SIAM J. NUMER. ANAL.
Vol. 47, No. 4, pp. 2455–2475
c 2009 Society for Industrial and Applied Mathematics
AN OPTIMAL ORDER ERROR ANALYSIS OF THE
ONE-DIMENSIONAL QUASICONTINUUM APPROXIMATION∗
MATTHEW DOBSON† AND MITCHELL LUSKIN†
Abstract. We derive a model problem for quasicontinuum approximations that allows a simple, yet insightful, analysis of the optimal-order convergence rate in the continuum limit for both
the energy-based quasicontinuum approximation and the quasi-nonlocal quasicontinuum approximation. For simplicity, the analysis is restricted to the case of second-neighbor interactions and is
linearized about a uniformly stretched reference lattice. The optimal-order error estimates for the
quasi-nonlocal quasicontinuum approximation are given for all strains up to the continuum limit
strain for fracture. The analysis is based on an explicit treatment of the coupling error at the
atomistic-to-continuum interface, combined with an analysis of the error due to the atomistic and
continuum schemes using the stability of the quasicontinuum approximation.
Key words. quasicontinuum, error analysis, atomistic-to-continuum
AMS subject classifications. 65Z05, 70C20
DOI. 10.1137/08073723X
1. Introduction. The quasicontinuum (QC) method is a technique for deriving
approximations of fully atomistic models for crystalline solids that reduce the degrees
of freedom necessary to compute a deformation to a desired accuracy [6,8,9,14,15,16,
17,18,20,21,24,27,30,32]. The QC method first removes atomistic degrees of freedom
by using a piecewise linear approximation of the atom deformations with respect to
a possibly much smaller number of representative atoms. This approximation is still
not computationally feasible since the atoms near element boundaries interact with
atoms in adjacent elements. To obtain an efficient method, a strain energy density is
utilized that is compatible with the atomistic model for uniform strain (the Cauchy–
Born rule), and the energy of the atoms in an element is computed by the product of
the element volume and the element strain energy density.
In this paper, we will analyze two QC variants that approximate the total atomistic energy by using a continuum approximation in a portion of the material called
the continuum region. The deformation gradient is assumed to be slowly varying
in the continuum region, making the continuum approximation accurate. The more
computationally intensive atomistic model is used for the remainder of the computational domain, which is called the atomistic region. In this region, all of the atoms
are representative atoms, so that there is no restriction on the types of deformations
in the atomistic region. To maintain accuracy, the atomistic region must contain all
regions of highly varying deformation, such as material defects. Adaptive methods
that determine what portion of the domain should be assigned to the atomistic region
in order to achieve the required accuracy have been considered in [1, 2, 3, 23, 24, 26].
Other approaches to atomistic-to-continuum coupling have been developed and ana∗ Received by the editors October 6, 2008; accepted for publication (in revised form) April 13,
2009; published electronically July 2, 2009. This work was supported in part by DMS-0757355,
DMS-0811039, the Institute for Mathematics and Its Applications, the University of Minnesota Supercomputing Institute, and the University of Minnesota Doctoral Dissertation Fellowship. This work
is also based on work supported by the Department of Energy under award DE-FG02-05ER25706.
http://www.siam.org/journals/sinum/47-4/73723.html
† School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455
([email protected], [email protected]).
2455
2456
MATTHEW DOBSON AND MITCHELL LUSKIN
lyzed in [4, 25], for example.
In section 2, we derive a model problem for QC approximations and describe
the energy-based quasicontinuum (QCE) approximation [21] and the quasi-nonlocal
quasicontinuum (QNL) approximation [30]. These two approximations use the same
continuum approximation but differ in how they couple the atomistic and continuum
regions. We have derived our model QC energies from general QC energies by expanding each interaction to second order about a uniformly stretched configuration
in order to be able to present a simple, but illuminating, analysis. This model differs
from a standard harmonic approximation by keeping certain first-order terms. These
are a source of leading-order coupling error and reflect the behavior in the nonlinear
case. We have also chosen to analyze periodic boundary conditions to maintain the
simplicity of the analysis. In addition to deriving the QCE and QNL approximations
in section 2, we give new stability results that are used to obtain the optimal-order
error estimates described below.
The goal of this paper is to give an error analysis with respect to the continuum
limit, which is the limit in which interatomic spacing and interatomic interactions are
scaled so that the total energy converges while the number of atoms per unit length
increases to infinity. The truncation error at atoms in the coupling interface is then
of order O(1/h) and O(1) for QCE and QNL, respectively, where h is the interatomic
spacing. This is lower order than the truncation error in either the atomistic or
continuum region, which is O(h2 ). We show that the corresponding coupling error
depends primarily on the sum of the truncation error at the atoms in the atomisticto-continuum coupling interface, and this sum has the higher order O(h) due to the
cancellation of the lowest-order terms when the truncation error is summed across the
interface.
In section 3, we split the truncation error for the QCE approximation into one
part due to approximating the continuum limit using second-order finite differences
(a five-point rule in the atomistic region and a three-point rule in the continuum
region) and another, lower-order part due to coupling the atomistic and continuum
regions. Our stability result for the QCE approximation and our O(h2 ) estimate
for the discretization error of approximating the continuum limit using second-order
finite differences combine to give an optimal-order bound for this contribution to the
error. We then derive an explicit representation of the coupling error, and we observe
that the coupling error converges at the rate O(h) in the discrete l∞ norm and the
rate O(h1/p ) in the w1,p norms for 1 ≤ p ≤ ∞. Combining the two error bounds,
we obtain an overall convergence analysis for QCE with rate O(h) in the l∞ norm
and rate O(h1/p ) in the w1,p norms. Thus, despite an O(1/h) truncation error in the
max-norm, the displacement still converges in the continuum limit. Related work [22]
has shown that the error is O(1) in the w1,∞ norm for the QCE method applied to a
problem with harmonic interactions and Dirichlet boundary conditions. Our analysis
holds for more general interatomic potentials and for a larger class of strains. We
note, however, that we have recently given sharp stability results in [12] that show
that the QCE approximation is not stable for all strains up to the continuum limit
strain for fracture.
In section 4, we present the analysis for the QNL case. Here we show that the
improved order of accuracy in the coupling interface serves to nearly balance the order
of the discretization error, and we are consequently able to give higher-order optimal
error estimates for the QNL approximation than for the QCE approximation. We
show that the displacement now converges at the rate O(h2 ) in the discrete l∞ norm
and the rate O(h1+1/p ) in the w1,p norms, where h is the interatomic spacing. Ming
ERROR ANALYSIS OF THE QUASICONTINUUM APPROXIMATION
2457
and Yang [22] have obtained O(h) estimates in the w1,∞ norm for the Lennard–Jones
potential and for strains that are restricted to being bounded away from the continuum
limit strain for fracture. We have obtained optimal-order QNL error estimates for the
discrete l∞ and w1,p norms for more general interatomic potentials and for all strains
up to the continuum limit strain for fracture, which gives a theoretical validation for
the use of the QNL method for defect motion. Thus, QNL has the dual advantages
of one full order higher rate of convergence in the w1,p norm of the displacement and
that this convergence is proven when expanding around any uniform strain up to the
continuum fracture limit.
This paper extends our analysis of the effect of atomistic-to-continuum model
coupling on the total error in the QCE approximation [11, section 3.2] to include external forcing. The analysis of the error due to interfacial coupling is expanded in this
paper to include curvature of the strain field and to include the QNL approximation.
The construction of the interfacial error terms demonstrates that the error estimates
are of optimal order; in particular, choosing f = 0 in the QCE case and choosing a
solution with nonzero curvature in the interface in the QNL case correspond to the
given convergence rates.
This paper treats two of the several different QC methods developed in the engineering and mathematical literature. While it is beyond the scope of the article
to compare all of the methods, we will briefly mention a few other results in the
mathematical literature. We have previously derived the force-based quasicontinuum
(QCF) method [9], which directly generates forces that do not correspond to any total
energy. We showed that this is the actual approximation being made when the “ghost
force correction” [29] is applied iteratively [9, section 2.6], and that QCE acts as an
effective preconditioner for iteratively solving the QCF equations [9, Theorem 5.1].
Recently, we have also proven an O(h2 ) error estimate for QCF [13] by overcoming
the noncoercivity of the approximation, which we also prove. Rather than employ
a continuum approximation, cluster-based QC methods [16] simplify the energy (respectively, force) computation by approximating the total energy (respectively, force)
using a cluster of atoms around each node of the piecewise linear mesh. However, the
energy-based and force-based cluster approximations have been shown to be inaccurate in [19].
2. One-dimensional linear QC approximation. We consider a one-dimensional
reference lattice with spacing h = 1/N, and we denote the positions of the atoms in
the reference lattice by
xj := jh,
−∞ < j < ∞.
We will derive and analyze the linearization about the uniform deformation
(2.1)
yjF := jF h,
−∞ < j < ∞,
which has uniform lattice spacing F h. We will then consider perturbations uj of the
lattice yjF which are 2N periodic in j; that is, we will consider deformations yj , where
yj := yjF + uj ,
−∞ < j < ∞,
for
(2.2)
uj+2N = uj ,
−∞ < j < ∞.
2458
MATTHEW DOBSON AND MITCHELL LUSKIN
We will often describe the perturbations uj satisfying (2.2) as displacements (which
they are if yjF is considered the reference lattice). We thus have that the deformation
satisfies
yj+2N = yj + 2F,
(2.3)
−∞ < j < ∞.
We note that neither the reference lattice spacing h nor the uniform lattice spacing
F h needs be the equilibrium lattice constant nor correspond to a well of the scaled
interatomic potential given below in (2.6).
2.1. Notation. Before introducing the models, we fix the following notation.
We define the backward differentiation operator, Du, on periodic displacements by
(Du)j :=
uj − uj−1
h
for − ∞ < j < ∞.
Then (Du)j is also 2N periodic in j. We will use the shorthand Duj := (Du)j .
For periodic displacements, u, we define the discrete norms
⎛
up := ⎝h
h
u∞ :=
h
⎞1/p
N
|uj |p ⎠
,
1 ≤ p < ∞,
j=−N +1
max
−N +1≤j≤N
|uj |.
Since the expressions above range over a full period, they are indeed norms (in particular, up = 0 implies u = 0). We restrict the sum to a single period in order
h
to make the norms finite. We will also consider periodic functions u(x) : R → R
satisfying
(2.4)
u(x + 2) = u(x)
for x ∈ R.
We define corresponding continuous norms
uLp :=
1
−1
1/p
|u(x)| dx
,
p
1 ≤ p < ∞,
uL∞ := ess sup |u(x)|.
x∈(−1,1)
We let u denote the weak derivative of the periodic function u. We note that if
u Lp < ∞, then u(x) is continuous for all x in R and u(−1) = u(1). We will
similarly denote higher-order weak derivatives of the periodic function u as u , u ,
and u(4) .
2.2. Atomistic model. We first consider the total energy per period
E tot,h (y) := E a,h (y) − F(y),
(2.5)
for deformations y satisfying (2.3), where the total atomistic energy per period is
(2.6)
E
a,h
(y) =
N
j=−N +1
yj − yj−1
yj − yj−2
h φ
+φ
h
h
ERROR ANALYSIS OF THE QUASICONTINUUM APPROXIMATION
2459
for a two-body interatomic energy density φ (assumptions on φ are given in section
2.5), and where the total external potential energy per period is
F (y) =
(2.7)
N
hfj yj
j=−N +1
for periodic dead loads f such that fj+2N = fj and N
j=−N +1 fj = 0.
The interatomic potential energy density φ(r) can be constructed by rescaling
an interatomic potential φ̃(r̃) to have units of energy per reference unit length. The
y −y
energy contribution of two nearest neighbor particles is then hφ( j h j−1 ) and of two
y −y
next-nearest neighbor particles is hφ( j h j−2 ). The scaling of the atomistic energy
per bond, hφ(r/h), and the external force per atom, hfi , permit a continuum limit as
h → 0. If y, f ∈ C ∞ (R) satisfy y(x + 2) = y(x) + 2F, y (x) > 0, f (x + 2) = f (x), and
1
f (x) dx = 0; if φ(r) is locally Lipschitz for r ∈ (0, ∞); and if we set yj = y(xj )
−1
and fj = f (xj ), then the energy per period (2.5) converges to (see [5])
1
(2.8)
φ̂ (y (x)) − f (x)y(x) dx
−1
as N → ∞ (which implies h → 0), where φ̂(r) = φ(r) + φ(2r). In the following,
we linearize the atomistic model, which leads to a corresponding linearized continuum model. This paper analyzes the convergence of two QC approximations to the
minimizer of the linearized continuum model’s total energy.
2.3. Linearized atomistic model. We will henceforth consider the linearized
version of the above energies while reusing the notation E a,h and E tot,h . The total
atomistic energy (2.6) becomes (a similar energy density that includes external forces
is in [11, section 2.1])
2
N
uj − uj−1
uj − uj−1
1
a,h
h φF
E (u) :=
+ φF
h
2
h
j=−N +1
(2.9)
2 uj − uj−2
1 uj − uj−2
+ φ2F
+ φ2F
h
2
h
for displacements, u, satisfying the periodic boundary conditions (2.2). Here φF :=
φ (F ), φF := φ (F ), φ2F := φ (2F ), φ2F := φ (2F ), where φ is the interatomic
energy density in (2.6). We have removed the additive constant 2φ(F ) + 2φ(2F )
from the quadratic expansion of the energy, and we will remove the additive conF
stant −h N
j=−N +1 fj yj from F (u) when computing the external potential of the
displacement u. We note that the first-order terms in (2.9) sum to zero by the periodic boundary conditions and thus do not contribute to the total energy or the
equilibrium equations. However, we retain the first-order terms in the model (2.9)
since they do not sum to zero when the atomistic model is coupled to the continuum
approximation in the QC energy. The atomistic energy (2.9) has the equilibrium
equations
(2.10)
(La,h u)j =
−φ2F uj+2 − φF uj+1 + 2(φF + φ2F )uj − φF uj−1 − φ2F uj−2
= fj ,
h2
uj+2N = uj
2460
MATTHEW DOBSON AND MITCHELL LUSKIN
for −∞ < j < ∞.
2.4. Linearized continuum model. For periodic u ∈ C ∞ (R) and uj = u(xj ),
the total linearized atomistic energy
(2.11)
E tot,h (u) := E a,h (u) − F(u)
converges to
1
(2.12)
−1
[W (u (x)) − f (x)u(x)] dx
as N → ∞, where the continuum strain energy density, W (), is given by
W () := (φF + 2φ2F ) + 12 (φF + 4φ2F )2 .
(2.13)
The equilibrium equations (2.10) are a consistent five-point difference approximation
of the equilibrium equations of the continuum model (2.12), which are [11, section
2.1]
−(φF + 4φ2F )ue = f,
ue (x + 2) = ue (x)
(2.14)
for x ∈ R.
QC approximations couple an approximation of the continuum model with the
atomistic model. The continuum approximation consists of a finite element discretization of the continuum model’s elastic energy. The discretization uses a continuous,
piecewise linear displacement u with the atom positions x as nodes. The external
force term is applied as a point force at each node, so that (2.12) becomes
N
(2.15)
h[W (Dul ) − fl ul ].
l=−N +1
The continuum approximation has equilibrium equations [11, section 2.1]
− (φF + 4φ2F )
ul+1 − 2ul + ul−1
= fl ,
h2
ul+2N = ul ,
−∞ < l < ∞,
which is a three-point consistent difference approximation of the equilibrium equations for the continuum model (2.14). In one dimension, the above is actually the
standard finite difference approximation of (2.12); however, it is framed in finite element terminology for flexibility in coarsening, adaptivity, and higher-dimensional
modeling.
2.5. Assumptions. We assume that
(2.16)
φF + 4φ2F > 0,
which implies that the total linearized atomistic energy (2.9) is positive definite (up
to uniform translation of the displacement). Under this assumption, both (2.10)
and (2.14) have a unique solution (up to uniform translation), provided that
(2.17)
N
j=−N +1
fj = 0.
ERROR ANALYSIS OF THE QUASICONTINUUM APPROXIMATION
2461
For simplicity, we assume in the following that f is odd in addition to being periodic;
that is,
(2.18)
f (x) = −f (−x) and f (x + 2) = f (x)
for − ∞ < x < ∞,
which implies that fj := f (xj ) satisfies
(2.19)
fj = −f−j
and fj+2N = fj
for
− ∞ < j < ∞.
We obtain a unique, odd periodic solution satisfying the mean value condition
(2.20)
N
uj = 0.
j=−N +1
We further assume that
(2.21)
φF > 0.
Typically, φ2F < 0, such as for the Lennard–Jones potential, though we do not require
this.
2.6. Energy-based QC approximation. The QCE approximation [21] of
E a,h (u) decomposes the reference lattice into an atomistic region and a continuum
region. It computes a total energy (2.23) by using the atomistic energy (2.9) in the
atomistic region and by using the continuum approximation (2.15) to sum the energy
of the continuum region.
For our analysis, we will consider an atomistic region defined by the atoms with
reference positions xj for j = −K, . . . , K, and a continuum region containing the
remaining atoms, j = −N + 1, . . . , −K − 1 and j = K + 1, . . . , N. All atoms in the
continuum region, along with the two atoms on the boundary, j = ±K, will act as
nodes for the continuum approximation. The continuum region can be decomposed
into elements (x(l−1) , xl ) for l = −N + 1, . . . , −K and l = K + 1, . . . , N. (In general,
elements can contain many atoms of the reference lattice, but in this paper we do not
consider coarsening in the continuum region.)
To construct the contribution of the atomistic region to the total QC energy, it is
convenient to construct an energy associated with each atom by splitting equally the
energy of each bond to obtain
2
uj+1 − uj
h
1 uj+1 − uj
a,h
φ
+ φF
Ej (u) :=
2 F
h
2
h
2 uj+2 − uj
1 uj+2 − uj
+ φ2F
+ φ2F
h
2
h
(2.22)
2
uj − uj−1
uj − uj−1
h
1
+ φF
+ φF
2
h
2
h
2 uj − uj−2
1 uj − uj−2
+ φ2F
.
+ φ2F
h
2
h
The continuum energy (2.15) is split into energy per element hW (Dul ), where W is
given in (2.13), and h = xl − xl−1 is the length of the continuum element (xl−1 , xl ).
In order for QCE to exactly conserve the energy of atomistic model (2.9) for
lattices yjF given by a uniform deformation y F (see (2.1)), the elements (x−K−1 , x−K )
2462
MATTHEW DOBSON AND MITCHELL LUSKIN
and (xK , xK+1 ) on the border of the atomistic region should contribute only one half
of the continuum energy associated with that element. The QCE energy is then
E qce,h (u) :=
−K−1
l=−N +1
(2.23)
K
1
hW (Dul ) + hW (Du−K ) +
Eja,h (u)
2
j=−K
1
+ hW (DuK+1 ) +
2
N
hW (Dul ).
l=K+2
The equilibrium equations for the total QCE energy, E qce,h (u) − F(u), then take
the form [9, section 2.4], [11, section 2.4]
Lqce,h uqce − g = f ;
(2.24)
here, for 0 ≤ j ≤ N, we have
(2.25)
−uj+1 + 2uj − uj−1
(Lqce,h u)j = φF
h2
⎧
−u
+
2u
−
u
j+2
j
j−2
⎪
,
⎪
⎪4φ2F
2
⎪
4h
⎪
⎪
⎪ −uj+2 + 2uj − uj−2
φ2F uj+2 − uj
⎪
⎪
,
+
4φ2F
⎪
⎪
⎪
4h2
h
2h
⎪
⎪
⎪
φ2F uj+2 − uj
−uj+2 + 2uj − uj−2
2φ2F uj+1 − uj
⎪
⎪
⎨4φ2F
+
,
−
4h2
h
h
h
2h
+
⎪
φ uj − uj−2
−uj+1 + 2uj − uj−1
2φ uj − uj−1
⎪
⎪4φ2F
+ 2F
,
− 2F
⎪
2
⎪
h
h
h
h
2h
⎪
⎪
⎪
⎪
−uj+1 + 2uj − uj−1
φ uj − uj−2
⎪
⎪
4φ2F
,
+ 2F
⎪
2
⎪
h
h
2h
⎪
⎪
⎪
⎪
⎩4φ −uj+1 + 2uj − uj−1 ,
2F
h2
0 ≤ j ≤ K − 2,
j = K − 1,
j = K,
j = K + 1,
j = K + 2,
K + 3 ≤ j ≤ N,
with g given by
(2.26)
⎧
0,
0 ≤ j ≤ K − 2,
⎪
⎪
⎪
⎪
1 ⎪
−
φ
,
j = K − 1,
⎪
2h 2F
⎪
⎪
⎨ 1 φ , j = K,
2h 2F
gj =
1 ⎪
⎪
2h φ2F , j = K + 1,
⎪
⎪
1 ⎪
⎪
⎪− 2h φ2F , j = K + 2,
⎪
⎩
0,
K + 3 ≤ j ≤ N.
For space reasons, we list only the entries for 0 ≤ j ≤ N. The equations for all other
j ∈ Z follow from symmetry and periodicity. Due to the symmetry in the definition
= Lqce,h
of the atomistic and continuum regions, we have that Lqce,h
i,j
−i,−j and gj = −g−j
for −N + 1 ≤ i, j ≤ 0. To see this, we define the involution operator (Su)j = −u−j
and observe that E qce,h (Su) = E qce,h (u). It then follows from the chain rule that
S T Lqce,h Su − S T g = Lqce,h u − g
for all periodic u.
Since S T = S, we can conclude that
(2.27)
SLqce,h S = Lqce,h
and Sg = g.
ERROR ANALYSIS OF THE QUASICONTINUUM APPROXIMATION
2463
Under the assumptions of Lemma 2.2, there is a unique mean zero solution (2.20) to
the equilibrium equations (2.24). This solution is odd because S −1 = S and (2.27)
which together imply that Su is a solution if and only if u is. Because S preserves
the mean zero property, we conclude that uqce is odd.
2.7. Stability of the QC operator. Our analysis of the QCE error will utilize
the following stability results for the operator Lqce,h .
Lemma 2.1. If φF + 5φ2F > 0, then
2
hv · Lqce,h v ≥ ν Dv2 ,
(2.28)
h
min(φF
5φ2F , φF ).
where ν :=
+
Proof. The stability result (2.28) follows from the identity
1
hv · Lqce,h v =
2
−K−1
l=−N +1
K
(Dv−K ) +
(Dvl ) + 1 hW
Eja,h (v)
hW
2
j=−K
1 (DvK+1 ) +
+ hW
2
where
Eˆja,h (v)
(2.29)
N
(Dvl ),
hW
l=K+2
2
2 h 1 vj+1 − vj
1 vj+2 − vj
φ
:=
+ φ2F
2 2 F
h
2
h
2
2 h 1 vj − vj−1
1 vj − vj−2
φ
+
+ φ2F
2 2 F
h
2
h
and
(2.30)
If φ2F
giving
() := 1 (φF + 4φ2F )2 .
W
2
≥ 0, we can drop all the second-neighbor terms since they are nonnegative,
hv · Lqce,h v ≥ h
N
φF (Dvj )2 .
j=−N +1
If
φ2F
< 0, then
hv · Lqce,h v
≥h
N
j=−N +1
1 φ (Dvj+1 )2 + (Dvj )2 + h
2 F
K
+ 2hφ2F (Dv−K )2 + h
−K−1
j=−N +1
φ2F (Dvj+2 )2 + (Dvj+1 )2 + (Dvj )2 + (Dvj−1 )2
j=−K
+ 2hφ2F (DvK+1 )2 + h
⎡
≥ (φF + 5φ2F ) ⎣h
N
4φ2F (Dvj )
j=K+2
N
⎤
(Dvj )2 ⎦ .
j=−N +1
2
4φ2F (Dvj )
2
2464
MATTHEW DOBSON AND MITCHELL LUSKIN
Remark 1. Note that the above stability result is not sharp, in the sense that
there are stable configurations where φF +5φ2F < 0. Having precise stability estimates
is essential when attempting to approximate stable solutions that are close to failure.
Such estimates are the subject of [12].
Lemma 2.1 and the discrete Poincaré inequality for 0 < h ≤ 1,
(2.31)
v2 ≤
h
h
1
Dv2 ≤ Dv2
πh
h
h
2
2 sin 2
N
if
vj = 0,
j=−N +1
give the following stability result in the ·2 norm. To see (2.31), we let X denote
h
the space of periodic mean zero vectors and note that
(2.32)
min Dv2 =
v∈X
v2 =1
h
h
min DT Dv, v1/2 = 4 sin2
πh v∈X
v2 =1
2
/h2
h
2
since 4 sin2 ( πh
2 )/h is the smallest eigenvalue of the three-point discrete Laplacian
T
D D restricted to mean zero periodic vectors [31, Exercise 13.9]. We can see this by
observing that a basis of eigenvectors for DT D restricted to periodic vectors is given
by vk , where
vjk = exp(iπkjh) for k = −N + 1, . . . , N,
with corresponding eigenvalues
4 sin2
πkh /h2 .
2
Lemma 2.2. If φF + 5φ2F > 0 and b satisfies
N
j=−N +1 bj
= 0, then
Lqce,h v = b
(2.33)
has a unique mean zero solution v which satisfies
Dv2 ≤
(2.34)
h
1
b2 ,
h
2ν
where ν := min(φF + 5φ2F , φF ).
Proof. By Lemma 2.1, the nullspace of Lqce,h consists of constant functions. Since
b has mean zero, we conclude that there is a unique solution v up to a constant.
By taking the inner product of (2.33) with v, applying (2.28) to the left-hand
side, and applying the Cauchy–Schwarz inequality and the Poincaré inequality (2.31),
we have
2
ν Dv2 ≤ hv · Lqce,h v
h
= hv · b
≤ b2 v2
h
h
1
≤ b2 Dv2 ,
h
h
2
which gives the desired result.
2465
ERROR ANALYSIS OF THE QUASICONTINUUM APPROXIMATION
2.8. Quasi-nonlocal QC approximation. The QNL approximation [30] is
similar to the QCE approximation, but it introduces “quasi-nonlocal” atoms at the
interface in order to remove g from the elastic force. The quasi-nonlocal atoms are
located at ±K, ±(K + 1), and they interact directly with any atoms in the atomistic
region within the next-nearest neighbor cut-off but interact as in the continuum region
with all other atoms. That is, unlike the atomistic model and continuum approximation, the form of energy contributions for quasi-nonlocal atoms depends on the type
(atomistic, continuum, or quasi-nonlocal) of the neighboring atoms. For example, the
energy contribution for j = K is
q,h
(u)
EK
2 uK+1 − uK
uK+1 − uK
h
1 (φF + 2φ2F )
:=
+ (φF + 4φ2F )
2
h
2
h
2
uK − uK−1
h uK − uK−1
1
+
φ
+ φF
2 F
h
2
h
2 uK − uK−2
1 uK − uK−2
+ φ2F
,
+ φ2F
h
2
h
and the energy contribution for j = K + 1 is
q,h
EK+1
(u)
2 h
uK+2 − uK+1
uK+2 − uK+1
1 (φF + 2φ2F )
:=
+ (φF + 4φ2F )
2
h
2
h
2
uK+1 − uK
h uK+1 − uK
1
φ
+
+ φF
2 F
h
2
h
2 uK+1 − uK−1
1 uK+1 − uK−1
+ φ2F
.
+ φ2F
h
2
h
The QNL energy is then
(2.35)
E qnl,h (u)
:=
−K−2
l=−N +1
+
K+1
j=K
1
hW (Dul ) + hW (Du−K−1 ) +
2
Ejq,h (u) + 12 hW (DuK+2 ) +
−K
Ejq,h (u) +
j=−K−1
N
l=K+3
The QNL equilibrium equations are
Lqnl,h uqnl = f ,
hW (Dul ).
K−1
j=−K+1
Eja,h (u)
2466
MATTHEW DOBSON AND MITCHELL LUSKIN
where
−uj+1 + 2uj − uj−1
(Lqnl,h u)j = φF
h2
⎧
−uj+2 + 2uj − uj−2
⎪
⎪
4φ
,
⎪
⎪ 2F
4h2
⎪
⎪
⎪
−uj+2 + 2uj − uj−2
−uj+2 + 2uj+1 − uj
⎪
⎪
⎨4φ2F
− φ2F
,
2
4h
h2
+
−uj+1 + 2uj − uj−1
−uj + 2uj−1 − uj−2
⎪
⎪
+ φ2F
,
4φ
⎪
⎪
⎪ 2F
h2
h2
⎪
⎪
⎪
⎪
⎩4φ2F −uj+1 + 2uj − uj−1 ,
h2
0 ≤ j ≤ K − 1,
j = K,
j = K + 1,
K + 2 ≤ j ≤ N.
We note that the QNL energy satisfies the symmetry condition E qnl,h (Su) = E qnl,h (u),
so the QNL operator Lqnl,h is defined for j < 0 by the identity SLqnl,h S = Lqnl,h .
While we have successfully removed the ghost force terms g, QNL is also not a pointwise consistent approximation of the continuum equations (2.14) at the interfacial
atoms j = −K − 1, −K, K, and K + 1. We will give a more detailed analysis of the
approximation at the interface in section 4.
Our analysis of the QNL error will utilize the following stability result for the
operator Lqnl,h .
N
Lemma 2.3. If φF + 4φ2F > 0 and b satisfies j=−N +1 bj = 0, then
Lqnl,h v = b
has a unique mean zero solution v which satisfies
hv · Lqnl,h v ≥ ν Dv22 ,
(2.36)
h
Dv2
h
1
b2 ,
≤
h
2ν
where ν := min(φF + 4φ2F , φF ).
Proof. The proof of the stability result (2.36) follows the proof of the stability results for the QCE approximation in Lemmas 2.1 and 2.2 with the appropriate
modification.
Remark 2. The basic formulation of the QNL method removes the ghost force
terms only for second-neighbor interactions in the one-dimensional case. A matching
reconstruction method is proposed in [14] that removes ghost forces for longer-range
interactions by modifying the energy contribution of more atoms at the interface. To
do this, they generalize the modification used for quasi-nonlocal atoms.
In two and three dimensions, there are similar restrictions on the interaction
length that QNL corrects. Additional ghost forces arise when the QNL energy is
extended to allow nonplanar atomistic-to-continuum interfaces and coarsening in the
continuum region [14]. So far, no extension of QNL has removed all ghost forces.
3. Convergence of the QCE solution. We now analyze the QCE error and
obtain estimates for its convergence rate by splitting the truncation error into two
parts. One portion contains the low-order terms, has support only near the atomisticto-continuum interface, and is oscillatory. The remainder is higher-order, and its
influence will be bounded using the stability results. We recall that the QCE solution,
uqce , is an odd periodic solution of
(3.1)
Lqce,h uqce = g + f ,
ERROR ANALYSIS OF THE QUASICONTINUUM APPROXIMATION
2467
and the continuum model solution is an odd periodic function ue (x) satisfying
−(φF + 4φ2F )ue = f.
(3.2)
Let ue denote the vector satisfying ue,j = ue (xj ). We will now derive estimates for
the QC error e = ue − uqce .
It follows from the QCE equilibrium equation (3.1) that
(3.3)
Lqce,h e = Lqce,h ue − Lqce,h uqce = Lqce,h ue − g − f .
We split the truncation error Lqce,h e as
Lqce,h e := ρ + σ,
(3.4)
where ρ contains the three lowest-order truncation error terms in the interface. In (2.25)
we split Lqce,h into the sum of an O(h2 ) approximation to the continuum operator
plus extra interface terms in the interface. We set ρ to be the sum of g and the three
lowest-order terms in the expansion of these extra interface terms around the point
xK+1/2 . This gives
(3.5)⎧
⎪
0,
⎪
⎪
⎪
⎪
⎪
1 ⎪
φ
⎪
⎪
⎪ 2 2F
⎪
⎪
⎨− 1 φ
2 2F
ρ=
⎪
⎪
− 12 φ2F
⎪
⎪
⎪
⎪
⎪
1 ⎪
⎪
2 φ2F
⎪
⎪
⎪
⎩0,
0 ≤ j ≤ K − 2,
7 + φ2F uK+1/2 h1 − 12 φ2F uK+1/2 + 24
φ2F u
K+1/2 h,
5 + φ2F uK+1/2 h1 + 12 φ2F uK+1/2 + 24
φ2F u
K+1/2 h,
1
1
5
+ φ2F uK+1/2 h − 2 φ2F uK+1/2 + 24 φ2F u
K+1/2 h,
7 + φ2F uK+1/2 h1 + 12 φ2F uK+1/2 + 24
φ2F u
K+1/2 h,
j = K − 1,
j = K,
j = K + 1,
j = K + 2,
K + 3 ≤ j ≤ N,
and ρj = −ρ−j . Although ρj = O(1/h) in the interface j = K − 1, . . . , K + 2, we
will prove that the effect of ρ on the error is small away from the interface because it
oscillates and the lowest-order terms cancel in the sum
(3.6)
Δρ :=
K+2
ρj = hφ2F u
K+1/2 .
j=K−1
The truncation error term ρ represents the inconsistency of the operator Lqce,h as a
second-order finite difference approximation of the differential equation (3.2). This
inconsistency is located only in the interface because the atomistic and continuum
models themselves are second-order approximations.
The truncation error term σ accounts for the error in approximating the continuum model (2.14) by a second-order finite difference approximation. We can estimate
the truncation error σ from Taylor’s theorem to obtain
(3.7)
σ2 ≤ Ch2 u(4)
e
h
L2
.
Note that since ue , f , g, and ρ are odd, σ is odd as well. Therefore, we can split
the error e as
e = eρ + eσ
2468
MATTHEW DOBSON AND MITCHELL LUSKIN
such that
(3.8)
Lqce,h eρ = ρ,
L
qce,h
eρ,j = −eρ,−j ,
eσ = σ,
eσ,j = −eσ,−j .
3.1. Global discretization error, eσ . We now have by the stability (2.34) of
Lqce,h and the estimate of the truncation error (3.7) that
Deσ 2 ≤ Ch2 u(4)
e
(3.9)
h
L2
.
We can extend the bound to other norms using the Poincaré and Hölder inequalities.
Lemma 3.1. For eσ defined in (3.8), we have
√
,
eσ ∞ ≤ 2 Deσ 2 ≤ Ch2 u(4)
(3.10)
e
L2
h
(3.11)
h
Deσ p ≤
h
⎧
⎨Ch2 u(4)
e
⎩Ch
3
1
2+p
L2
1 ≤ p ≤ 2,
,
(4)
ue
L2
, 2 ≤ p ≤ ∞.
Proof. We obtain the Poincaré inequality [7]
√
(3.12)
v∞ ≤ Dv1 ≤ 2 Dv2
h
h
h
for all odd periodic v from the identity
!
j
if j > 0,
=1 h(Dv )
vj =
− 0=j−1 h(Dv ) if j < 0,
which gives the first inequality in (3.12). The second follows from Hölder’s inequality.
We can then obtain the error estimate (3.10) for eσ ∞ from the Poincaré inequalh
ity (3.12) and the bound (3.9).
The “inverse” estimate [7],
(3.13)
Dv∞ ≤ h−1/2 Dv2
h
h
for all v, and the Hölder estimates [28],
! 2−p
2 2p Dv2 ,
h
(3.14)
Dvp ≤
2/p
1−2/p
h
Dv2 Dv∞ ,
h
h
1 ≤ p ≤ 2,
2 ≤ p ≤ ∞,
combine to prove (3.11) by taking v = eσ .
3.2. Interfacial coupling error, eρ . In the following, we will bound the error,
eρ , by constructing and estimating an explicit odd solution of
(3.15)
Lqce,h eρ = ρ.
Since ρj is zero for all j except j ∈ ±{K −1, K, K +1, K +2}, eρ satisfies a secondorder, homogeneous recurrence relation in the interior of the continuum region and a
fourth-order, homogeneous recurrence relation in the interior of the atomistic region.
One can thus show that eρ,j is linear for j ≥ K +3 or j ≤ −K −3 and that it is the sum
ERROR ANALYSIS OF THE QUASICONTINUUM APPROXIMATION
2469
of a linear solution and exponential solution for −K + 2 ≤ j ≤ K − 2. The coefficients
for these solutions are determined by the equations in the atomistic-to-continuum
interface.
The homogeneous atomistic difference scheme
−φ2F uj+2 − φF uj+1 + (2φF + 2φ2F )uj − φF uj−1 − φ2F uj−2 = 0
(3.16)
has characteristic equation
0 = −φ2F Λ2 − φF Λ + (2φF + 2φ2F ) − φF Λ−1 − φ2F Λ−2
= −φ2F Λ−2 (Λ2 − 1)2 − φF Λ−1 (Λ − 1)2
with roots
1
1, 1, λ, ,
λ
(3.17)
where
λ=
"
(φF + 2φ2F ) + (φF )2 + 4φF φ2F
.
−2φ2F
General solutions of the homogeneous atomistic equations (3.16) have the form uj =
C1 + C2 hj + C3 λj + C4 λ−j , but seeking an odd solution reduces this to the form
uj = C2 hj + C3 (λj − λ−j ).
The odd solution of the approximate error equations (3.15) has the form uj =
j
−j
m1 hj + β λ −λ
in the atomistic region and uj = m2 hj − m2 in the continuum region,
λK
for some m1 , m2 , β ∈ R. The four equations with indices j = K − 1, K, K + 1, K + 2
will act as boundary conditions for the two regions, determining m1 , m2 , and β, and
requiring one more degree of freedom to be solvable in general. We therefore guess a
solution of the form
⎧
j −j λ −λ
⎪
, 0 ≤ j ≤ K,
⎪
⎨m1 hj + β
λK
(3.18)
eρ,j = m2 hj − m2 + êK+1 , j = K + 1,
⎪
⎪
⎩m hj − m ,
K + 2 ≤ j ≤ N,
2
2
where expressing the unknown eρ,K+1 using a perturbation of the linear solution,
êK+1 , simplifies the solution of the equilibrium equations. The coefficients m1 , m2 ,
êK+1 , and β can be found by satisfying the equilibrium equations in the interface,
j = K − 1, . . . , K + 2. Summing the equilibrium equations across the interface gives
Δρ =
K+2
j=K−1
K+2
(Lqce,h eρ )j
j=K−1
eρ,K−1 − eρ,K−2
eρ,K + eρ,K−1 − eρ,K−2 − eρ,K−3
+
4φ
2F
h2
4h2
eρ,K+3 − eρ,K+2
− (φF + 4φ2F )
h2
m
m2
1
−
.
= (φF + 4φ2F )
h
h
= φF
ρj =
The cancellation of the exponential terms in the final equality holds because
φ2F (λK −λ−K )+(φF +φ2F )(λK−1 −λ−K+1 −λK−2 +λ−K+2 )+φ2F (−λK−3 +λ−K+3 ) = 0,
2470
MATTHEW DOBSON AND MITCHELL LUSKIN
which can be seen by summing (3.16) with the homogeneous solution yj = −λj for
j = −K + 2, . . . , K − 2. Thus, we have from summing the equilibrium equations (3.15)
across the interface that
m1 = m2 +
(3.19)
φF
hΔρ
.
+ 4φ2F
The equality (3.19) can be interpreted as saying that the interfacial truncation error
ρ acts as a source f = Δρ in the continuum equations (3.2) at x = xK .
Lemma 3.2. For eρ defined in (3.8), we have that
eρ ∞ ≤ Ch(1 + |uK+1/2 | + h|uK+1/2 | + h|u
K+1/2 |),
h
(3.20)
Deρ p ≤ Ch1/p (1 + |uK+1/2 | + h|uK+1/2 | + h|u
K+1/2 |),
h
where C > 0 is independent of h, K, and p, 1 ≤ p ≤ ∞.
Proof. We will set up the system of equations for the coefficients in (3.18) and
bound the decay of the coefficients. We split the interface equations as (A+ hB)x = b
and show that b is O(h) and A has a uniformly bounded inverse. We have
(3.21)
⎡
0
⎢ 0
⎢
A=⎢
⎣ 0
⎡
0
⎢ (K
⎢
B=⎢
⎣
⎡
⎤
1 − 12 φ2F
φ2F γK+1 − 12 φ2F γK−1
2 φ2F
φF + 52 φ2F −φF − 2φ2F φF γK+1 + φ2F γK+2 + 32 φ2F γK ⎥
⎥
⎥,
1 ⎦
−φF − 52 φ2F 2φF + 13
φ
−φ
γ
−
2φ
γ
−
φ
γ
F K
2F K
2 2F
2 2F K−1
1 1 − 2 φ2F −φF − 4φ2F
− 2 φ2F γK
⎤
1
3
1
1
−( 2 K + 2 )φ2F 0 0
( 2 K + 2 )φ2F
+ 1)φF + ( 52 K + 2)φ2F −(K + 1)φF − ( 52 K + 3)φ2F 0 0 ⎥
⎥
⎥,
5
1
5
3
−KφF − ( 2 K − 2 )φ2F
KφF − ( 2 K − 2 )φ2F 0 0 ⎦
− 12 Kφ2F
( 12 K + 1)φ2F 0 0
⎤
m1
⎢ m2 ⎥
⎥
x=⎢
⎣ êK+1 ⎦ ,
β
⎡
⎤
ρK−1
⎢ ρK
⎥
⎥
b = h2 ⎢
⎣ ρK+1 ⎦ .
ρK+2
% where
% x = b,
%K + hB)%
Using the equality (3.19), we rewrite the above as (A
⎤
⎡
1 − 21 φ2F
φ2F γK+1 − 12 φ2F γK−1
2 φ2F
⎥
%K = ⎢
A
−φF γK − 2φ2F γK − 12 φ2F γK−1 ⎦ ,
⎣ −φF − 52 φ2F 2φF + 13
2 φ2F
− 12 φ2F −φF − 4φ2F
− 21 φ2F γK
⎤
⎡
⎤
⎡
m2
φ2F 0 0
%
⎣
% = ⎣ êK+1 ⎦ ,
x
B = −φ2F 0 0 ⎦ ,
(3.22)
φ2F 0 0
β
⎡
⎤
⎡
⎤
( 12 K + 32 )φ2F
ρK−1
Δρ
⎢
⎥
% = h2 ⎣ ρK+1 ⎦ − h2
b
⎣ −KφF − ( 52 K − 12 )φ2F ⎦ ,
φF + 4φ2F
ρK+2
− 1 Kφ
2
j
−j
2F
. We have omitted the second equation, as the full system is linearly
and γj = λ −λ
λK
dependent after the elimination of m1 by (3.19).
ERROR ANALYSIS OF THE QUASICONTINUUM APPROXIMATION
2471
%K and B
% do not depend on h directly, though A
%K may have
We note that A
%
indirect dependence if K scales with h. Therefore, we can neglect B for sufficiently
%−1 exists and is bounded uniformly in K. The following
small h, provided that A
K
%K and is a slight extension of [11, Lemma 3.1] that
lemma gives such a bound for A
no longer restricts to the case φ2F < 0.
%K is nonsingular
Lemma 3.3. For all K satisfying 2 ≤ K ≤ N − 2, the matrix A
−1
%
and ||AK || ≤ C, where C > 0 is independent of K and h.
Due to the definition of ρ in (3.5) and Δρ in (3.6), we have that
2 2 % ∞ ≤ C(h + h|u
||b||
K+1/2 | + h |uK+1/2 | + h |uK+1/2 |).
3
The |u
K+1/2 | contribution from Δρ does not have h as a coefficient since K may
scale linearly with N = 1/h. In general, we only have that hK ≤ 1.
% is O(h), and by (3.19), so is x. From (3.18) and
Applying Lemma 3.3, we see that x
the fact that |γj | ≤ 1 for −K ≤ j ≤ K, we see that the error is pointwise O(h), giving
the ∞
h estimate. The derivative of e is pointwise O(h) inside the continuum region,
pointwise O(1) in the interface, and decreases from O(1) to O(h) in the atomistic
region over a distance of O(h| log h|). This allows us to conclude (3.20). Additional
details may be found in [11, Theorem 3.1].
%K is singular whenever φ = 0, the coupling
Remark 3. Note that although A
2F
error solutions are well-behaved as φ2F → 0. The form of the solution becomes linear
in both the atomistic and continuum regions, with a jump at j = K and j = K + 1.
Remark 4. Note that in the nonlinear case, one usually requires an estimate in the
Du∞ norm to control the long-range decay of defects. Thus, it is not immediately
h
clear how the coupling error will decay in the atomistic region.
3.3. Total error. Combining the estimates (3.11) and (3.10) given in Lemma 3.1
for eσ with the estimate (3.20) in Lemma 3.2 for eρ , we obtain from the triangle
inequality the following claim.
Theorem 3.4. Let e denote the QCE error. Then for 1 ≤ p ≤ ∞, 2 ≤ K ≤ N −2,
and h sufficiently small, the error can be bounded by
(4)
|
+
h
u
,
e∞ ≤ Ch 1 + |uK+1/2 | + h|uK+1/2 | + h|u
e
K+1/2
L2
h
(4)
.
Dep ≤ Ch1/p 1 + |uK+1/2 | + h|uK+1/2 | + h|u
2
K+1/2 | + h ue
L
h
Remark 5. From the explicit construction of eρ , we can see that the order of h
in the above result is sharp. In particular, the above are optimal order results when
f = 0. See [11] for plots of the interfacial error in the homogeneous case.
4. Convergence of the QNL solution. For the quasi-nonlocal approximation,
we split the truncation error as
(4.1)
where
(4.2)
Lqnl,h e = Lqnl,h ue − f = ρ + σ,
⎧
0,
⎪
⎪
⎪
⎨−φ u
1 2F K+1/2 − 2 φ2F uK+1/2 h,
ρ=
⎪
φ2F uK+1/2 − 12 φ2F u
⎪
K+1/2 h,
⎪
⎩
0,
0 ≤ j ≤ K − 1,
j = K,
j = K + 1,
K + 2 ≤ j ≤ N,
2472
MATTHEW DOBSON AND MITCHELL LUSKIN
and where
σ2 ≤ Ch2 u(4)
e
L2
h
.
The maximum norm of the truncation error, ρ∞ , here is O(1) as opposed to
h
the QCE, which has an O(1/h) truncation error maximum norm. However, the sum
of ρ is still O(h); that is,
Δρ = −hφ2F u
K+1/2 .
(4.3)
A similar argument as in the QCE case follows. We split the error as
e = eρ + eσ ,
where
Lqnl,h eρ = ρ,
eρ,j = −eρ,−j ,
Lqnl,h eσ = σ,
eσ,j = −eσ,−j .
The same arguments apply to give the bounds (3.11) and (3.10) on eσ . Thus, we
need to work through the modified argument to bound eρ . Since ρ is nonzero only
for j ∈ ±{K, K + 1}, the odd solution eρ has the form
!
j
−j
), 0 ≤ j ≤ K,
m1 hj + β( λ −λ
λK
(4.4)
eρ,j =
m2 hj − m2 ,
K + 1 ≤ j ≤ N.
Summing across the interface again gives
Δρ :=
K+2
K+2
ρj =
j=K−1
=
(φF
+
(Lqce,h eρ )j
j=K−1
4φ2F )
m
1
h
−
m2 .
h
Thus, we have again that
m1 = m2 +
(4.5)
φF
hΔρ
.
+ 4φ2F
We focus on the equations at j = K − 1, K, and K + 1 and split the interface
equations as (A + hB)x = b, where we have that b is O(h2 ) and A has a uniformly
bounded inverse. We have
⎤
⎡
φ2F γK+1
0
φ2F
⎢
φF + 2φ2F φF γK+1 + φ2F γK+2 + φ2F γK ⎥
A=⎣ 0
⎦,
0
⎡
(4.6)
⎢
B=⎣
−φF − 3φ2F
−φF γK − 2φ2F γK − φ2F γK−1
(K + 1)φ2F
(K + 1)(φF + φ2F )
⎤
−(K + 1)φ2F 0
−(K + 1)(φF + φ2F ) 0 ⎥
⎦,
K(φF + 3φ2F ) − φ2F 0
⎤
−K(φF + 3φ2F ) + φ2F
⎤
⎡
0
m1
⎦.
b = h 2 ⎣ ρK
x = ⎣ m2 ⎦ ,
β
ρK+1
⎡
ERROR ANALYSIS OF THE QUASICONTINUUM APPROXIMATION
2473
% where
%K x
% = b,
Using the equality (4.5), we rewrite the above as A
(4.7)
%K =
A
φ2F
φ2F γK+1
,
φF + 2φ2F φF γK+1 + φ2F γK+2 + φ2F γK
(K + 1)φ2F
Δρ
m2
0
% = h2
x=
,
b
.
+ h2 β
(K + 1)(φF + 4φ2F )
ρK
φF + 4φ2F
We have omitted the third equation, as the full system is linearly dependent after
% ∞ ≤ Ch2 (|u
%K has full rank and ||b||
substitution of m1 . We have that A
K+1/2 | +
|uK+1/2 |), so that we obtain the following error estimate for QNL.
Theorem 4.1. Let e be the solution to the QNL error equation (4.1). Then for
1 ≤ p ≤ ∞, 2 ≤ K ≤ N − 2, and h sufficiently small, the error can be bounded by
,
| + u(4)
e∞ ≤ Ch2 |uK+1/2 | + |u
2
e
K+1/2
L
h
(4)
u
Dep ≤ Ch1+1/p |uK+1/2 | + |u
,
|
+
e
K+1/2
L2
h
where C > 0 is independent of h, K, and p.
Remark 6. From the explicit construction of eρ , we can see that the order of h
in the above result is sharp. In particular, the above are optimal-order results when
cos(πx)
f (x) = cos(πx), which has the continuum limit solution ue (x) = π2 (φ
+4φ ) with
F
2F
nonzero second derivatives in the interface.
5. Conclusion. In the above, we have presented convergence results for two fundamental QC energy approximations. They both exhibit a lack of pointwise consistency in the atomistic-to-continuum interface, with the truncation error being O(1/h)
in the QCE case and O(1) in the QNL case. Despite this, we show O(h1/p ) convergence for Deqce p and O(h1+1/p ) convergence for Deqnl p . The extra order of
h
h
convergence exhibited by QNL, combined with the fact that Lqnl,h is stable whenever
the continuum problem is stable, recommends this method. Sharp stability results
are investigated in more detail in [12], where it is proven that QCE does not remain
stable for all strains up to the continuum limit strain, even though our results above
show that it converges for strains bounded away from the continuum limit. The stability of a QC approximation for strains up to the continuum limit strain, such as
proven in this paper for QNL, is essential for the accurate approximation of fracture
or dislocation nucleation [12].
We note that our analysis has been restricted to nearest neighbor and secondneighbor interactions. QNL does not remove all O(1/h) truncation error beyond
second-neighbor interactions [30], and while this has been corrected in one dimension,
no extension has yet been demonstrated that does so in higher dimensions [14]. The
QCF method [9, 10] offers a pointwise consistent QC approximation for which O(h2 )
convergence has been recently demonstrated for Deqcf ∞ [13], although the QCF
h
method is a nonconservative approximation and is thus not derived from an energy [9].
REFERENCES
[1] M. Arndt and M. Luskin, Goal-oriented adaptive mesh refinement for the quasicontinuum
approximation of a Frenkel-Kontorova model, Comput. Methods Appl. Mech. Engrg., 196
(2007), pp. 3759–3770.
2474
MATTHEW DOBSON AND MITCHELL LUSKIN
[2] M. Arndt and M. Luskin, Goal-oriented atomistic-continuum adaptivity for the quasicontinuum approximation, Int. J. Multiscale Comput. Engrg., 5 (2007), pp. 407–415.
[3] M. Arndt and M. Luskin, Error estimation and atomistic-continuum adaptivity for the quasicontinuum approximation of a Frenkel–Kontorova model, Multiscale Model. Simul., 7
(2008), pp. 147–170.
[4] S. Badia, M. Parks, P. Bochev, M. Gunzburger, and R. Lehoucq, On atomistic-tocontinuum coupling by blending, Multiscale Model. Simul., 7 (2008), pp. 381–406.
[5] X. Blanc, C. L. Bris, and P.-L. Lions, From molecular models to continuum mechanics,
Arch. Ration. Mech. Anal., 164 (2002), pp. 341–381.
[6] X. Blanc, C. Le Bris, and F. Legoll, Analysis of a prototypical multiscale method coupling
atomistic and continuum mechanics, M2AN Math. Model. Numer. Anal., 39 (2005), pp.
797–826.
[7] S. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 2nd ed.,
Springer, New York, 2002.
[8] W. Curtin and R. Miller, Atomistic/continuum coupling in computational materials science,
Model. Simul. Mater. Sci. Eng., 11 (2003), pp. R33–R68.
[9] M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum method, M2AN Math.
Model. Numer. Anal., 42 (2008), pp. 113–139.
[10] M. Dobson and M. Luskin, Iterative solution of the quasicontinuum equilibrium equations
with continuation, J. Sci. Comput., 37 (2008), pp. 19–41.
[11] M. Dobson and M. Luskin, An analysis of the effect of ghost force oscillation on quasicontinuum error, M2AN Math. Model. Numer. Anal., 43 (2009), pp. 591–604.
[12] M. Dobson, M. Luskin, and C. Ortner, Sharp Stability Estimates for the Accurate Prediction
of Instabilities by the Quasicontinuum Method, preprint, 2009, arXiv:0905.2914.
[13] M. Dobson, M. Luskin, and C. Ortner, Stability, Instability, and Error of the Force-Based
Quasicontinuum Approximation, preprint, 2009, arXiv:0903.0610.
[14] W. E, J. Lu, and J. Yang, Uniform accuracy of the quasicontinuum method, Phys. Rev. B,
2006, 74 (2006), paper 214115.
[15] W. E and P. Ming, Analysis of the local quasicontinuum method, in Frontiers and Prospects
of Contemporary Applied Mathematics, T. Li and P. Zhang, eds., Higher Education Press,
World Scientific, River Edge, NJ, 2005, pp. 18–32.
[16] J. Knap and M. Ortiz, An analysis of the quasicontinuum method, J. Mech. Phys. Solids, 49
(2001), pp. 1899–1923.
[17] P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model, Math. Comp., 72 (2003), pp. 657–675.
[18] P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects, SIAM J. Numer. Anal., 45 (2007), pp. 313–332.
[19] M. Luskin and C. Ortner, An analysis of node-based cluster summation rules in the quasicontinuum method, SIAM J. Numer. Anal., to appear.
[20] R. Miller, L. Shilkrot, and W. Curtin, A coupled atomistic and discrete dislocation plasticity simulation of nano-indentation into single crystal thin films, Acta Mater., 52 (2003),
pp. 271–284.
[21] R. Miller and E. Tadmor, The quasicontinuum method: Overview, applications and current
directions, J. Comput. Aided Mater. Des., 9 (2002), pp. 203–239.
[22] P. Ming and J. Z. Yang, Analysis of a one-dimensional nonlocal quasicontinuum method,
Multiscale Model. Simul., to appear.
[23] J. T. Oden, S. Prudhomme, A. Romkes, and P. T. Bauman, Multiscale modeling of physical
phenomena: Adaptive control of models, SIAM J. Sci. Comput., 28 (2006), pp. 2359–2389.
[24] C. Ortner and E. Süli, Analysis of a quasicontinuum method in one dimension, M2AN Math.
Model. Numer. Anal., 42 (2008), pp. 57–91.
[25] M. L. Parks, P. B. Bochev, and R. B. Lehoucq, Connecting atomistic-to-continuum coupling
and domain decomposition, Multiscale Model. Simul., 7 (2008), pp. 362–380.
[26] S. Prudhomme, P. T. Bauman, and J. T. Oden, Error control for molecular statics problems,
Int. J. Multiscale Comput. Engrg., 4 (2006), pp. 647–662.
[27] D. Rodney and R. Phillips, Structure and strength of dislocation junctions: An atomic level
analysis, Phys. Rev. Lett., 82 (1999), pp. 1704–1707.
[28] W. Rudin, Real and Complex Analysis, McGraw–Hill, New York, 1986.
[29] V. Shenoy, R. Miller, E. Tadmor, D. Rodney, R. Phillips, and M. Ortiz, An adaptive
finite element approach to atomic-scale mechanics—The quasicontinuum method, J. Mech.
Phys. Solids, 47 (1999), pp. 611–642.
[30] T. Shimokawa, J. Mortensen, J. Schiotz, and K. Jacobsen, Matching conditions in the
quasicontinuum method: Removal of the error introduced at the interface between the
coarse-grained and fully atomistic regions, Phys. Rev. B, 69 (2004), paper 214104.
ERROR ANALYSIS OF THE QUASICONTINUUM APPROXIMATION
2475
[31] E. Süli and D. F. Mayers, An Introduction to Numerical Analysis, Cambridge University
Press, Cambridge, UK, 2003.
[32] E. Tadmor, M. Ortiz, and R. Phillips, Quasicontinuum analysis of defects in solids, Phil.
Mag. A, 73 (1996), pp. 1529–1563.