.2) is $0$.

We are left with the interaction bonds crossing the atomistic-continuum
interface and the boundaries of the elements in the continuum region. Again
because of its tediousness, we leave the detail of this analysis to the
Appendix and only give the results here:

$\displaystyle\mathcal{E}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})-\mathcal{E}_{{\rm
qc}}(y_{h})=\sum_{i=1}^{M}\sum_{j=-1}^{1}[[\phi]]_{\ell_{La_{i}}+j}+\sum_{i=1}^{M}\sum_{j=-1}^{1}[[\phi]]_{\ell_{Ra_{i}}+j}+\sum_{k\in\mathcal{K}^{\prime}_{c}}\sum_{j=-1}^{1}[[\phi]]_{\ell_{k}+j}.$
(5.15)

For $k=L_{a_{i}}$ where $i\in\\{1,2,\ldots,M\\}$, we have

$\displaystyle[[\phi]]_{\ell_{k}}=$
$\displaystyle\varepsilon\big{\\{}\phi((1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})\big{\\}},$
(5.16)
$\displaystyle[[\phi]]_{\ell_{k-1}}=2\varepsilon\big{\\{}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})\theta_{k}y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})-(1+\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})\big{\\}}.$
(5.17)

and

$\displaystyle[[\phi]]_{\ell_{k+1}}=$ $\displaystyle
2\varepsilon\big{\\{}\phi(y_{h}^{\prime}|_{T_{k+2}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}+(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}})$
$\displaystyle-(2-\theta_{k})\phi\big{(}\frac{2}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k+2}}+\frac{2(1-\theta_{k})}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k+1}}\big{)}-\theta_{k}\phi(2y_{h}^{\prime}|_{T_{k}})\big{\\}}.$
(5.18)

For $k=R_{a_{i}}$ where $i\in\\{1,2,\ldots,M\\}$, we have

$\displaystyle[[\phi]]_{\ell_{k}}=\varepsilon\big{\\{}\phi((1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})\big{\\}},$
(5.19) $\displaystyle[[\phi]]_{\ell_{k-1}}=$ $\displaystyle
2\varepsilon\big{\\{}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}+y_{h}^{\prime}|_{T_{k-1}})$
$\displaystyle-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})-(1+\theta_{k})\phi\big{(}\frac{2\theta_{k}}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k}}+\frac{2}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k-1}}\big{)}\big{\\}},$
(5.20)

and

$\displaystyle[[\phi]]_{\ell_{k+1}}=2\varepsilon\big{\\{}\phi((2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-(2-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})-\theta_{k}\phi\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}\big{\\}}.$
(5.21)

For $k\in\mathcal{K}^{\prime}_{c}$, we have

$\displaystyle[[\phi]]_{\ell_{k}}=\varepsilon\big{\\{}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})\big{\\}},$
(5.22)
$\displaystyle[[\phi]]_{\ell_{k}-1}=\frac{1}{2}\varepsilon\big{\\{}2\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})-(1+\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})\big{\\}},$
(5.23)

and

$\displaystyle[[\phi]]_{\ell_{k}+1}=\frac{1}{2}\varepsilon\big{\\{}2\phi((2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-\theta_{k}\phi(2y_{h}^{\prime}|_{T_{k+1}})-(2-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})\big{\\}}.$
(5.24)

We then analyze the energy difference caused by the external forces. The
energy difference is given by

$\langle f,J_{\mathcal{U}_{{\rm qc}}}u_{h}\rangle_{\varepsilon}-\langle
f,u_{h}\rangle_{h}=\langle f,u_{h}\rangle_{\varepsilon}-\langle
f,u_{h}\rangle_{h},$

since $J_{\mathcal{U}_{{\rm qc}}}u_{h}=u_{h}+C$ for some contant $C$ and
$\langle f,C\rangle_{\varepsilon}=0\ \forall C$. We decompose this energy
difference to each element and write it as

$\big{|}\langle f,u_{h}\rangle_{\varepsilon}-\langle
f,u_{h}\rangle_{h}\big{|}\leq\sum_{k=1}^{K}{\eta_{E}^{f}}_{k},$ (5.25)

where

$\displaystyle{\eta_{E}^{f}}_{k}=$
$\displaystyle\bigg{|}(1-\theta_{k})\frac{1}{2}\varepsilon(f_{\ell_{k-1}}u_{\ell_{k-1}}+f_{\ell_{k-1}+1}u_{\ell_{k-1}+1})+\frac{1}{2}\sum_{\ell=\ell_{k-1}+2}^{\ell_{k}-1}\varepsilon(f_{\ell}u_{\ell}+f_{\ell+1}u_{\ell+1})$
$\displaystyle+\theta\frac{1}{2}\varepsilon(f_{\ell_{k}}u_{\ell_{k}}+f_{\ell_{k}+1}u_{\ell_{k}+1})\big{\\}}-\sum_{k=1}^{K}\frac{1}{2}(x_{k}-x_{k-1})\big{[}f(x_{k-1})y(x_{k-1})+f(x_{k})y(x_{k})\big{]}\bigg{|},$
(5.26)

where $f_{\ell}=f(\ell\varepsilon)$ and $u_{\ell}=u(\ell\varepsilon)$. ∎

##  6\. Numerical Experiments

In this section, we present numerical experiments to illustrate our analysis.
Throughout this section we fix $F=1$, $N=8193$, and let $\phi$ be the Morse
potential

$\phi(r)=\exp(-2\alpha(r-1))-2\exp(-\alpha(r-1)),$

with the parameter $\alpha=5$.

For our benchmark problem, we defined the external force $\boldsymbol{f}$ to
be

$\displaystyle
f_{\ell}=\left\\{\begin{array}[]{rl}-0.1\big{\\{}1-\frac{|\ell-\frac{N-1}{2}|}{\frac{N-1}{2}}\big{\\}}\frac{N}{|\ell-\frac{N-1}{2}-0.5|},&\text{for
$\ell\leq\frac{N-1}{2}$},\\\
0.1\big{\\{}1-\frac{\ell-\frac{N-1}{2}-1}{\frac{N-1}{2}}\big{\\}}\frac{N}{|\ell-\frac{N-1}{2}-0.5|},&\text{for
$\ell\geq\frac{N-1}{2}+1$.}\end{array}\right.$

We briefly explain the meaning of the external force. On each atom, the
external force is a product of three components. The third component, namely
$\frac{N}{|\ell-\frac{N-1}{2}-0.5|}$ is essentially $\frac{1}{r_{\ell}}$ where
$r_{\ell}$ is the distance between an atom and the center of this atomistic
chain located at $\frac{N-1}{2}+0.5$. This non-linear force will create a
defect in the middle of the chain but affect little in the far field. The
second component, namely $1-\frac{|\ell-\frac{N-1}{2}|}{\frac{N-1}{2}}$, adds
a decay of the first component and in particular, it is $0$ when $\ell=N$,
which prevents the ’kink’ of the force on the boundary due to a rapid change
of the sign of the force that will leads to non-smooth deformation gradient
that should be contained in the atomistic region. The first component, which
is the constant $0.1$, is to rescale the force so that the solution of this
problem is stable.

We solve for the atomistic problem and consider the solution to be the
accurate solution. We then solve for the QC problem on different meshes
generated by the mesh refinement schemes.

We show two relative errors against the number of degrees of freedom. The
first one is the error of the deformation gradient in $L_{2}$-norm over the
$L_{2}$-norm of the difference between the deformation gradient of the
atomistic solution and the homogeneous state, which is defined by

$e_{deformation}:=\frac{\|y_{{\rm qc}}^{\prime}-y_{\rm
a}^{\prime}\|_{L^{2}[0,1]}}{\|y_{\rm a}^{\prime}-Fx\|_{L^{2}[0,1]}}.$ (6.1)

The second relative error is the absolute value of the energy difference of
the atomistic solution and the QC solution over the absolute value of the
energy change of the atomistic solution from the homogeneous state, which is
defined by

$e_{energy}:=\frac{|E_{\rm a}(y_{\rm a})-E_{{\rm qc}}(y_{{\rm qc}})|}{|E_{\rm
a}(y_{\rm a})-E_{\rm a}(Fx)|}.$ (6.2)

Before we present the plots of the errors, we first introduce the mesh
generating schemes.

###  6.1. Mesh Construction

To avoid unnecessary technical difficulty in the mesh refinement algorithm, we
assume that the defect core is already captured in the middle of the chain.
There are three mesh generating schemes we use.

The first mesh generating scheme is derived in Section 7.1 of [6] using
calculus of variations. From this analysis, we get that the (quasi-)optimal
mesh size in the continuum region, with the restriction that the atomistic
region is symmetric and has $K$ atoms on each side, is given by

$h(r)=\big{(}\frac{f(K\varepsilon)}{f(r)}\frac{r}{K\varepsilon}\big{)}^{\frac{2}{3}}.$
(6.3)

Since the mesh size can not change continuously and we restrict the smallest
mesh size in the continuum region to be $2\varepsilon$, we use the following
algorithm to generate this mesh (we only list the case on the right hand side
of the atomistic region):

Algorithm 1.

  1. (1)

Set atom $\frac{N-1}{2}+1$ to be the middle of the atomistic region.

  2. (2)

Choose $K$ so that there are $K$ atoms on each side of the atomistic region.

  3. (3)

Choose $h$ to be $2\varepsilon$ for every element on the right hand side of
the atomistic region until $h(r)>2\varepsilon$, where $r$ is the distance
between the right boundary of the previous element and the middle of the
atomistic region.

  4. (4)

Choose $h$ according to (6.3) until the right boundary of the newly created
element is out of the right limit of the chain.

∎

The second mesh generating scheme is essentially a mesh refinement process
according to the error estimator with respect to the deformation gradient
according to Lemma 3.3, Lemma 3.3 and Lemma 3.3. The mesh refinement algorithm
is stated as follows:

Algorithm 2.

  1. (1)

Set atom $\frac{N-1}{2}+1$ to be the middle of the atomistic region.

  2. (2)

Choose $5$ atoms on each side of the atomistic region.

  3. (3)

Divide the left and the right part of the continuum region into two equally
large element

  4. (4)

Compute the QC solution on this mesh and then compute the squared error
indicator of each element $\eta_{i}$ and sort these indicators according to
its value.

  5. (5)

Bisect the first $M$ sorted elements such that

$\sum_{i=1}^{M-1}\eta^{2}_{i}\leq 0.5\eta^{2}\text{ and
}\sum_{i=1}^{M}\eta^{2}_{i}\geq 0.5\eta^{2},$ (6.4)

where $\eta_{i}$ is the error estimator of each element defined by

$\eta^{deformation}_{i}=\big{[}(\eta^{e}_{i})^{2}+(\eta^{f}_{i})^{2}\big{]})^{\frac{1}{2}}\big{/}{\textstyle\frac{A_{*}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})}{2}}.$ (6.5)

If the element is near the atomistic region, merge the element into the
atomistic region.

  6. (6)

If the resulting mesh reaches the maximal number of degrees of freedom, stop
the process, else, go to Step 4.

∎

The third mesh generating scheme is the mesh refinement process according to
the error estimator with respect to the energy which is defined by

$\eta^{energy}_{i}=C^{E}_{Lip}\big{(}\eta^{deformation}_{i}\big{)}^{2}+{\eta^{e}_{E}}_{k}+{\eta^{f}_{E}}_{k},$
(6.6)

for each element and the refinement algorithm is exactly the same.

In short, the first and second mesh generating schemes tend to minimize the
error in the deformation gradient and the third one tends to minimize the
error in the total energy.

###  6.2. Numerical Results

We compare the relative errors defined in 6.1 and 6.2. We plot the relative
errors against the number of degrees of freedom with respect to the meshes
generated.

Figure 1. Relative Error of the Gradient Figure 2. Efficiency Factor of the
Gradient

Figure 1 shows that the pre-defined optimal mesh performs better than the two
mesh refinement strategies for a fix number of degrees of freedom. The
possible reason for this is that, due to some technical difficulty in coding,
both of the mesh refinement algorithms tend to produce larger atomistic region
by merging the elements in the continuum region to the atomistic region and
create some unnecessary degrees of freedom. For the two mesh refinement
strategies, the one according to the gradient error indicator perform better
asymptotically.

Figure 2 shows the efficiency factor of the error estimator of the deformation
gradient. It shows that the efficiency factor is comparatively large but
decreases as the number of degrees of freedom increases and finally become
stable. The reason for this phenomenon lies in the form of the external force.
One can show that if the external force takes the form of $f(r)=\frac{1}{r}$,
where $r$ is the distance to the centre of the defect, then the residual due
to the external force is of order $h^{2}$ as opposed to order $h$ in general
which is achieved by our analysis. As a result, our estimate exaggerate the
real error by $\frac{1}{h}$ for this particular external force. This
phenomenon gradually disappear as the continuum region moves apart from the
centre of the defect since the influence of this exaggeration is eliminated as
the external force tends to $0$ when it is away from the centre of the defect,
which makes the residual of the sotred energy become the leading error term.
It can also well explain the fact that the efficiency of the estimate is
better for the mesh refinement strategies than the pre-defined mesh for a
certain number of degrees of freedom, as the two mesh refinement algorithms
tend to put more atoms in the atomistic region, i.e., the continuum region is
further away from the centre of defect than that of the pre-defined mesh.

Figure 3. Relative Error of the Total Energy Figure 4. Efficiency Factor of
the Energy

Figure 3 shows that the refinement based on the energy error performs the best
among all the three mesh generating schemes.

Figure 4 shows the efficiency factor of the error estimator of the energy. For
the same reason, this factor decreases as the number of degrees of freedom
increases and finally becomes stable.

##  7\. Conclusion

We have presented the a posteriori error estimates for the Consistent Energy-
Based QC method in one dimension. The procedure of the estimate is the same as
that in [8]. However, since the formulation of the QC problem is newly
developed and is totally different from previous ones, new techniques have
been developed and applied to deal with the difficulty in the analysis.
Several results derived may be of independent interest and usefulness. In
addition, the error estimate of the total energy is also derived. Numerical
experiments are also implemented to illustrate our analysis.

Particular interesting future work are the extension and the implementation of
the a posteriori error estimate in higher dimensional problems. The difficulty
lies in the complication of the formulation and the varied location of the
interaction bonds. However, since a priori analysis for the two dimensional
problem has been proposed [6], ways of circumventing these difficulties could
be a source of reference.

##  Appendix A Detailed Analysis for the Residuals of the Stored Energy

In this section, we provide the omitted detailed analysis for the residuals of
the stored energy, namely

$\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})[v]-\mathcal{E}^{\prime}_{{\rm qc}}(y_{h})[J_{\mathcal{U}}[v]\text{
and }\mathcal{E}_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{h})-\mathcal{E}_{{\rm
qc}}(y_{h}),$

where $y_{h}\in\mathcal{Y}_{{\rm qc}}$, $y^{\prime}_{h}(x)>0\ \forall
x\in\mathbb{R}$ and $v\in\mathcal{U}$.

The idea is to find the differences defined by

$\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x,$ (A.1)

and

$\varepsilon\phi(r_{b}D_{b}y_{h})-|b\cap\Omega_{\rm
a}|{r_{b}}\phi\big{(}r_{b}D_{b\cap\Omega_{\rm
a}}y\big{)}-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x,$ (A.2)

for each interaction bond $b$.

We have analyzed the cases that $b\in\Omega_{\rm a}$ and $b\in
T_{k}\cap\Omega_{c}$ and are left with the analysis for the cases that $b$ is
across the atomistic-continuum interface and the boundaries of the elements in
the continuum region. There are three cases and in each case there are three
subcases to be considered.

Case 1: $b$ is across two adjacent elements $T_{k},T_{k+1}\in\Omega_{c}$. In
this case $|b\cap\Omega_{\rm a}|=0$ and the atomistic contribution of the
interaction bond in the QC energy is $0$.

Subcase 1: If $b=\big{(}\ell_{k}\varepsilon,(\ell_{k}+1)\varepsilon\big{)}$,
then $r_{b}=1$, $b\cap T_{k}=[\ell_{k}\varepsilon,x^{h}_{k}]$, $b\cap
T_{k+1}=[x^{h}_{k},(\ell_{k}+1)\varepsilon]$,
$r_{b}D_{b}v=v^{\prime}_{\ell_{k}+1}$ and

$r_{b}D_{b}y_{h}=\frac{y_{h}((\ell_{k}+1)\varepsilon)-y_{h}(\ell_{k}\varepsilon)}{\varepsilon}=(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}.$

We have

$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}v^{\prime}_{\ell_{k}+1}-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k+1}})\int_{b\cap
T_{k}}r_{b}v^{\prime}\,{\rm
d}x-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k}})\int_{b\cap
T_{k+1}}r_{b}v^{\prime}\,{\rm d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\phi^{\prime}(r_{b}D_{b}y_{h})-(1-\theta_{k})\phi^{\prime}(y_{h}^{\prime}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(\theta_{k}y_{h}^{\prime}|_{T_{k}})\bigg{\\}}v^{\prime}_{\ell_{k}+1},$

and

$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-\int_{b\cap
T_{k}}\phi(y_{h}^{\prime}|_{T_{k}})\,{\rm d}x-\int_{b\cap
T_{k+1}}\phi(y_{h}^{\prime}|_{T_{k+1}})\,{\rm d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{(}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})\bigg{)}.$
(A.3)

Subcase 2: If
$b=\big{(}(\ell-1)_{k}\varepsilon,(\ell_{k}+1)\varepsilon\big{)}$, then
$r_{b}=2$, $b\cap T_{k}=\big{[}(\ell_{k}-1)\varepsilon,x^{h}_{k}\big{]}$,
$b\cap T_{k+1}=\big{[}x^{h}_{k},(\ell_{k}+1)\varepsilon\big{]}$,
$r_{b}D_{b}v=v^{\prime}_{\ell_{k}+1}+v^{\prime}_{\ell_{k}}$ and

$r_{b}D_{b}y_{h}=\frac{y_{h}(\ell_{k}\varepsilon)-y_{h}((\ell_{k}-1)\varepsilon)}{\varepsilon}=(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}.$

We have

$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}(v^{\prime}_{\ell_{k}+1}+v^{\prime}_{\ell_{k}})$
$\displaystyle-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k+1}})\int_{b\cap
T_{k}}r_{b}v^{\prime}\,{\rm
d}x-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k}})\int_{b\cap
T_{k+1}}r_{b}v^{\prime}\,{\rm d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(2y^{\prime}_{h}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}+1}\bigg{\\}},$

and

$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon\bigg{(}2\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}})-(1+\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})\bigg{)}.$
(A.4)

Subcase 3: If $b=\big{(}\ell_{k}\varepsilon,(\ell_{k}+2)\varepsilon\big{)}$,
then $r_{b}=2$, $b\cap T_{k}=\big{[}\ell_{k}\varepsilon,x^{h}_{k}\big{]}$,
$b\cap T_{k+1}=\big{[}x^{h}_{k},(\ell_{k}+2)\varepsilon\big{]}$,
$r_{b}D_{b}v=v^{\prime}_{\ell_{k}+2}+v^{\prime}_{\ell_{k}+1}$ and

$r_{b}D_{b}y_{h}=\frac{y_{h}((\ell_{k}+2)\varepsilon)-y_{h}(\ell_{k}\varepsilon)}{\varepsilon}=(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}.$

We have

$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})(v^{\prime}_{\ell_{k}+2}+v^{\prime}_{\ell_{k}+1})$
$\displaystyle-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k+1}})\int_{b\cap
T_{k}}r_{b}v^{\prime}\,{\rm
d}x-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k}})\int_{b\cap
T_{k+1}}r_{b}v^{\prime}\,{\rm d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(2y^{\prime}_{h}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}+1}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}(2y^{\prime}_{h}|_{T_{k+1}})\big{]}v^{\prime}_{\ell_{k}+2}\bigg{\\}},$

and

$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon\bigg{(}2\phi((2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-\theta_{k}\phi(2y_{h}^{\prime}|_{T_{k+1}})-(2-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})\bigg{)}.$
(A.5)

Case 2: $b$ is across the left atomistic-continuum interface of an atomistic
region.

Subcase 1: If $b=(\ell_{k}\varepsilon,\ell_{k+1}\varepsilon)$, then $r_{b}=1$,
$b\cap\Omega_{c}=(\ell_{k}\varepsilon,x^{h}_{k})$, $b\cap\Omega_{\rm
a}=\big{(}x^{h}_{k},(\ell_{k}+1)\varepsilon\big{)}$,
$r_{b}D_{b}v=v_{\ell_{k}+1}^{\prime}$ and

$r_{b}D_{b}y_{h}=(1-\theta_{k})y_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}.$

We have

$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(y^{\prime}_{h}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(y^{\prime}_{h}|_{T_{k}})\bigg{]}v^{\prime}_{\ell_{k}+1},$
(A.6)

and

$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-|b\cap\Omega_{\rm
a}|{r_{b}}\phi\big{(}r_{b}D_{b\cap\Omega_{\rm
a}}y\big{)}-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\big{(}\phi((1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})\big{)}$
(A.7)

Subcase 2: If
$b=\big{(}(\ell_{k}-1)\varepsilon,(\ell_{k}+1)\varepsilon\big{)}$, then
$r_{b}=2$, $b\cap\Omega_{c}=\big{(}(\ell_{k}-1)\varepsilon,x^{h}_{k})$,
$b\cap\Omega_{\rm a}=\big{(}x^{h}_{k},(\ell_{k}+1)\varepsilon\big{)}$,
$r_{b}D_{b}v=v_{\ell_{k}+1}^{\prime}+v_{\ell_{k}}^{\prime}$ and

$r_{b}D_{b}y_{h}=(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+(1+\theta_{k})y^{\prime}_{h}|_{T_{k}},\quad
r_{b}D_{b\cap\Omega_{\rm a}}y_{h}=2y^{\prime}_{h}|_{T_{k+1}}$

We have

$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+(1+\theta_{k})y^{\prime}_{h}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(2y^{\prime}_{h}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+(1+\theta_{k})y^{\prime}_{h}|_{T_{k}}\big{)}-\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}}-\theta_{k}\phi^{\prime}(y^{\prime}_{h}|_{T_{k}})v^{\prime}_{\ell_{k}+1},$
(A.8)

and

$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-|b\cap\Omega_{\rm
a}|{r_{b}}\phi\big{(}r_{b}D_{b\cap\Omega_{\rm
a}}y\big{)}-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$ $\displaystyle
2\varepsilon\big{(}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})\theta_{k}y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})-(1+\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})\big{)}.$
(A.9)

Subcase 3: If $b=(\ell_{k}\varepsilon,(\ell_{k}+2)\varepsilon)$, then
$r_{b}=2$, $b\cap\Omega_{c}=\big{(}\ell_{k}\varepsilon,x^{h}_{k})$,
$b\cap\Omega_{\rm a}=\big{(}x^{h}_{k},(\ell_{k}+2)\varepsilon\big{)}$,
$r_{b}D_{b}v=v_{\ell_{k}+2}^{\prime}+v_{\ell_{k}+1}^{\prime}$,
$D_{b\cap\Omega_{\rm
a}}v=\frac{1}{2-\theta_{k}}v^{\prime}_{\ell_{k}+2}+\frac{1-\theta_{k}}{2-\theta_{k}}v^{\prime}_{\ell_{k}+1}$,
and

$r_{b}D_{b}y_{h}=y^{\prime}_{h}|_{T_{k+2}}+(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\quad
r_{b}D_{b\cap\Omega_{\rm
a}}y_{h}=\frac{2}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+2}}+\frac{2(1-\theta_{k})}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+1}}.$

We have

$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}y^{\prime}_{h}|_{T_{k+2}}+(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-\phi^{\prime}(\frac{2}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+2}}+\frac{2(1-\theta_{k})}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+1}})\big{]}v^{\prime}_{\ell_{k}+2}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}y^{\prime}_{h}|_{T_{k+2}}+(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(\frac{2}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+2}}+\frac{2(1-\theta_{k})}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+1}})$
$\displaystyle\quad\
-\theta_{k}\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}+1},$
(A.10)

Case 2: $b$ is across the right atomistic-continuum interface of an atomistic
region.

Subcase 1: If $b=(\ell_{k}\varepsilon,\ell_{k+1}\varepsilon)$, then $r_{b}=1$,
$b\cap\Omega_{c}=\big{(}x^{h}_{k},(\ell_{k}+1)\varepsilon\big{)}$,
$b\cap\Omega_{\rm a}=(\ell_{k}\varepsilon,x^{h}_{k})$,
$r_{b}D_{b}v=v_{\ell_{k}+1}^{\prime}$ and

$r_{b}D_{b}y_{h}=(1-\theta_{k})y_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}.$

We have

$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(y^{\prime}_{h}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(y^{\prime}_{h}|_{T_{k}})\bigg{]}v^{\prime}_{\ell_{k}+1},$
(A.11)

and

$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-\int_{b\cap
T_{k}}\phi(y_{h}^{\prime}|_{T_{k}})\,{\rm d}x-\int_{b\cap
T_{k+1}}\phi(y_{h}^{\prime}|_{T_{k+1}})\,{\rm d}x$ $\displaystyle=$
$\displaystyle\varepsilon\big{(}\phi((1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})\big{)}.$
(A.12)

Subcase 2: If $b=\big{(}\ell_{k}\varepsilon,(\ell_{k}+2)\varepsilon\big{)}$,
then $r_{b}=2$,
$b\cap\Omega_{c}=\big{(}x^{h}_{k},(\ell_{k}+2)\varepsilon\big{)}$,
$b\cap\Omega_{\rm a}=(\ell_{k}\varepsilon,x^{h}_{k})$,
$r_{b}D_{b}v=v^{\prime}_{\ell_{k}+2}+v^{\prime}_{\ell_{k}+1}$,
$r_{b}D_{b\cap\Omega_{\rm a}}v=v^{\prime}_{\ell_{k}+1}$ and

$\displaystyle
r_{b}D_{b}y_{h}=(2-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\text{
and }r_{b}D_{b\cap\Omega_{\rm a}}y_{h}=y^{\prime}_{h}|_{T_{k}}.$ (A.13)

We have

$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}(2-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-\theta_{k}\phi^{\prime}\big{(}2y^{\prime}_{h}|_{T_{k}})-(1-\theta_{k})\phi^{\prime}\big{(}2y^{\prime}_{h}|_{T_{k+1}}\big{]}v^{\prime}_{\ell_{k}+1}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}(2-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-\phi^{\prime}(2y^{\prime}_{h}|_{T_{k+1}})\big{]}v^{\prime}_{\ell_{k}+2},$
(A.14)

and

$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\big{(}\phi((1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})\big{)}.$
(A.15)

Subcase 3: If
$b=\big{(}(\ell_{k}-1)\varepsilon,(\ell_{k}+1)\varepsilon\big{)}$, then
$r_{b}=2$, $b\cap\Omega_{c}=\big{(}x^{h}_{k},(\ell_{k}+1)\varepsilon\big{)}$,
$b\cap\Omega_{\rm a}=\big{(}(\ell_{k}-1)\varepsilon,x^{h}_{k}\big{)}$,
$r_{b}D_{b}v=v^{\prime}_{\ell_{k}+1}+v^{\prime}_{\ell_{k}}$,
$r_{b}D_{b\cap\Omega_{\rm
a}}v=\frac{\theta_{k}}{1+\theta_{k}}v^{\prime}_{\ell_{k}+1}+\frac{1}{1+\theta_{k}}v^{\prime}_{\ell_{k}}$
and

$\displaystyle
r_{b}D_{b}y_{h}=(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}+y^{\prime}_{h}|_{T_{k-1}}\text{
and }r_{b}D_{b\cap\Omega_{\rm
a}}y_{h}=\frac{2\theta_{k}}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k}}+\frac{2}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k-1}}.$
(A.16)

We have

$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}+y^{\prime}_{h}|_{T_{k-1}}\big{)}-\phi^{\prime}(\frac{2\theta_{k}}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k}}+\frac{2}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k-1}})\big{]}v^{\prime}_{\ell_{k}}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}+y^{\prime}_{h}|_{T_{k-1}}\big{)}-\theta_{k}\phi^{\prime}(\frac{2\theta_{k}}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k}}+\frac{2}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k-1}})$
$\displaystyle-(1-\theta_{k})\phi^{\prime}(y^{\prime}_{h}|_{T_{k+1}})\big{]}v^{\prime}_{\ell_{k}+1},$
(A.17)

and

$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$ $\displaystyle
2\varepsilon\big{(}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}+y_{h}^{\prime}|_{T_{k-1}})$
$\displaystyle-(1+\theta_{k})\phi\big{(}\frac{2\theta_{k}}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k}}+\frac{2}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k-1}}\big{)}-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})\big{)}.$
(A.18)

##  Appendix B Approximation Properties

In this section, we prove some approximation properties which we have used but
are hardly found in standard text books.

Lemma 11.  Let $v\in C^{0}(\mathbb{R})\cap W^{1,2}(\mathbb{R})$ be a periodic
function with $[a,b]$ being one of its period. Let $v_{h}$ be a
$\mathcal{P}_{1}$ interpolation of $v$ with respect to the nodes $a\leq
x_{0}<x_{1}<\cdots<x_{n}\leq b\leq x_{n+1}=x_{0}+(b-a)$ in $[x_{0},x_{n+1}]$,
subject to a constant, i.e., $v_{h}(x_{k})=v(x_{k})+C$, for
$k=\\{1,2,\ldots,n+1\\}$, and is extended periodically with period $b-a$. Then
the following estimate holds:

$\|v_{h}^{\prime}\|_{L^{2}_{[a,b]}}\leq\|v^{\prime}\|_{L^{2}_{[a,b]}},$ (B.1)

where $v^{\prime}$ and $v_{h}^{\prime}$ denote the weak derivatives of $v$ and
$v_{h}$ respectively.

###### Proof.

First we note that, since $v\in C^{0}(\mathbb{R})$ and $v_{h}$ is a
$\mathcal{P}_{1}$ interpolation of $v$, the weak derivative of $v_{h}$ on
$(x_{k},x_{k+1})$ is defined by

$v_{h}^{\prime}(x)=\frac{v(x_{k+1})-v(x_{k})}{x_{k+1}-x_{k}}.$

Since $v\in C^{0}(\mathbb{R})$ is piecewise differentiable, we have

$v(x_{k+1})-v(x_{k})=\int_{x_{k}}^{x_{k+1}}v^{\prime}(t)\,{\rm d}t,$

where $v^{\prime}$ is the weak derivative of $v$. By the periodicity of
$v_{h}^{\prime}$ and $v^{\prime}$, and Cauchy-Schwarz Inequality, we have

$\displaystyle\|v_{h}^{\prime}\|^{2}_{L^{2}_{[a,b]}}$
$\displaystyle=\int_{a}^{b}\big{[}v_{h}^{\prime}(x)\big{]}^{2}\,{\rm d}x$
$\displaystyle=\int_{x_{0}}^{x_{n+1}}\big{[}v_{h}^{\prime}(x)\big{]}^{2}\,{\rm
d}x$
$\displaystyle=\sum_{k=0}^{n}\int_{x_{k}}^{x_{k+1}}(\frac{v(x_{k+1})-v(x_{k})}{x_{k+1}-x_{k}})^{2}\,{\rm
d}x$
$\displaystyle=\sum_{k=0}^{n}\int_{x_{k}}^{x_{k+1}}\frac{(\int_{x_{k}}^{x_{k+1}}v^{\prime}(t)\,{\rm
d}t)^{2}}{(x_{k+1}-x_{k})^{2}}\,{\rm d}x$
$\displaystyle\leq\sum_{k=0}^{n}\frac{1}{x_{k+1}-x_{k}}(\int_{x_{k}}^{x_{k+1}}\,{\rm
d}t)(\int_{x_{k}}^{x_{k+1}}|v^{\prime}(t)|^{2}\,{\rm d}t)$
$\displaystyle=\sum_{k=0}^{n}\int_{x_{k}}^{x_{k+1}}|v^{\prime}(t)|^{2}\,{\rm
d}t$ $\displaystyle=\int_{a}^{b}|v^{\prime}(t)|^{2}\,{\rm d}t$
$\displaystyle=\|v^{\prime}\|^{2}_{L^{2}_{[a,b]}}.$

Taking the square root on both sides gives the stated result. ∎

Lemma 12.  Let $v\in C^{0}([a,b])\cap W^{1,2}([a,b])$ and $I_{h}v$ is the
$\mathcal{P}_{1}$ function that interpolates $v$ at the points $a$ and $b$. We
have the following inequality:

$\|v^{\prime}-(I_{h}v)^{\prime}\|^{2}_{L^{2}_{(a,b)}}\leq\|v^{\prime}\|^{2}_{L^{2}_{(a,b)}}.$
(B.2)

###### Proof.

Since $v(a)=I_{h}v(a)$ and $v(b)=I_{h}v(b)$, by the definition of $I_{h}v$, we
have

$\int_{a}^{b}v^{\prime}\,{\rm d}x=\int_{a}^{b}(I_{h}v)^{\prime}dx,$

and equivalently,

$\int_{a}^{b}\big{(}v^{\prime}-(I_{h}v)^{\prime}\big{)}\cdot 1\,{\rm d}x=0,$

where $v^{\prime}$ denotes the weak derivative of $v$ on $[a,b]$. This shows
that $(I_{h}v)^{\prime}$ is the best $L^{2}$ approximation of $v^{\prime}$ in
the space of $\mathcal{P}_{0}$ functions as $(I_{h}v)^{\prime}$ is a constant.
Therefore, by the property of best approximation,

$\|v^{\prime}-(I_{h}v)^{\prime}\|^{2}_{L^{2}_{(a,b)}}\leq\|v^{\prime}-C\|^{2}_{L^{2}_{(a,b)}},$
(B.3)

for any constant $C$. In particular, if we choose $C$ to be $0$, the stated
result holds. ∎

##  Appendix C Discrete Sobolev Inequalities on Non-uniform mesh

In this section, we prove some discrete Soblev inequalities on non-uniform
mesh that are used in the residual analysis for the external force. These
results are extensions to the inequalities proved in [7, Lemma A.1, Lemma A.2,
Theorem A.4] on non uniform mesh.

Lemma 13.  Let $\boldsymbol{g}\in\mathbb{R}^{L}$,
$\boldsymbol{\varepsilon}^{0},\boldsymbol{\varepsilon}^{1}\in\mathbb{R}^{L}$
and $\varepsilon^{0}_{i},\varepsilon^{1}_{i}>0\ \forall i=1,\ldots,L$,
$\boldsymbol{g}^{\prime}=(g^{\prime}_{i})_{i=2}^{L}\in\mathbb{R}^{L-1}$,
$g_{i}^{\prime}:=\frac{g_{i}-g_{i-1}}{\varepsilon^{1}}\ i=2,\ldots,L$. If
$\sum_{i=1}^{L}\varepsilon^{0}_{i}g_{i}=0$, then

$|g_{i}|\leq\frac{1}{h}\sum_{i=2}^{L}\bar{\varepsilon}^{1}_{k}|g_{k}^{\prime}|\phi_{i,k},$
(C.1)

where, $h=\sum_{i=1}^{L}\varepsilon^{0}_{i}$,
$\phi_{i,k}=\sum_{\ell=1}^{k-1}\varepsilon^{0}_{\ell}$ for $k=2,\ldots,i$ and
$\phi_{i,k}=\sum_{\ell=k}^{L}\varepsilon^{0}_{\ell}$ for $k=i+1,\ldots,L$.

###### Proof.

Let $i\in\\{1,\ldots,L\\}$, then

$\displaystyle h|g_{i}|$
$\displaystyle=|hg_{i}-\sum_{j=1}^{L}\varepsilon^{0}_{j}g_{j}|$
$\displaystyle=|\sum_{j=1}^{L}\varepsilon^{0}_{j}g_{i}-\sum_{j=1}^{L}\varepsilon^{0}_{j}g_{j}|$
$\displaystyle\leq\sum_{j=1}^{i-1}\varepsilon^{0}_{j}|g_{i}-g_{j}|+\sum_{j=i+1}^{L}\varepsilon^{0}_{j}|g_{i}-g_{j}|.$

Since

$|g_{i}-g_{j}|=|\sum_{k=j+1}^{i}\varepsilon^{1}_{k}g^{\prime}_{k}|,$

we have

$\displaystyle h|g_{i}|$
$\displaystyle\leq\sum_{j=1}^{i-1}\varepsilon^{0}_{j}\sum_{k=j+1}^{i}\varepsilon^{1}_{k}|g^{\prime}_{k}|+\sum_{j=i+1}^{L}\varepsilon^{0}_{j}\sum_{k=i+1}^{j}\varepsilon^{1}_{k}|g^{\prime}_{k}|$
$\displaystyle=\sum_{k=2}^{i}\varepsilon^{1}_{k}|g^{\prime}_{k}|\big{(}\sum_{j=1}^{k-1}\varepsilon^{0}_{j}\big{)}+\sum_{k=i+1}^{L}\varepsilon^{1}_{k}|g^{\prime}_{k}|\big{(}\sum_{j=k}^{L}\varepsilon^{0}_{j}\big{)}$
$\displaystyle=\sum_{k=2}^{L}\varepsilon^{1}_{k}|g^{\prime}_{k}|\phi_{i,k}.$

Divide both sides by $h$, we obtain the stated result. ∎

Lemma 14.  (Discrete Poincare’s Inequality) Suppose that $L\geq 1$,
$\boldsymbol{\varepsilon}^{0},\boldsymbol{\varepsilon}^{1}\in\mathbb{R}^{L}$
with $\varepsilon^{0}_{i},\varepsilon^{1}_{i}>0$, $\forall i=1,\ldots,L$. Let
$\boldsymbol{g}\in\mathbb{R}^{L}$ such that
$\sum_{i=1}^{L}\varepsilon^{0}_{i}g_{i}=0$ and
$\boldsymbol{g}^{\prime}=(g^{\prime}_{i})_{i=2}^{L}\in\mathbb{R}^{L-1}$ such
that $g_{i}^{\prime}=\frac{g_{i}-g_{i-1}}{\varepsilon^{1}_{i}}$. Define
$\mathcal{D}_{0}$ to be the set $\\{1,\ldots,L\\}$ and $\mathcal{D}_{1}$ to be
the set $\\{2,\ldots,L\\}$, then

$\|\boldsymbol{g}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{0}(\mathcal{D}_{0})}}\leq\frac{1}{2}\frac{L^{2}\max\\{\max_{1\leq
i\leq L}\varepsilon^{0}_{i},\max_{2\leq k\leq
L}\varepsilon^{1}_{k}\\}^{2}}{h}\|\boldsymbol{g}^{\prime}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{1}(\mathcal{D}_{1})}},$
(C.2)

for $p\in\\{1,\infty\\}$, where $\ h=\sum_{i=1}^{L}\varepsilon^{0}_{i}$.

###### Proof.

Using the result of Lemma C, we have

$\displaystyle\sum_{i=1}^{L}\varepsilon^{0}_{i}|g_{i}|$
$\displaystyle\leq\sum_{i=1}^{L}\frac{\varepsilon^{0}_{i}}{h}\sum_{k=2}^{i}\varepsilon^{1}_{k}|g_{k}^{\prime}|\phi_{i,k}+\sum_{i=1}^{L}\frac{\varepsilon^{0}_{i}}{h}\sum_{k=i+1}^{L}\varepsilon^{1}_{k}|g_{k}^{\prime}|\phi_{i,k}$
$\displaystyle=\frac{1}{h}\bigg{[}\sum_{k=2}^{L}\big{(}\sum_{i=1}^{L}\varepsilon^{0}_{i}\phi_{i,k}\big{)}\varepsilon^{1}_{k}|g_{k}^{\prime}|\bigg{]}.$

Since

$\sum_{i=1}^{L}\varepsilon^{0}_{i}\phi_{i,k}\leq\max_{1\leq i\leq
L}\varepsilon^{0}_{i}\sum_{i=1}^{L}\phi_{i,k}=\max_{1\leq i\leq
L}\varepsilon^{0}_{i}\bigg{[}\sum_{i=1}^{k-1}\phi_{i,k}+\sum_{i=k}^{L}\phi_{i,k}\bigg{]},$

and

$\displaystyle\sum_{i=1}^{k-1}\phi_{i,k}+\sum_{i=k}^{L}\phi_{i,k}$
$\displaystyle\leq(k-1)\sum_{\ell=k}^{L}\varepsilon^{0}_{\ell}+\big{(}L-(k-1)\big{)}\sum_{\ell=1}^{k-1}\varepsilon^{0}_{\ell}$
$\displaystyle\leq\big{[}(k-1)\big{(}L-(k-1)\big{)}+\big{(}L-(k-1)\big{)}(k-1)\big{]}\max_{1\leq
i\leq L}\varepsilon^{0}_{i}$ $\displaystyle\leq\frac{1}{2}\max_{1\leq i\leq
L}\varepsilon^{0}_{i}L^{2}.$

Put these results together, we obtain the stated result for $p=1$. For
$p=\infty$,

$\displaystyle|g_{i}|$
$\displaystyle\leq\frac{1}{h}\sum_{k=2}^{L}\varepsilon^{1}_{k}|g_{k}^{\prime}|\phi_{i,k}$
$\displaystyle\leq\frac{1}{h}\bigg{[}\sum_{k=2}^{i}\varepsilon^{1}_{k}|g_{k}^{\prime}|\phi_{i,k}+\sum_{k=i+1}^{L}\varepsilon^{1}_{k}|g_{k}^{\prime}|\phi_{i,k}\bigg{]}$
$\displaystyle\leq\frac{1}{h}\sum_{k=2}^{L}\phi_{i,k}\max_{2\leq k\leq
L}\varepsilon^{1}_{k}|g_{k}^{\prime}|$
$\displaystyle\leq\frac{1}{2}\frac{L^{2}\max_{1\leq i\leq
L}\varepsilon^{0}_{i}}{h}\max_{2\leq k\leq
L}\varepsilon^{1}_{k}|g_{k}^{\prime}|.$

The stated result is obtained by taking the maximum of $\varepsilon^{0}_{i}$
and $\varepsilon^{1}_{k}$ over $\mathcal{D}_{0}$ and $\mathcal{D}_{1}$. ∎

Lemma 15.  (Discrete Friedrichs’ Inequality) Suppose that $L\geq 1$,
$\boldsymbol{\varepsilon}^{0}$, $\boldsymbol{\varepsilon}^{1}$,
$\mathcal{D}_{0}$, $\mathcal{D}_{2}$ are the same as in Lemma C. Let
$\boldsymbol{f}\in\mathbb{R}^{L}$ such that $f_{1}=f_{L}=0$, and
$\boldsymbol{f}^{\prime}=(f_{i}^{\prime})_{i=2}^{L}\in\mathbb{R}^{L-1}$ such
that $f_{i}^{\prime}=\frac{f_{i}-f_{i-1}}{\varepsilon^{1}_{i}}$, then

$\|\boldsymbol{f}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{0}(\mathcal{D}_{0})}}\leq\frac{1}{2}(L-1)\max_{2\leq
i\leq
L-1}\max\\{\varepsilon^{0}_{i},\varepsilon^{1}_{i}\\}\|\boldsymbol{f}^{\prime}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{1}(\mathcal{D}_{1})}},$
(C.3)

for $p\in\\{1,\infty\\}$.

###### Proof.

For $p=1$,

$\displaystyle\sum_{i=1}^{L}\varepsilon^{0}_{i}|f_{i}|$
$\displaystyle=\sum_{i=2}^{L-1}\varepsilon^{0}_{i}|f_{i}|$
$\displaystyle=\frac{1}{2}\sum_{i=2}^{L-1}\varepsilon^{0}_{i}\big{[}|\sum_{j=2}^{i}(f_{j}-f_{j-1})|+|\sum_{j=i+1}^{L}(f_{j}-f_{j-1})|\big{]}$
$\displaystyle\leq\frac{1}{2}\sum_{i=2}^{L-1}\varepsilon^{0}_{i}\big{[}\sum_{j=2}^{i}\varepsilon^{1}_{j}|f_{j}^{\prime}|+\sum_{j=i+1}^{L}\varepsilon^{1}_{j}|f_{j}^{\prime}|\big{]}$
$\displaystyle=\frac{1}{2}\sum_{i=2}^{L-1}\varepsilon^{0}_{i}\sum_{j=1}^{L}\varepsilon^{1}_{j}|f_{j}^{\prime}|$
$\displaystyle\leq\frac{1}{2}(L-1)\max_{2\leq i\leq
L-1}\varepsilon^{0}_{i}\sum_{j=1}^{L}\varepsilon^{1}_{j}|f_{j}^{\prime}|.$

For $p=\infty$,

$|f_{i}|\leq\sum_{j=2}^{i}\varepsilon^{1}_{j}|f_{j}^{\prime}|=(i-1)\max_{2\leq
j\leq L}\varepsilon^{1}_{j}\max_{2\leq j\leq L}|f_{j}^{\prime}|,$

and

$|f_{i}|\leq\sum_{j=i+1}^{L}\varepsilon^{1}_{j}|f_{j}^{\prime}|=(L-i)\max_{2\leq
j\leq L}\varepsilon^{1}_{j}\max_{2\leq j\leq L}|f_{j}^{\prime}|.$

Thus we have

$\displaystyle\max_{i\in\mathcal{D}_{0}}|f_{i}|$
$\displaystyle\leq\min(i-1,L-i)\max_{2\leq j\leq
L}\varepsilon^{1}_{j}\max_{2\leq j\leq L}|f_{j}^{\prime}|$
$\displaystyle\leq\frac{1}{2}(L-1)\max_{2\leq j\leq
L}\varepsilon^{1}_{j}\max_{2\leq j\leq L}|f_{j}^{\prime}|.$

∎

Remark 2. The bounds we have got here are not optimal as if
$\varepsilon_{i}$’s and $\bar{\varepsilon}_{j}$’s vary too much, taking the
maximum of them in the inequalities could significantly reduce the sharpness
of the estimate. However, for the analysis of this paper, such a bound is
optimal enough to produce efficient error estimators and we leave the work of
looking for optimal bounds to future work. ∎

Theorem 16.  (bounds on the interpolation error) Let $L\geq 1$,
$\boldsymbol{\varepsilon}^{0},\boldsymbol{\varepsilon}^{1},\boldsymbol{\varepsilon}^{2}\in\mathbb{R}^{L}$,
with $\varepsilon^{0}_{i},\varepsilon^{1}_{i},\varepsilon^{2}_{i}>0\ \forall
i=1,\ldots,L$. Let $\boldsymbol{f}\in\mathbb{R}^{L}$ and
$\boldsymbol{F}=\in\mathbb{R}^{L}$ such that $F_{1}=f_{1}$ and

$F_{i}=f_{1}+\frac{\sum_{j=2}^{i}\varepsilon^{0}_{i}}{h}(f_{L}-f_{1})\quad
i=2,\ldots,L,$ (C.4)

where $h=\sum_{i=2}^{L}\varepsilon^{0}_{i}$. Define
$\boldsymbol{f}^{\prime}=(f_{i}^{\prime})_{i=2}^{L}\in\mathbb{R}^{L-1}$ such
that $f_{i}^{\prime}=\frac{f_{i}-f_{i-1}}{\varepsilon^{1}_{i}}$ and
$\boldsymbol{f}^{\prime\prime}=(f_{i}^{\prime\prime})_{i=2}^{L-1}\in\mathbb{R}^{L-2}$
such that
$f_{i}^{\prime\prime}=\frac{f^{\prime}_{i+1}-f^{\prime}_{i}}{\varepsilon^{2}_{i}}$,
and $\boldsymbol{F}^{\prime}$ and $\boldsymbol{F}^{\prime\prime}$ are defined
in the same way. Let $\mathcal{D}_{0}$, $\mathcal{D}_{1}$ be the same sets
defined in Lellmma C and $\mathcal{D}_{2}$ be the set $\\{2,\ldots,L-1\\}$.
Then, for $p\in\\{1,\infty\\}$,

$\|\boldsymbol{f}-\boldsymbol{F}\|_{\ell_{\boldsymbol{\varepsilon}^{0}}^{p}(\mathcal{D}_{0})}\leq\frac{1}{4}\frac{L^{3}\max_{2\leq
i\leq L-1}\varepsilon_{i}^{0}\max_{2\leq j\leq
L-1}\varepsilon^{1}_{j}\max_{2\leq k\leq
L-1}\varepsilon^{2}_{j}}{h}\|\boldsymbol{f}^{\prime\prime}\|_{\ell_{\boldsymbol{\varepsilon}^{2}}^{p}(\mathcal{D}_{2})}.$
(C.5)

###### Proof.

Let $\boldsymbol{g}=\boldsymbol{f}-\boldsymbol{F}$, by the definition of
$\boldsymbol{F}$, we have $g_{1}=g_{L}=0$ and

$\sum_{i=2}^{L}\varepsilon_{i}g_{i}^{\prime}=\sum_{i=2}^{L}(f_{i}-f_{i-1})-\sum_{i=2}^{L}(F_{i}-F_{i-1})=0.$

By Lemma C,

$\|\boldsymbol{g}\|_{\ell_{\boldsymbol{\varepsilon}^{1}}^{p}(\mathcal{D}_{0})}\leq\frac{1}{2}(L-1)\max_{2\leq
i\leq
L-1}\max\\{\varepsilon^{0}_{i},\varepsilon^{1}_{i}\\}\|\boldsymbol{g}^{\prime}\|_{\ell_{\boldsymbol{\varepsilon}}^{p}(\mathcal{D}_{1})},$

as $g_{1}=g_{L}=0$, and by Lemma C,

$\|\boldsymbol{g}^{\prime}\|_{\ell_{\boldsymbol{\varepsilon}}^{p}(\mathcal{D}_{1})}\leq\frac{1}{2}\frac{L^{2}\max\\{\max_{1\leq
i\leq L}\varepsilon^{1}_{i},\max_{2\leq k\leq
L-1}\varepsilon^{2}_{k}\\}^{2}}{h}\|\boldsymbol{g}^{\prime\prime}\|_{\ell_{\bar{\boldsymbol{\varepsilon}}}^{p}(\mathcal{D}_{2})},$

as $\sum_{i=2}^{L}\varepsilon_{i}g_{i}^{\prime}=0$. Since
$\boldsymbol{F}^{\prime\prime}=0$, from which we know
$\boldsymbol{g}^{\prime\prime}=\boldsymbol{f}^{\prime\prime}$, the stated
estimate holds. ∎

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