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# Error Estimate of Multiscale Finite Element Method for Periodic Media
Revisited

Pingbing Ming and Siqi Song  LSEC, Institute of Computational Mathematics and
Scientific/Engineering Computing, AMSS
Chinese Academy of Sciences, No. 55, East Road Zhong-Guan-Cun, Beijing 100190,
China
and School of Mathematical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China
email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We derive the optimal energy error estimate for multiscale finite element
method with oversampling technique applying to elliptic system with rapidly
oscillating periodic coefficients under the assumption that the coefficients
are bounded and measurable, which may admit rough microstructures. As a by-
product of the energy estimate, we derive the rate of convergence in
L${}^{d/(d-1)}-$norm.

##  1\. Introduction

The multiscale finite element method (MsFEM) introduced by Hou an Wu [19] aims
for solving the boundary value problems with rapidly oscillating coefficients
without resolving the fine scale information. The main idea is to exploit the
multiscale basis functions that capture the fine scale information of the
underlying partial differential equations. MsFEM has been successfully applied
to many problems such as two phase flows, nonlinear homogenization problems,
convection-diffusion problems, elliptic interface problems with high-contrast
coefficients and Poisson problem with rough and oscillating boundary, we refer
to book [15] for a survey of MsFEM before 2009. More recent efforts for MsFEM
focus on extending the method to deal with more general media; cf., [11, 7,
6]. We also refer to [31, 32, 2, 5] for a summary of recent progress for
related methods.

In [20] and [16], the authors proved MsFEM converges for the scalar elliptic
boundary value problem in two dimension with periodic oscillating coefficients
in the energy norm, and the rate of convergence is
$\sqrt{\varepsilon}+h+\varepsilon/h$, where $h$ is the mesh size of the
triangulation, and $\varepsilon$ is the period of the oscillation. The
technical assumptions are

  1. (1)

The coefficient matrix of the elliptical problem is symmetric, and each entry
is a $C^{1}$ function;

  2. (2)

The homogenized solution $u_{0}\in W^{1,\infty}(\Omega)\cap H^{2}(\Omega)$;

  3. (3)

The corrector $\chi$ defined in (3.2) belong to $W^{1,\infty}$.

The first assumption excludes the rough microstructures, which frequently
appears in the realistic materials [36]; The second assumption is standard
except that $u_{0}\in W^{1,\infty}(\Omega)$, which may not be true even for
Poisson equation posed on a ball [12]. The last assumption on the corrector is
not realistic at all, though it may be true for certain special
microstructures such as laminates [10] and for problems with piecewise Hölder
continuous coefficients [25, 24]; We refer to [14] for an elaboration on this
assumption.

Nevertheless, there are some subsequent endeavor on proving the error
estimates for MsFEM under weaker assumptions; see, e.g., [8, 33, 9, 37], just
name a few, most of them concern the second assumption, while it is still
unknown whether the above assumptions may be removed or to what degree they
may be weakened. Moreover, though MsFEM has been successfully applied to
elliptic systems [15, 11], while it does not seem easy to extend the proof to
elliptic systems because the maximum principle has been exploited, which may
be invalid for elliptic systems [22].

The present work gives an affirmative answer to the above questions. Assuming
that $u_{0}\in W^{2,d}$ with the dimensionality $d=2,3$, we prove the optimal
energy error estimate of MsFEM with/without oversampling for elliptic system
with bounded, measurable and symmetric periodical coefficients; cf. Theorem
4.1 and Theorem 4.10. This means that MsFEM achieves optimal rate of
convergence for problems with rough microstructures. Furthermore, the same
rate of convergence is proved for the elliptic scalar problem without the
symmetrical assumption on the coefficient matrix. The practical implication of
such result is that the methods tailored to resolve the underlying rough
microstructure are in great demand to improve the overall accuracy of MsFEM.

As an application of the energy error estimate, we derive improved error
estimate of MsFEM in $L^{d/(d-1)}-$norm by resorting to the Aubin-Nitsche dual
argument [3, 30], naturally, this gives the $L^{2}-$error estimates for two-
dimensional problem and the elliptic scalar equation in three dimension. Such
estimate would be useful for analyzing MsFEM applying to the eigenvalue
problems in composites [21].

There are two ingredients in our proof. The one is a local version of the
multiplier estimates for periodic homogenization of elliptic system [38, 35],
which helps us to remove the boundedness assumption on the gradient of the
corrector. Another one is a local estimate of the gradient of the first order
approximation of the solution; see Lemma 4.7, which bypasses the maximum
principle in the proof, hence we may derive the error estimate for elliptic
system.

The remaining part of this paper is as follows. We formulate MsFEM with
oversampling in § 2. In § 3, we recall some quantitative estimates of the
periodic homogenization for elliptic system. The energy error estimate will be
given in § 4, from which we prove the error estimates in the $L^{d/d-1}$ norm.
As a direct consequence of these estimates, we prove the error estimates for
MsFEM without oversampling. In the last section, we summarize our results and
discuss certain extensions.

Throughout this paper, $C$ is a generic constant that may be different at
different occurrence, while it is independent of the mesh size $h$ and the
small parameter $\varepsilon$.

##  2\. Multiscale Finite Element Method with Oversampling

We firstly fix some notations. Let $\Omega$ be a bounded Lipschitz domain in
$\mathbb{R}^{d}$ (we focus on $d=2,3$). The standard Sobolev space
$W^{k,p}(\Omega)$ will be used [1], which is equipped with the norm
$\left\|\,\cdot\,\right\|_{W^{k,p}(\Omega)}$. We use the convention
$H^{k}(\Omega)=W^{k,2}(\Omega)$. We denote by $W^{k,p}(\Omega;\mathbb{R}^{m})$
the vector-valued function with each component belonging to $W^{k,p}(\Omega)$,
and define $\left\lvert D\right\rvert{:}=\text{mes}D$ for any measurable set
$D$.

We consider the second order elliptic system in divergence form

$\mathcal{L}_{\varepsilon}=-\operatorname{div}\left(\,A(x/\varepsilon)\nabla\,\right)$

with the coefficient $A$ is given by

$A(y)=a_{ij}^{\alpha\beta}(y)\qquad
i,j=1,\cdots,d\text{\;and\;}\alpha,\beta=1,\cdots,m.$

For $u=(u^{1},\cdots,u^{m})$,

$\left(\,\mathcal{L}_{\varepsilon}(u)\,\right)^{\alpha}{:}=-\dfrac{\partial}{\partial
x_{i}}\left(\,a_{ij}^{\alpha\beta}\left(\,\dfrac{x}{\varepsilon}\,\right)\dfrac{\partial
u^{\beta}}{\partial x_{j}}\,\right)\qquad\alpha=1,\cdots,m.$

We always assume that $A$ is bounded measurable and satisfies the Legendre-
Hardmard condition as

(2.1)
$\lambda\left\lvert\xi\right\rvert^{2}\left\lvert\eta\right\rvert^{2}\leq
a_{ij}^{\alpha\beta}(y)\xi_{i}\xi_{j}\eta_{\alpha}\eta_{\beta}\leq\Lambda\left\lvert\xi\right\rvert^{2}\left\lvert\eta\right\rvert^{2}\qquad\text{for
a.e.\;}y\in\mathbb{R}^{d},$

where $\xi=(\xi_{1},\cdots,\xi_{d})$ and $\eta=(\eta_{1},\cdots,\eta_{m})$.
The transpose of $A$ is understood as $A^{t}(y)=a_{ji}^{\beta\alpha}(y)$. We
assume that $A$ is $1-$periodic; i.e., for all $z\in\mathbb{Z}^{d}$,

$A(y+z)=A(y)\qquad\text{for a.e.\;}y\in\mathbb{R}^{d}.$

Considering the following homogeneous boundary value problem: Given $f\in
H^{-1}(\Omega;\mathbb{R}^{m})$, we find $u^{\varepsilon}\in
H_{0}^{1}(\Omega;\mathbb{R}^{d})$ satisfying

(2.2)
$\mathcal{L}_{\varepsilon}(u^{\varepsilon})=f\quad\text{in\quad}\Omega\qquad\text{and\quad}u^{\varepsilon}=0\quad\text{on\quad}\partial\Omega$

in the sense of distribution. The corresponding variational problem reads as:
Find $u^{\varepsilon}\in H_{0}^{1}(\Omega;\mathbb{R}^{d})$ such that

(2.3) $a_{{\Omega}}(u^{\varepsilon},v)=\left\langle
f,v\right\rangle_{\Omega}\quad\text{for all~{}}v\in
H_{0}^{1}(\Omega;\mathbb{R}^{d}),$

where for any measurable subset $\widetilde{\Omega}$ of $\Omega$,

$a_{\widetilde{\Omega}}(u,v){:}=\int_{\widetilde{\Omega}}\nabla v\cdot
A(x/\varepsilon)\nabla u\,\mathrm{d}x\qquad\text{and\quad}\left\langle
f,v\right\rangle_{\Omega}=\int_{\Omega}f(x)\cdot v(x)\,\mathrm{d}x.$

We shall drop the subscript when the subset is the whole domain $\Omega$.

$\Omega$ is triangulated by $\mathcal{T}_{h}$ that consists of simplices
$\tau$ with $h_{\tau}$ its diameter and
$h=\max_{\tau\in\mathcal{T}_{h}}h_{\tau}$. We assume that $\mathcal{T}_{h}$ is
shape-regular in the sense of Ciarlet-Raviart [13]: there exists a chunkiness
parameter $\sigma_{0}$ such that $h_{\tau}/\rho_{\tau}\leq\sigma_{0}$, where
$\rho_{\tau}$ is the diameter of the largest ball inscribed into $\tau$. We
also assume that $\mathcal{T}_{h}$ satisfies the inverse assumption: there
exists $\sigma_{1}>0$ such that $h/h_{\tau}\leq\sigma_{1}$.

For each element $\tau$, we firstly choose an over-sampling domain
$S=S(\tau)\supset\tau$, which is also a simplex. Let $\lambda_{i}$ be the
$i$th barycentric coordinate of the simplex $S$ and
$e^{\beta}=(0,\cdots,1\cdots,0)$ with $1$ in the $\beta$th position. Denote
$Q\in\mathbb{R}^{(d+1)\times m}$ with $Q_{i}^{\beta}=\lambda_{i}e^{\beta}$ for
$i=1,\cdots,d+1$ and $\beta=1,\cdots,m$, we find
$\psi_{i}^{\beta}-Q_{i}^{\beta}\in H_{0}^{1}(S;\mathbb{R}^{m})$ such that

(2.4) $a_{S}(\psi_{i}^{\beta},\varphi)=0\quad\text{for all\quad}\varphi\in
H_{0}^{1}(S;\mathbb{R}^{m}).$

Next, the basis function $\phi_{i}^{\beta}$ associated with the node $x_{i}$
of $\tau$ is defined as

(2.5) $\phi_{i}^{\beta}=c_{ij}^{\beta}\psi_{j}^{\beta}\qquad
i=1,\cdots,d+1\quad\text{and\;}\beta=1,\cdots,m,$

where the coefficients $c_{ij}^{\beta}$ are determined by
$c_{ik}^{\beta}Q_{k}^{\beta}(x_{j})=\delta_{ij}e^{\beta}$ for any node $x_{j}$
of $\tau$. The matrix $c^{\beta}=(c_{ij}^{\beta})$ is invertible because
$\\{\psi_{i}^{\beta}\\}_{i=1}^{d+1}$ are linear independent over $S$. The
multiscale finite element space is defined by

$V_{h}:=\text{Span}\\{\phi_{i}\quad\text{for all
nodes\;}x_{i}\text{\;of\;}\mathcal{T}_{h}\\}.$

Note that $V_{h}\subsetneq H^{1}(\Omega;\mathbb{R}^{m})$ because the functions
in $V_{h}$ may not be continuous across the element boundary. The bilinear
form $a_{h}$ is defined for any $v,w\in V_{h}$ in a piecewise manner as
$a_{h}(v,w):=\sum_{\tau\in\mathcal{T}_{h}}a_{\tau}(v,w).$ The approximation
problem reads as: Find $u_{h}\in V_{h}^{0}$ such that

(2.6) $a_{h}(u_{h},v)=\left\langle f,v\right\rangle\qquad\text{for
all\quad}v\in V_{h}^{0},$

where $V_{h}^{0}{:}=\\{v\in V_{h}|\ \text{the degrees of freedom of the nodes
on~{}}\partial\Omega\text{~{}are zero}\\}$. It follows from [16, Appendix B]
that

(2.7) $\left\|\,v\,\right\|_{h}:=\left(\,\sum_{\tau\in\mathcal{T}_{h}}\|\nabla
v\|_{L^{2}(\tau)}^{2}\,\right)^{1/2}$

is a norm over $V_{h}^{0}$.

######  Remark 2.1.

The authors in [18] introduced a new MsFEM that allows for the over-sampling
domain of more general shape, e.g. an element star, which facilitates the
implementation of MsFEM, while it is equivalent to the original version [16]
if the over-sampling domain is a simplex.

##  3\. Quantitative Estimates for Periodic Homogenization of Elliptic System

By the theory of H-convergence [29], the solution $u^{\varepsilon}$ of (2.2)
converges weakly to the homogenized solution $u_{0}$ in
$H^{1}(\Omega;\mathbb{R}^{m})$ as $\varepsilon\to 0$, and $u_{0}$ satisfies

(3.1) $\mathcal{L}_{0}(u_{0})=f\quad\text{in\quad}\Omega,\qquad
u_{0}=0\quad\text{on\quad}\partial\Omega,$

where $\mathcal{L}_{0}=\operatorname{div}(\widehat{A}\nabla)$ with the
homogenized coefficients $\widehat{A}=\widehat{a}_{ij}^{\alpha\beta}$ given by

$\widehat{a}_{ij}^{\alpha\beta}={\int\negthickspace\negthickspace\negthickspace\negthinspace-}_{Y}\left(\,a_{ij}^{\alpha\beta}(y)+a_{ik}^{\alpha\gamma}\dfrac{\partial\chi_{j}^{\gamma\beta}}{\partial
y_{k}}\,\right)\,\mathrm{d}y,$

where the unit cell $Y{:}=[0,1)^{d}$, and the corrector
$\chi(y)=\left(\,\chi_{j}^{\beta}(y)\,\right)=\left(\,\chi_{j}^{\alpha\beta}\,\right)$
for $j=1,\cdots,d$ and $\alpha,\beta=1,\cdots,m$ satisfies the following cell
problem: Find $\chi_{j}^{\beta}\in H^{1}_{\text{per}}(Y;\mathbb{R}^{m})$ such
that $\int_{Y}\chi_{j}^{\beta}\,\mathrm{d}y=0$ and

(3.2) $a_{Y}(\chi_{j}^{\beta},\psi)=-a_{Y}(P_{j}^{\beta},\psi)\qquad\text{for
all\quad}\psi\in H^{1}_{\text{per}}(Y;\mathbb{R}^{m}),$

where $P_{j}^{\beta}=y_{j}e^{\beta}$, and for all $\phi,\psi\in
H^{1}_{\text{per}}(Y;\mathbb{R}^{m})$,

$a_{Y}(\phi,\psi){:}=\int_{Y}a_{ij}^{\alpha\beta}(y)\dfrac{\partial\phi^{\beta}}{\partial
y_{j}}\dfrac{\partial\psi^{\alpha}}{\partial y_{i}}\,\mathrm{d}y.$

The existence and uniqueness of the solution of (3.2) follows from the
ellipticity of $A$ and the Lax-Milgram theorem. Moreover,

$\left\|\,\nabla\chi_{j}^{\beta}\,\right\|_{L^{2}(Y)}\leq\Lambda/\lambda\qquad\text{and}\quad\left\|\,\chi_{j}^{\beta}\,\right\|_{H^{1}(Y)}\leq
C_{p}\Lambda/\lambda,$

where $C_{p}$ is the constant arising from Poincaré’s inequality:

$\left\|\,\psi\,\right\|_{H^{1}(Y)}\leq
C_{p}\left\|\,\nabla\psi\,\right\|_{L^{2}(Y)}\quad\text{for all\quad}\psi\in
H^{1}_{\text{per}}(Y)\quad\text{and}\quad\int_{Y}\psi\,\mathrm{d}y=0.$

By Meyers’ regularity result [27, 28], there exists $p>2$ such that

(3.3) $\left\|\,\nabla\chi_{j}^{\beta}\,\right\|_{L^{p}(Y)}\leq C,$

where the index $p$ and the constant $C$ depend only on $\lambda$ and
$\Lambda$. This inequality implies that $\chi$ is Hölder continuous when $d=2$
by the Sobolev embedding theorem [1]. By the De Giorgi-Nash theorem, $\chi$ is
also Hölder continuous when $d=3$ and $m=1$. However, for $d=3$ and $m\geq 2$,
we only have

(3.4) $\left\|\,\chi_{j}^{\beta}\,\right\|_{L^{q}(Y)}\leq C\qquad\text{for
certain\quad}q\geq 6,$

which is a direct consequence of (3.3) and the Sobolev embedding theorem.

Another frequently used estimate for the corrector matrix is: For any
measurable set $D$, and for $1\leq p\leq\infty$, there exists $C$ depends on
$d$ and $p$ such that

(3.5) $\left\|\,\chi(x/\varepsilon)\,\right\|_{L^{p}(D)}\leq C\left\lvert
D\right\rvert^{1/p}\left\|\,\chi\,\right\|_{L^{p}(Y)}.$

Given the corrector $\chi$, the first order approximation of $u^{\varepsilon}$
is defined by

(3.6) $u_{1}^{\varepsilon}(x){:}=u_{0}(x)+\varepsilon\chi(x/\varepsilon)\nabla
u_{0}(x).$

We summarize the convergence rate of $u_{1}^{\varepsilon}$ in the following
theorem.

######  Theorem 3.1.

Assume that $A$ is $1-$periodic and satisfies (2.1). Let $\Omega$ be a bounded
Lipschitz domain in $\mathbb{R}^{d}$. Let $u^{\varepsilon}$ and $u_{0}$ be the
weak solutions of (2.2) and (3.1), respectively.

  1. (1)

If $u_{0}\in W^{2,d}(\Omega;\mathbb{R}^{m})$, then

(3.7)
$\left\|\,u^{\varepsilon}-u_{1}^{\varepsilon}\,\right\|_{H^{1}(\Omega)}\leq
C\sqrt{\varepsilon}\left\|\,\nabla u_{0}\,\right\|_{W^{1,d}(\Omega)},$

where $C$ depends on $\lambda,\Lambda$ and $\Omega$.

  2. (2)

If the corrector $\chi$ is bounded and $u_{0}\in
H^{2}(\Omega;\mathbb{R}^{m})$, then

(3.8)
$\left\|\,u^{\varepsilon}-u_{1}^{\varepsilon}\,\right\|_{H^{1}(\Omega)}\leq
C\sqrt{\varepsilon}\left\|\,\nabla u_{0}\,\right\|_{H^{1}(\Omega)},$

where $C$ depends $\lambda,\Lambda,\left\|\,\chi\,\right\|_{L^{\infty}}$ and
$\Omega$.

The estimates (3.7) and (3.8) are taken from [35, Theorem 3.2.7].

We also need the following estimate in certain $L^{p}-$norm.

######  Theorem 3.2.

Under the same assumption of Theorem 3.1, and assume that $A=A^{\ast}$ for
$m\geq 2$. Suppose that $u_{0}\in W^{2,q}(\Omega;\mathbb{R}^{m})$ for
$q=2d/(d+1)$. Then

(3.9) $\left\|\,u^{\varepsilon}-u_{0}\,\right\|_{L^{p}(\Omega)}\leq
C\varepsilon\left\|\,\nabla u_{0}\,\right\|_{W^{1,q}(\Omega)},$

where $p=2d/(d-1)$ and $C$ depends on $\lambda,\Lambda$ and $\Omega$.

This theorem was proved in [34]; See also [35, Theorem 3.4.5] with

$\left\|\,u^{\varepsilon}-u_{0}\,\right\|_{L^{p}(\Omega)}\leq
C\varepsilon\left\|\,u_{0}\,\right\|_{W^{2,q}(\Omega)}.$

The present form may be obtained by pursuing the proof. Moreover, using a
scaling argument, we rewrite (3.9) as

(3.10) $\left\|\,u^{\varepsilon}-u_{0}\,\right\|_{L^{p}(\Omega)}\leq
C\varepsilon\left(\,(\text{diam\;}\Omega)^{-1}\left\|\,\nabla
u_{0}\,\right\|_{L^{q}(\Omega)}+\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{q}(\Omega)}\,\right),$

where $C$ is independent of the diameter of $\Omega$.

##  4\. Error Estimates for the Periodic Media

Before stating the main result, we make an assumption on the size of the over-
sampling domain $S$ [8]. Assumption A: There exist constants $\gamma_{1}$ and
$\gamma_{2}$ independent of $h$ such that

$\text{\;diam\;}S\leq\gamma_{1}h_{\tau}\qquad\text{and\quad}\text{dist}(\partial\tau,\partial
S)\geq\gamma_{2}h_{\tau}.$

###  4.1. H1 error estimate

The main result of this work is

######  Theorem 4.1.

Assume that $A$ is $1-$periodic and satisfies (2.1). For $m\geq 2$, we assume
$A^{t}=A$. Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d}$, and
let $u^{\varepsilon}$ and $u_{h}$ be the solutions of Problems (2.3) and
(2.6), respectively.

For $m=1$ or $d=2$, if $u_{0}\in H^{2}(\Omega;\mathbb{R}^{m})$, then

(4.1) $\left\|\,u^{\varepsilon}-u_{h}\,\right\|_{h,\Omega}\leq
C\left(\,\sqrt{\varepsilon}+\varepsilon/h+h\,\right)\left(\,\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\Omega)}+\left\|\,f\,\right\|_{L^{2}(\Omega)}\,\right),$

where $C$ depends on
$\lambda,\Lambda,\left\|\,\chi\,\right\|_{L^{\infty}},\Omega$ and the mesh
parameters $\sigma_{0},\sigma_{1},\gamma_{1},\gamma_{2}$.

For $m\geq 2$ and $d=3$, if $u_{0}\in W^{2,3}(\Omega;\mathbb{R}^{m})$, then

(4.2) $\left\|\,u^{\varepsilon}-u_{h}\,\right\|_{h,\Omega}\leq
C\left(\,\sqrt{\varepsilon}+\varepsilon/h+h\,\right)\left(\,\left\|\,\nabla
u_{0}\,\right\|_{W^{1,3}(\Omega)}+\left\|\,f\,\right\|_{L^{2}(\Omega)}\,\right),$

where $C$ depends on $\lambda,\Lambda,\Omega$ and the mesh parameters
$\sigma_{0},\sigma_{1},\gamma_{1},\gamma_{2}$.

The implication of the above theorem is as follows.

  1. (1)

The convergence rate of MsFEM proved above is the same with that in [16] for
the scalar elliptic problem in two dimension, while we remove the technical
assumptions on the coefficient $a^{\varepsilon}$, the homogenized solution
$u_{0}$ and the correctors $\chi$.

  2. (2)

The convergence rate of MsFEM is new for elliptic system as well as problems
in three dimension.

  3. (3)

We clarify the dependence of the right-hand side of the energy error estimates
on $u_{0}$ and $f$ in the natural Sobolev norms, which together with the
Aubin-Nitsche dual argument yields the convergence rate of MsFEM in
L${}^{d/(d-1)}-$norm. In particular, we obtain the L2 error estimate for
problem in $d=2$ and scalar elliptic problem in $d=3$, cf. Theorem (4.9).

The proof of Theorem 4.1 is based on the second lemma of Strang [4] because
MsFEM with oversampling is a nonconforming method.

(4.3) $\left\|\,u^{\varepsilon}-u_{h}\,\right\|_{h}\leq C\left(\,\inf_{v\in
V_{h}^{0}}\left\|\,u^{\varepsilon}-v\,\right\|_{h}+\sup_{w\in
V_{h}^{0}}\dfrac{\left\lvert\left\langle f,w\right\rangle-
a_{h}(u^{\varepsilon},w)\right\rvert}{\left\|\,w\,\right\|_{h}}\,\right),$

where $C$ depends on $\lambda,\Lambda,\gamma_{1}$ and $\gamma_{2}$.

To bound the right-hand side of (4.3), we start with a local multiplier
estimate that control the L${}^{2}-$norm of $(\nabla\chi)\psi$ for certain
$\psi$.

######  Lemma 4.2.

Let $\chi$ be defined in (3.2) and suppose that $D$ is a convex polyhedron.
For any $\psi\in W^{1,d}(D;\mathbb{R}^{m})$, there exists $C$ independent of
the size of $D$ such that

(4.4)
$\varepsilon\left\|\,\nabla\chi(x/\varepsilon)\psi\,\right\|_{L^{2}(D)}\leq
C\left\lvert
D\right\rvert^{1/2-1/d}\left(\,\left\|\,\psi\,\right\|_{L^{d}(\tau)}+\varepsilon\left\|\,\nabla\psi\,\right\|_{L^{d}(D)}\,\right).$

If $\left\|\,\chi\,\right\|_{L^{\infty}}$ is bounded, then for any $\psi\in
H^{1}(D;\mathbb{R}^{m})$, there exists $C$ independent of the size of $D$ such
that

(4.5)
$\varepsilon\left\|\,\nabla\chi(x/\varepsilon)\psi\,\right\|_{L^{2}(D)}\leq
C(1+\left\|\,\chi\,\right\|_{L^{\infty}})\left(\,\left\|\,\psi\,\right\|_{L^{2}(D)}+\varepsilon\left\|\,\nabla\psi\,\right\|_{L^{2}(D)}\,\right).$

The proof depends on the following multiplier estimates proved in [35, Lemma
3.2.8]: For any $\psi\in W^{1,d}(\Omega;\mathbb{R}^{m})$,

(4.6)
$\varepsilon\left\|\,\nabla\chi(x/\varepsilon)\psi\,\right\|_{L^{2}(\Omega)}\leq
C\left(\,\left\|\,\psi\,\right\|_{L^{d}(\Omega)}+\varepsilon\left\|\,\nabla\psi\,\right\|_{L^{d}(\Omega)}\,\right),$

and for any $\psi\in H^{1}(\Omega;\mathbb{R}^{m})$,

(4.7)
$\varepsilon\left\|\,\nabla\chi(x/\varepsilon)\psi\,\right\|_{L^{2}(\Omega)}\leq
C(1+\left\|\,\chi\,\right\|_{L^{\infty}})\left(\,\left\|\,\psi\,\right\|_{L^{2}(\Omega)}+\varepsilon\left\|\,\nabla\psi\,\right\|_{L^{2}(\Omega)}\,\right),$

where $C$ depends on $\lambda,\Lambda$ and $\Omega$. These multiplier
estimates are crucial to prove the error bounds (3.7) and (3.8). We clarify
the dependence of the constant $C$ on the size of the domain.

###### Proof.

Denote $L=\text{diam\;}D$, and we apply the scaling $x^{\prime}=x/L$ to $D$ so
that the rescaled element $\widehat{D}$ has diameter $1$. Note that

$x/\varepsilon=x^{\prime}/\varepsilon^{\prime}\qquad\text{with}\quad\varepsilon^{\prime}=\varepsilon/L.$

Hence
$\varepsilon\nabla\chi(x/\varepsilon)=\varepsilon^{\prime}\;\nabla_{x^{\prime}}\chi(x^{\prime}/\varepsilon^{\prime})$
and $\psi(x)=\psi(Lx^{\prime})=\widehat{\psi}(x^{\prime})$. Applying (4.6) to
$\widehat{D}$, we obtain that there exists $C$ depends only on $\widehat{D}$
such that

$\displaystyle\varepsilon\left\|\,\nabla\chi(x/\varepsilon)\psi\,\right\|_{L^{2}(D)}$
$\displaystyle\leq\left(\,\left\lvert
D\right\rvert/\left\lvert\widehat{D}\right\rvert\,\right)^{1/2}\varepsilon^{\prime}\left\|\,\nabla_{x^{\prime}}\chi(x^{\prime}/\varepsilon^{\prime})\widehat{\psi}\,\right\|_{L^{2}(\widehat{D})}$
$\displaystyle\leq C\left\lvert
D\right\rvert^{1/2}\left(\,\left\|\,\widehat{\psi}\,\right\|_{L^{d}(\widehat{D})}+\varepsilon^{\prime}\left\|\,\nabla_{x^{\prime}}\widehat{\psi}\,\right\|_{L^{d}(\widehat{D})}\,\right)$
$\displaystyle\leq C\left\lvert
D\right\rvert^{1/2-1/d}\left(\,\left\|\,\psi\,\right\|_{L^{d}(D)}+\varepsilon\left\|\,\nabla\psi\,\right\|_{L^{d}(D)}\,\right).$

This yields (4.4).

Replacing (4.6) by (4.7) and proceeding along the same line that leads to
(4.4), we obtain (4.5). ∎

The second ingredient of the error estimate is the quantitative estimates for
the MsFEM functions in $V_{h}$, which has been used in all the previous study.
For any $w\in V_{h}$, we may write, on each element $\tau\in\mathcal{T}_{h}$,

$w(x)|_{\tau}{:}=\sum_{i=1}^{d+1}w_{i}\phi_{i}(x)=\sum_{i=1}^{d+1}\sum_{\beta=1}^{m}w_{i}^{\beta}c_{ik}^{\beta}\psi_{k}^{\beta}(x)$

for certain coefficients $w_{i}\in\mathbb{R}^{m}$. It is well-defined over
$S$, and

$\mathcal{L}_{\varepsilon}(w)=0\quad\text{in\quad}S\qquad\text{and}\quad
w=w_{0}\quad\text{on\quad}\partial S,$

where
$w_{0}=\sum_{i=1}^{d+1}\sum_{\beta=1}^{m}w_{i}^{\beta}c_{ik}^{\beta}Q_{k}^{\beta}(x)$.
It is clear that the homogenization limit of $w$ is $w_{0}$. And there exists
$C$ depending on $\lambda,\Lambda,\gamma_{1}$ and $\gamma_{2}$ such that

(4.8) $\left\|\,\nabla w_{0}\,\right\|_{L^{2}(\tau)}\leq C\left\|\,\nabla
w\,\right\|_{L^{2}(\tau)}\qquad\text{for all\quad}\tau\in\mathcal{T}_{h}.$

The first order approximation of $w$ is defined as
$w_{1}^{\varepsilon}{:}=w_{0}+\varepsilon(\chi\cdot\nabla)w_{0}.$

######  Lemma 4.3.

Suppose that Assumption A is true, and let $w\in V_{h}$, there exists $C$
independent of $h_{\tau}$ such that

(4.9) $\left\|\,w-w_{0}\,\right\|_{L^{2}(S)}\leq C\varepsilon\left\|\,\nabla
w_{0}\,\right\|_{L^{2}(S)},$

and

(4.10) $\left\|\,\nabla(w-w_{1}^{\varepsilon})\,\right\|_{L^{2}(\tau)}\leq
C\dfrac{\varepsilon}{h_{\tau}}\left\|\,\nabla w_{0}\,\right\|_{L^{2}(S)}.$

###### Proof.

Applying Theorem 3.2 to $w$, using (3.10) and the fact that $w_{0}$ is linear
over $S$, we obtain

$\displaystyle\left\|\,w-w_{0}\,\right\|_{L^{2}(S)}$
$\displaystyle\leq\left\lvert
S\right\rvert^{1/2-1/p}\left\|\,w-w_{0}\,\right\|_{L^{p}(S)}$
$\displaystyle\leq C\varepsilon\left\lvert
S\right\rvert^{1/2-1/p}\left(\,(\text{diam\,}S)^{-1}\left\|\,\nabla
w_{0}\,\right\|_{L^{q}(S)}+\left\|\,\nabla^{2}w_{0}\,\right\|_{L^{q}(S)}\,\right)$
$\displaystyle=C\dfrac{\varepsilon}{\text{diam\,}S}\left\lvert
S\right\rvert^{1/2-1/p+1/q}\left\lvert\nabla w_{0}\right\rvert$
$\displaystyle\leq C\varepsilon\left\|\,\nabla w_{0}\,\right\|_{L^{2}(S)},$

where we have used $1/q-1/p=1/d$ in the last step. This gives (4.9).

Note that

$a_{S}(w-w_{1}^{\varepsilon},v)=0\qquad\text{for all\quad}v\in
H_{0}^{1}(S;\mathbb{R}^{m}).$

By the Caccioppoli inequality [17, Corollary 1.37] and Assumption A, there
exists $C$ that depends on $\lambda,\Lambda,\gamma_{1}$ and $\gamma_{2}$ such
that

(4.11)
$\left\|\,\nabla(w-w_{1}^{\varepsilon})\,\right\|_{L^{2}(\tau)}\leq\dfrac{C}{h_{\tau}}\left\|\,w-w_{1}^{\varepsilon}\,\right\|_{L^{2}(S)}.$

Using the fact that $\nabla w_{0}$ is a piecewise constant matrix and (3.5)
with $p=2$, we obtain

$\displaystyle\left\|\,w_{1}^{\varepsilon}-w_{0}\,\right\|_{L^{2}(S)}$
$\displaystyle=\varepsilon\left\|\,\chi(x/\varepsilon)\nabla
w_{0}\,\right\|_{L^{2}(S)}=\varepsilon\left\|\,\chi(x/\varepsilon)\,\right\|_{L^{2}(S)}\left\lvert\nabla
w_{0}\right\rvert$ $\displaystyle\leq C\varepsilon\left\lvert
S\right\rvert^{1/2}\left\|\,\chi\,\right\|_{L^{2}(Y)}\left\lvert\nabla
w_{0}\right\rvert=C\varepsilon\left\|\,\chi\,\right\|_{L^{2}(Y)}\left\|\,\nabla
w_{0}\,\right\|_{L^{2}(S)},$

which together with (4.9) and the triangle inequality gives

$\left\|\,w-w_{1}^{\varepsilon}\,\right\|_{L^{2}(S)}\leq\left\|\,w-w_{0}\,\right\|_{L^{2}(S)}+\left\|\,w_{1}^{\varepsilon}-w_{0}\,\right\|_{L^{2}(S)}\leq
C\varepsilon\left\|\,\nabla w_{0}\,\right\|_{L^{2}(S)}.$

Substituting the above inequality into (4.11), we obtain (4.10). ∎

Another useful tool is the following inequality for a tubular domain defined
as follows. Let $\tau\in\mathcal{T}_{h}$ be an element, for any $\delta>0$, we
define

$\tau_{\delta}{:}=\\{\,x\in\tau\,\mid\,\text{dist}(x,\partial\tau)\leq\delta\,\\}.$

######  Lemma 4.4.

Let $1\leq p<\infty$, for any $v\in W^{1,p}(\tau)$, there exists $C$ depending
on $p$ and $\sigma_{0}$ such that

(4.12) $\left\|\,v\,\right\|_{L^{p}(\tau_{\delta})}\leq
C(\delta/h_{\tau})^{1/p}\left\|\,v\,\right\|_{W^{1,p}(\tau)}.$

This inequality has appeared in many occurrences, and we give a proof for the
readers’ convenience.

###### Proof.

For any $0<s<\delta$, we let $\tau_{s}^{c}=\tau\backslash\tau_{s}$. It is
clear that $\tau_{s}^{c}$ is also a simplex. For any face $f$ of
$\tau_{s}^{c}$, we define a vector

$m(x)=\dfrac{\left\lvert
f\right\rvert}{d\left\lvert\tau_{s}^{c}\right\rvert}(x-a_{f}),$

where $a_{f}$ is the vertex opposite to $f$. A direct calculation gives that
$m(x)\cdot n_{f}=1$ for any $x\in f$, while $m(x)\cdot n_{f}$ vanishes on the
remaining faces of $\tau_{s}^{c}$, where $n_{f}$ is the outward normal of $f$.
Using the divergence theorem, we obtain

$\displaystyle\int_{f}\left\lvert v(x)\right\rvert^{p}\mathrm{d}\sigma(x)$
$\displaystyle=\int_{f}\left\lvert v(x)\right\rvert^{p}m(x)\cdot
n_{f}\mathrm{d}\sigma(x)=\int_{\tau_{s}^{c}}\operatorname{div}\left(\left\lvert
v(x)\right\rvert^{p}m(x)\right)\,\mathrm{d}x$
$\displaystyle=\int_{\tau_{s}^{c}}\left(\,\left(\,m(x)\cdot\nabla\,\right)\left\lvert
v(x)\right\rvert^{p}+\left\lvert
v(x)\right\rvert^{p}\operatorname{div}m(x)\,\right)\,\mathrm{d}x.$

A direct calculation gives

$\max_{x\in\tau_{s}^{c}}\left\lvert
m(x)\right\rvert\leq\sigma_{0}\qquad\operatorname{div}m(x)=\dfrac{\left\lvert
f\right\rvert}{d\left\lvert\tau_{s}^{c}\right\rvert}\leq\dfrac{\sigma_{0}}{h_{\tau}}.$

A combination of the above two inequalities leads to

$\displaystyle\int_{f}\left\lvert v(x)\right\rvert^{p}\mathrm{d}\sigma(x)$
$\displaystyle\leq\sigma_{0}\left(\,\dfrac{1}{h_{\tau}}\int_{\tau_{s}^{c}}\left\lvert
v(x)\right\rvert^{p}\,\mathrm{d}x+p\int_{\tau_{s}^{c}}\left\lvert
v(x)\right\rvert^{p-1}\left\lvert\nabla
v(x)\right\rvert\,\mathrm{d}x\,\right)$
$\displaystyle\leq\dfrac{\sigma_{0}}{h_{\tau}}\left(\,\int_{\tau}\left\lvert
v(x)\right\rvert^{p}\,\mathrm{d}x+ph_{\tau}\int_{\tau}\left\lvert
v(x)\right\rvert^{p-1}\left\lvert\nabla
v(x)\right\rvert\,\mathrm{d}x\,\right).$

Summing up all faces $f\in\partial\tau_{s}^{c}$, we obtain

$\int_{\partial\tau_{s}^{c}}\left\lvert
v(x)\right\rvert^{p}\mathrm{d}\sigma(x)\leq\dfrac{d\sigma_{0}}{h_{\tau}}\left(\,\int_{\tau}\left\lvert
v(x)\right\rvert^{p}\,\mathrm{d}x+ph_{\tau}\int_{\tau}\left\lvert
v(x)\right\rvert^{p-1}\left\lvert\nabla
v(x)\right\rvert\,\mathrm{d}x\,\right).$

Integration with respect to $s$ from $0$ to $\delta$, we obtain

$\int_{\tau_{\delta}}\left\lvert
v(x)\right\rvert^{p}\mathrm{d}\sigma(x)\leq\dfrac{d\sigma_{0}\delta}{h_{\tau}}\left(\,\int_{\tau}\left\lvert
v(x)\right\rvert^{p}\,\mathrm{d}x+ph_{\tau}\int_{\tau}\left\lvert
v(x)\right\rvert^{p-1}\left\lvert\nabla
v(x)\right\rvert\,\mathrm{d}x\,\right).$

Using Hölder’s inequality, we obtain

$\left\|\,v\,\right\|_{L^{p}(\tau_{\delta})}\leq\left(\,\delta/h_{\tau}\,\right)^{1/p}(d\sigma_{0})^{1/p}\left(\,\left\|\,v\,\right\|_{L^{p}(\tau)}+(ph_{\tau})^{1/p}\left\|\,v\,\right\|_{L^{p}(\tau)}^{1-1/p}\left\|\,\nabla
v\,\right\|_{L^{p}(\tau)}^{1/p}\,\right).$

This gives (4.12) for $p>1$.

The proof for $p=1$ is the same, we omit the details. ∎

####  4.1.1. Approximation error

We firstly define an MsFEM interpolant on each element
$\tau\in\mathcal{T}_{h}$ as

(4.13)
$\widetilde{u}(x)|_{\tau}{:}=\sum_{i=1}^{d+1}u_{0}(x_{i})\cdot\phi_{i}(x),$

which may be written as
$\widetilde{u}=\sum_{i=1}^{d+1}\sum_{\beta=1}^{m}u_{0}^{\beta}(x_{i})c_{ik}^{\beta}\psi_{k}^{\beta}(x).$
It is well-defined over $S$, and

$\mathcal{L}_{\varepsilon}(\widetilde{u})=0\quad\text{in\quad}S\qquad\text{and}\quad\widetilde{u}=\widetilde{u}_{0}\quad\text{on\quad}\partial
S,$

where
$\widetilde{u}_{0}=\sum_{i=1}^{d+1}\sum_{\beta=1}^{m}u_{0}^{\beta}(x_{i})c_{ik}^{\beta}Q_{k}^{\beta}(x)$.
It is clear that the homogenization limit of $\widetilde{u}$ is
$\widetilde{u}_{0}$. By definition, $\widetilde{u}_{0}|_{\tau}=\pi u_{0}$ with
$\pi u_{0}$ the linear Lagrange interpolant of $u_{0}$ over $\tau$. The first
order approximation of $\widetilde{u}$ is defined as

$\widetilde{u}_{1}^{\varepsilon}{:}=\widetilde{u}_{0}+\varepsilon(\chi\cdot\nabla)\widetilde{u}_{0}\quad\text{and\quad}\widetilde{u}_{1}^{\varepsilon}|_{\tau}=\pi
u_{0}+\varepsilon(\chi\cdot\nabla)\pi u_{0}.$

The approximation error of the MsFEM interpolant is given by

######  Lemma 4.5.

Under the same assumptions in Theorem 4.1, if $m=1$ or $d=2$, then

(4.14) $\left\|\,u^{\varepsilon}-\widetilde{u}\,\right\|_{h}\leq
C\left(\,(\sqrt{\varepsilon}+h)\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\Omega)}+\dfrac{\varepsilon}{h}\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\Omega)}\,\right),$

where $C$ depends on
$\lambda,\Lambda,\left\|\,\chi\,\right\|_{L^{\infty}},\Omega$ and the mesh
parameters $\sigma_{0},\sigma_{1},\gamma_{1},\gamma_{2}$.

Furthermore, if $m\geq 2$ and $d=3$, then

(4.15) $\left\|\,u^{\varepsilon}-\widetilde{u}\,\right\|_{h}\leq
C\left(\,(\sqrt{\varepsilon}+h)\left\|\,\nabla
u_{0}\,\right\|_{W^{1,3}(\Omega)}+\dfrac{\varepsilon}{h}\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\Omega)}\,\right),$

where $C$ depends on $\lambda,\Lambda,\Omega$ and the mesh parameters
$\sigma_{0},\sigma_{1},\gamma_{1},\gamma_{2}$.

######  Remark 4.6.

The interpolation estimate (4.15) is new, while (4.14) with $m=1$ and $d=2$
was proved in [16] by assuming that $\nabla\chi$ is bounded. The proof therein
does not apply to elliptic system because the maximum principle used in the
proof may fail for elliptic system [26]. We shall use the local multiplier
estimates in Lemma 4.2 to remove the boundedness assumption on $\nabla\chi$.

###### Proof.

Using the triangle inequality, we have

(4.16) $\displaystyle\left\|\,u^{\varepsilon}-\widetilde{u}\,\right\|_{h}$
$\displaystyle\leq\left\|\,u^{\varepsilon}-u_{1}^{\varepsilon}\,\right\|_{h}+\left\|\,\widetilde{u}-\widetilde{u}_{1}^{\varepsilon}\,\right\|_{h}+\left\|\,u_{1}^{\varepsilon}-\widetilde{u}_{1}^{\varepsilon}\,\right\|_{h}$
$\displaystyle=\left\|\,\nabla(u^{\varepsilon}-u_{1}^{\varepsilon})\,\right\|_{L^{2}(\Omega)}+\left\|\,\widetilde{u}-\widetilde{u}_{1}^{\varepsilon}\,\right\|_{h}+\left\|\,u_{1}^{\varepsilon}-\widetilde{u}_{1}^{\varepsilon}\,\right\|_{h}.$

Applying Lemma 4.3 to $\widetilde{u}$, using (4.10) and Assumption A, we
obtain

$\displaystyle\left\|\,\nabla(\widetilde{u}-\widetilde{u}_{1}^{\varepsilon})\,\right\|_{L^{2}(\tau)}$
$\displaystyle\leq
C\dfrac{\varepsilon}{h_{\tau}}\left\|\,\nabla\widetilde{u}_{0}\,\right\|_{L^{2}(S)}=C\dfrac{\varepsilon}{h_{\tau}}\left\lvert
S\right\rvert^{1/2}\left\lvert\nabla\widetilde{u}_{0}\right\rvert$
$\displaystyle=C\dfrac{\varepsilon}{h_{\tau}}\left\lvert
S\right\rvert^{1/2}\left\lvert\nabla\pi
u_{0}\right\rvert=C\dfrac{\varepsilon}{h_{\tau}}\dfrac{\left\lvert
S\right\rvert^{1/2}}{\left\lvert\tau\right\rvert^{1/2}}\left\|\,\nabla\pi
u_{0}\,\right\|_{L^{2}(\tau)}$ $\displaystyle\leq
C\dfrac{\varepsilon}{h_{\tau}}\left\|\,\nabla\pi
u_{0}\,\right\|_{L^{2}(\tau)}.$

Summing up all $\tau\in\mathcal{T}_{h}$, using the shape-regular and inverse
assumption of $\mathcal{T}_{h}$, we obtain that there exists $C$ depending on
$\sigma_{0}$ and $\sigma_{1}$ such that

$\displaystyle\left\|\,\widetilde{u}-\widetilde{u}_{1}^{\varepsilon}\,\right\|_{h}$
$\displaystyle\leq C\dfrac{\varepsilon}{h}\left\|\,\nabla\pi
u_{0}\,\right\|_{L^{2}(\Omega)}\leq
C\dfrac{\varepsilon}{h}\left(\,\left\|\,\nabla(u_{0}-\pi
u_{0})\,\right\|_{L^{2}(\Omega)}+\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\Omega)}\,\right)$ (4.17) $\displaystyle\leq
C\left(\,\varepsilon\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{2}(\Omega)}+\dfrac{\varepsilon}{h}\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\Omega)}\,\right).$

On each element
$\tau,u_{1}^{\varepsilon}-\widetilde{u}_{1}^{\varepsilon}=u_{0}-\pi
u_{0}+\varepsilon\chi(x/\varepsilon)\nabla(u_{0}-\pi u_{0})$ and

$\nabla(u_{1}^{\varepsilon}-\widetilde{u}_{1}^{\varepsilon})=\nabla(u_{0}-\pi
u_{0})+\varepsilon\nabla\chi(x/\varepsilon)\nabla(u_{0}-\pi
u_{0})+\varepsilon\chi(x/\varepsilon)\nabla^{2}(u_{0}-\pi u_{0}).$

For $m=1$ or $d=2$, $\chi$ is bounded, using the local multiplier inequality
(4.5), we obtain

$\displaystyle\varepsilon\left\|\,\nabla\chi(x/\varepsilon)\nabla(u_{0}-\pi
u_{0})\,\right\|_{L^{2}(\tau)}$ $\displaystyle\leq
C\left(\,\left\|\,\nabla(u_{0}-\pi
u_{0})\,\right\|_{L^{2}(\tau)}+\varepsilon\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{2}(\tau)}\,\right)$
$\displaystyle\leq
C(\varepsilon+h_{\tau})\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{2}(\tau)}.$

It follows from the above two equations that

$\displaystyle\left\|\,\nabla(u_{1}^{\varepsilon}-\widetilde{u}_{1}^{\varepsilon})\,\right\|_{L^{2}(\tau)}$
$\displaystyle\leq\left\|\,\nabla(u_{0}-\pi
u_{0})\,\right\|_{L^{2}(\tau)}+\varepsilon\left\|\,\nabla\chi(x/\varepsilon)\nabla(u_{0}-\pi
u_{0})\,\right\|_{L^{2}(\tau)}$
$\displaystyle\quad+\varepsilon\left\|\,\chi(x/\varepsilon)\nabla^{2}u_{0}\,\right\|_{L^{2}(\tau)}$
$\displaystyle\leq
C\left(\,1+\left\|\,\chi\,\right\|_{L^{\infty}(Y)}\,\right)(\varepsilon+h_{\tau})\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{2}(\tau)}.$

Summing up all $\tau\in\mathcal{T}_{h}$, we get

(4.18)
$\left\|\,u_{1}^{\varepsilon}-\widetilde{u}_{1}^{\varepsilon}\,\right\|_{h}\leq
C(\varepsilon+h)\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{2}(\Omega)}.$

Substituting the above inequality, (3.8) and (4.1.1) into (4.16), we obtain
(4.14).

For $m\geq 2$ and $d=3$, by (3.4), we have $\chi\in L^{6}(Y)$. Using the local
multiplier estimate (4.4) and the standard interpolation estimate for $\pi
u_{0}$, we obtain

$\displaystyle\varepsilon\left\|\,\nabla\chi(x/\varepsilon)\nabla(u_{0}-\pi
u_{0})\,\right\|_{L^{2}(\tau)}$ $\displaystyle\leq
C\left\lvert\tau\right\rvert^{1/6}\left(\,\left\|\,\nabla(u_{0}-\pi
u_{0})\,\right\|_{L^{3}(\tau)}+\varepsilon\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{3}(\tau)}\,\right)$
$\displaystyle\leq
C(\varepsilon+h_{\tau})\left\lvert\tau\right\rvert^{1/6}\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{3}(\tau)}.$

Using Hölder’s inequality and (3.5) with $p=6$ we obtain

$\varepsilon\left\|\,\chi(x/\varepsilon)\nabla^{2}u_{0}\,\right\|_{L^{2}(\tau)}\leq\varepsilon\left\|\,\chi(x/\varepsilon)\,\right\|_{L^{6}(\tau)}\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{3}(\tau)}\leq
C\varepsilon\left\lvert\tau\right\rvert^{1/6}\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{3}(\tau)}.$

Proceeding along the same line that leads to (4.18), and using Hölder’s
inequality, we have

$\left\|\,u_{1}^{\varepsilon}-\widetilde{u}_{1}^{\varepsilon}\,\right\|_{h}\leq
C(\varepsilon+h)\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{3}(\Omega)}.$

Substituting the above inequality, (3.7) and (4.1.1) into (4.16), we obtain
(4.15). ∎

####  4.1.2. Consistency error

To bound the consistency error, we need a local estimate of $\nabla
u_{1}^{\varepsilon}$, which helps us to remove the extra smoothness assumption
on $\chi$.

######  Lemma 4.7.

There exists $C$ independent of $h_{\tau}$ such that

(4.19) $\left\|\,\nabla
u_{1}^{\varepsilon}\,\right\|_{L^{2}(\tau_{\delta})}\leq
C\left(\,\varepsilon+\sqrt{\delta/h_{\tau}}\,\right)\left\lvert\tau\right\rvert^{1/2-1/d}\left\|\,\nabla
u_{0}\,\right\|_{W^{1,d}(\tau)}.$

If $\chi$ is bounded, then

(4.20) $\left\|\,\nabla
u_{1}^{\varepsilon}\,\right\|_{L^{2}(\tau_{\delta})}\leq
C\left(\,\varepsilon+\sqrt{\delta/h_{\tau}}\,\right)\left(\,1+\left\|\,\chi\,\right\|_{L^{\infty}(Y)}\,\right)\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\tau)}.$

###### Proof.

Since $\tau$ is a simplex, we may decompose $\tau_{\delta}$ into $d+1$
disjoint convex domains $\\{\tau_{\delta}^{i}\\}_{i=1}^{d+1}$. Over each
$\tau_{\delta}^{i}$, using the local multiplier estimate (4.4), we obtain

$\varepsilon\left\|\,\nabla\chi(x/\varepsilon)\nabla
u_{0}\,\right\|_{L^{2}(\tau_{\delta}^{i})}\leq
C\left\lvert\tau_{\delta}^{i}\right\rvert^{1/2-1/d}\left(\,\left\|\,\nabla
u_{0}\,\right\|_{L^{d}(\tau_{\delta}^{i})}+\varepsilon\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{d}(\tau_{\delta}^{i})}\,\right).$

Summing up the above estimate for $i=1,\ldots,d+1$ and using the scaled trace
inequality (4.12) with $p=d$, we obtain

$\displaystyle\varepsilon\left\|\,\nabla\chi(x/\varepsilon)\nabla
u_{0}\,\right\|_{L^{2}(\tau_{\delta})}$ $\displaystyle\leq
C\left\lvert\tau_{\delta}\right\rvert^{1/2-1/d}\left(\,\left\|\,\nabla
u_{0}\,\right\|_{L^{d}(\tau_{\delta})}+\varepsilon\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{d}(\tau_{\delta})}\,\right)$
$\displaystyle\leq
C\left\lvert\tau_{\delta}\right\rvert^{1/2-1/d}(\delta/h_{\tau})^{1/d}\left\|\,\nabla
u_{0}\,\right\|_{W^{1,d}(\tau)}+C\varepsilon\left\lvert\tau\right\rvert^{1/2-1/d}\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{d}(\tau)}$
$\displaystyle\leq
C(\varepsilon+\sqrt{\delta/h_{\tau}})\left\lvert\tau\right\rvert^{1/2-1/d}\left\|\,\nabla
u_{0}\,\right\|_{W^{1,d}(\tau)}.$

Invoking (4.12) with $p=2$ and using Hölder’s inequality, we obtain

$\left\|\,\nabla u_{0}\,\right\|_{L^{2}(\tau_{\delta})}\leq
C\sqrt{\delta/h_{\tau}}\left\|\,\nabla u_{0}\,\right\|_{H^{1}(\tau)}\leq
C\sqrt{\delta/h_{\tau}}\left\lvert\tau\right\rvert^{1/2-1/d}\left\|\,\nabla
u_{0}\,\right\|_{W^{1,d}(\tau)}.$

Using Hölder’s inequality with $1/q=1/2-1/d$ and (3.5) with $p=q$, we obtain

$\displaystyle\varepsilon\left\|\,\chi(x/\varepsilon)\nabla^{2}u_{0}\,\right\|_{L^{2}(\tau_{\delta})}$
$\displaystyle\leq\varepsilon\left\|\,\chi(x/\varepsilon)\nabla^{2}u_{0}\,\right\|_{L^{2}(\tau)}\leq\varepsilon\left\|\,\chi(x/\varepsilon)\,\right\|_{L^{q}(\tau)}\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{d}(\tau)}$
$\displaystyle\leq
C\varepsilon\left\lvert\tau\right\rvert^{1/2-1/d}\left\|\,\chi\,\right\|_{L^{q}(Y)}\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{d}(\tau)}.$

A combination of the above three inequalities leads to (4.19).

If $\chi$ is bounded, then we sum up the local multiplier estimate (4.5) over
$\tau_{\delta}^{i}$ for $i=1,\ldots,d+1$ and obtain

$\varepsilon\left\|\,\nabla\chi(x/\varepsilon)\nabla
u_{0}\,\right\|_{L^{2}(\tau_{\delta})}\leq
C(1+\left\|\,\chi\,\right\|_{L^{\infty}(Y)})\left(\,\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\tau_{\delta})}+\varepsilon\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{2}(\tau_{\delta})}\,\right).$

Invoking the inequality (4.12), we obtain

$\displaystyle\left\|\,\nabla
u_{1}^{\varepsilon}\,\right\|_{L^{2}(\tau_{\delta})}$
$\displaystyle\leq\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\tau_{\delta})}+{\varepsilon}\left\|\,\nabla\chi(x/\varepsilon)\nabla
u_{0}\,\right\|_{L^{2}(\tau_{\delta})}+\varepsilon\left\|\,\chi\nabla^{2}u_{0}\,\right\|_{L^{2}(\tau_{\delta})}$
$\displaystyle\leq
C(1+\left\|\,\chi\,\right\|_{L^{\infty}(Y)})\left(\,\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\tau_{\delta})}+\varepsilon\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{2}(\tau)}\,\right)$
$\displaystyle\leq
C\left(\,\varepsilon+\sqrt{\delta/h_{\tau}}\,\right)(1+\left\|\,\chi\,\right\|_{L^{\infty}(Y)})\left\|\,\nabla^{2}u_{0}\,\right\|_{L^{2}(\tau)}.$

This gives (4.20) and finishes the proof. ∎

The next lemma concerns the estimate of the consistency error.

######  Lemma 4.8.

Under the same assumptions in Theorem 4.1, for $m=1$ or $d=2$, there holds

(4.21) $\sup_{w\in V_{h}^{0}}\dfrac{\left\lvert\left\langle f,w\right\rangle-
a_{h}(u^{\varepsilon},w)\right\rvert}{\left\|\,w\,\right\|_{h}}\leq
C\left(\,\varepsilon+\varepsilon/h\,\right)\left(\,\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\Omega)}+\left\|\,f\,\right\|_{L^{2}(\Omega)}\,\right).$

where $C$ depends on
$\lambda,\Lambda,\left\|\,\chi\,\right\|_{L^{\infty}},\Omega$ and the mesh
parameters $\sigma_{0},\sigma_{1},\gamma_{1},\gamma_{2}$.

For $m\geq 2$ and $d=3$, there holds

(4.22) $\sup_{w\in V_{h}^{0}}\dfrac{\left\lvert\left\langle f,w\right\rangle-
a_{h}(u^{\varepsilon},w)\right\rvert}{\left\|\,w\,\right\|_{h}}\leq
C\left(\,\varepsilon+\varepsilon/h\,\right)\left(\,\left\|\,\nabla
u_{0}\,\right\|_{W^{1,3}(\Omega)}+\left\|\,f\,\right\|_{L^{2}(\Omega)}\,\right),$

where $C$ depends on $\lambda,\Lambda,\Omega$ and the mesh parameters
$\sigma_{0},\sigma_{1},\gamma_{1},\gamma_{2}$.

###### Proof.

For $w\in V_{h}^{0}$, over each over-sampling domain $S$, let $w_{0}$ be its
homogenized part over $S$. By $w_{0}\in H_{0}^{1}(\Omega;\mathbb{R}^{m})$,
there holds

$a_{h}(u^{\varepsilon},w_{0})=\left\langle f,w_{0}\right\rangle.$

Therefore, we write the consistency error functional as

$\displaystyle\left\langle f,w\right\rangle-a_{h}(u^{\varepsilon},w)$
$\displaystyle=\left\langle f,w-w_{0}\right\rangle-
a_{h}(u^{\varepsilon},w-w_{0})$ $\displaystyle=\left\langle
f,w-w_{0}\right\rangle-
a_{h}(u^{\varepsilon},w-w_{1}^{\varepsilon})-a_{h}(u^{\varepsilon},w_{1}^{\varepsilon}-w_{0}).$

Using Lemma 4.3, (4.9), (4.8) and Assumption A, we obtain

$\displaystyle\left\|\,w-w_{0}\,\right\|_{L^{2}(\tau)}$
$\displaystyle\leq\left\|\,w-w_{0}\,\right\|_{L^{2}(S)}\leq
C\varepsilon\left\|\,\nabla w_{0}\,\right\|_{L^{2}(S)}$ $\displaystyle\leq
C\varepsilon\left\|\,\nabla w_{0}\,\right\|_{L^{2}(\tau)}\leq
C\varepsilon\left\|\,\nabla w\,\right\|_{L^{2}(\tau)},$

which immediately implies

(4.23) $\left\lvert\left\langle f,w-w_{0}\right\rangle\right\rvert\leq
C\varepsilon\left\|\,f\,\right\|_{L^{2}(\Omega)}\left\|\,w\,\right\|_{h}.$

Using (4.10), (4.8) again, and the inverse assumption of $\mathcal{T}_{h}$, we
obtain

$\displaystyle\left\lvert
a_{h}(u^{\varepsilon},w-w_{1}^{\varepsilon})\right\rvert$
$\displaystyle\leq\Lambda\sum_{\tau\in\mathcal{T}_{h}}\left\|\,\nabla
u^{\varepsilon}\,\right\|_{L^{2}(\tau)}\left\|\,\nabla(w-w_{1}^{\varepsilon})\,\right\|_{L^{2}(\tau)}$
$\displaystyle\leq
C\sum_{\tau\in\mathcal{T}_{h}}\dfrac{\varepsilon}{h_{\tau}}\left\|\,\nabla
u^{\varepsilon}\,\right\|_{L^{2}(\tau)}\left\|\,\nabla
w_{0}\,\right\|_{L^{2}(\tau)}$ $\displaystyle\leq
C\dfrac{\varepsilon}{h}\sum_{\tau\in\mathcal{T}_{h}}\left\|\,\nabla
u^{\varepsilon}\,\right\|_{L^{2}(\tau)}\left\|\,\nabla
w\,\right\|_{L^{2}(\tau)}$ $\displaystyle\leq
C\dfrac{\varepsilon}{h}\left\|\,\nabla
u^{\varepsilon}\,\right\|_{L^{2}(\Omega)}\left\|\,w\,\right\|_{h}.$

Combining the above two estimates, we obtain

(4.24) $\left\lvert\left\langle f,w-w_{0}\right\rangle-
a_{h}(u^{\varepsilon},w-w_{1}^{\varepsilon})\right\rvert\leq
C\left(\,\varepsilon+\varepsilon/h\,\right)\left\|\,f\,\right\|_{L^{2}(\Omega)}\left\|\,w\,\right\|_{h},$

where we have used the a-priori estimate $\left\|\,\nabla
u^{\varepsilon}\,\right\|_{L^{2}(\Omega)}\leq
C\left\|\,f\,\right\|_{L^{2}(\Omega)}$.

It remains to bound $a_{h}(u^{\varepsilon},w_{1}^{\varepsilon}-w_{0})$. On
each element $\tau$, we introduce a cut-off function $\rho_{\varepsilon}\in
C_{0}^{\infty}(\tau)$ such that $0\leq\rho_{\varepsilon}\leq 1$ and
$\left\lvert\nabla\rho_{\varepsilon}\right\rvert\leq C/\varepsilon$, moreover,

$\rho_{\varepsilon}=\left\\{\begin{aligned}
1\quad&\text{dist}(x,\partial\tau)\geq 2\varepsilon,\\\
0\quad&\text{dist}(x,\partial\tau)\leq\varepsilon.\end{aligned}\right.$

Denote
$\widehat{w}^{\varepsilon}=(w_{1}^{\varepsilon}-w_{0})(1-\rho_{\varepsilon})$,
which is the oscillatory part of $w_{1}^{\varepsilon}$ supported inside the
strip $\tau_{\varepsilon}$. We write

$\displaystyle a_{\tau}(u^{\varepsilon},w_{1}^{\varepsilon}-w_{0})=$
$\displaystyle
a_{\tau}(u^{\varepsilon},(w_{1}^{\varepsilon}-w_{0})\rho_{\varepsilon})+a_{\tau}(u^{\varepsilon},\widehat{w}^{\varepsilon})$
$\displaystyle=$ $\displaystyle\left\langle
f,(w_{1}^{\varepsilon}-w_{0})\rho_{\varepsilon}\right\rangle_{\tau}+a_{\tau}(u^{\varepsilon},\widehat{w}^{\varepsilon}).$

Using (3.5) with $p=2$, we obtain

(4.25) $\displaystyle\left\lvert\left\langle
f,(w_{1}^{\varepsilon}-w_{0})\rho_{\varepsilon}\right\rangle_{\tau}\right\rvert$
$\displaystyle\leq\varepsilon\left\|\,f\,\right\|_{L^{2}(\tau)}\left\|\,\chi(x/\varepsilon)\,\right\|_{L^{2}(\tau)}\left\lvert\nabla
w_{0}\right\rvert$ $\displaystyle\leq
C\varepsilon\left\lvert\tau\right\rvert^{1/2}\left\|\,f\,\right\|_{L^{2}(\tau)}\left\|\,\chi\,\right\|_{L^{2}(Y)}\left\lvert\nabla
w_{0}\right\rvert$
$\displaystyle=C\varepsilon\left\|\,f\,\right\|_{L^{2}(\tau)}\left\|\,\chi\,\right\|_{L^{2}(Y)}\left\|\,\nabla
w_{0}\,\right\|_{L^{2}(\tau)}.$

A direct calculation gives111We may also refer to [14, Lemma 3.1] for a proof
of (4.26).

(4.26)
$\left\|\,\nabla\widehat{w}^{\varepsilon}\,\right\|_{L^{2}(\tau_{2\varepsilon})}\leq
C\sqrt{\varepsilon/h_{\tau}}\left\|\,\nabla w_{0}\,\right\|_{L^{2}(\tau)},$

which together with the local estimate (4.19) implies that, for $m\geq 2$ and
$d=3$, there holds

$\displaystyle\left\lvert
a_{\tau}(u^{\varepsilon},\widehat{w}^{\varepsilon})\right\rvert\leq\left\lvert
a_{\tau}(u_{1}^{\varepsilon},\widehat{w}^{\varepsilon})\right\rvert+\left\lvert
a_{\tau}(u^{\varepsilon}-u_{1}^{\varepsilon},\widehat{w}^{\varepsilon})\right\rvert$
$\displaystyle\leq
C\left(\,\left(\,\varepsilon+\dfrac{\varepsilon}{h_{\tau}}\,\right)\left\lvert\tau\right\rvert^{1/6}\left\|\,\nabla
u_{0}\,\right\|_{W^{1,3}(\tau)}+\sqrt{\dfrac{\varepsilon}{h_{\tau}}}\left\|\,\nabla(u^{\varepsilon}-u_{1}^{\varepsilon})\,\right\|_{L^{2}(\tau)}\,\right)\left\|\,\nabla
w_{0}\,\right\|_{L^{2}(\tau)}.$

This estimate together with (4.25) implies

$\displaystyle\left\lvert
a_{\tau}(u^{\varepsilon},w_{1}^{\varepsilon}-w_{0})\right\rvert$
$\displaystyle\leq
C\Bigl{(}\left(\,\varepsilon+\dfrac{\varepsilon}{h_{\tau}}\,\right)\left\lvert\tau\right\rvert^{1/6}\left\|\,\nabla
u_{0}\,\right\|_{W^{1,3}(\tau)}+\sqrt{\dfrac{\varepsilon}{h_{\tau}}}\left\|\,\nabla(u^{\varepsilon}-u_{1}^{\varepsilon})\,\right\|_{L^{2}(\tau)}$
$\displaystyle\qquad\quad+\varepsilon\left\|\,f\,\right\|_{L^{2}(\tau)}\Bigr{)}\left\|\,\nabla
w_{0}\,\right\|_{L^{2}(\tau)}.$

Summing up the above estimates for all $\tau\in\mathcal{T}_{h}$, using (4.8),
(3.8), the inverse assumption of $\mathcal{T}_{h}$ and Hölder’s inequality, we
obtain

$\displaystyle\left\lvert
a_{h}(u^{\varepsilon},w_{1}^{\varepsilon}-w_{0})\right\rvert$
$\displaystyle\leq
C\Bigl{(}\left(\,\varepsilon+\dfrac{\varepsilon}{h}\,\right)\left\|\,\nabla
u_{0}\,\right\|_{W^{1,3}(\Omega)}+\sqrt{\dfrac{\varepsilon}{h}}\left\|\,\nabla(u^{\varepsilon}-u^{\varepsilon}_{1})\,\right\|_{L^{2}(\Omega)}$
$\displaystyle\qquad\quad+\varepsilon\left\|\,f\,\right\|_{L^{2}(\Omega)}\Bigr{)}\left\|\,w\,\right\|_{h}$
$\displaystyle\leq
C\left(\,\varepsilon+\dfrac{\varepsilon}{h}\,\right)\left(\,\left\|\,\nabla
u_{0}\,\right\|_{W^{1,3}(\Omega)}+\left\|\,f\,\right\|_{L^{2}(\Omega)}\,\right)\left\|\,w\,\right\|_{h}.$

This inequality together with (4.24) implies (4.22).

For $m=1$ or $d=2$, $\chi$ is bounded. Replacing (4.19) by (4.20) and
proceeding along the same line that leads to (4.22), we obtain (4.21).

∎

Proof of Theorem 4.1 A combination of (4.3), Lemma 4.5 and Lemma 4.8 gives
Theorem 4.1.∎

###  4.2. Ld/(d-1) error estimate

We exploit the Aubin-Nitsche trick to obtain the error estimate of MsFEM in
$L^{d/(d-1)}-$norm with $d=2,3$.

######  Theorem 4.9.

Under the same assumption of Theorem 4.1, and suppose that $\varphi\in
H_{0}^{1}(\Omega;\mathbb{R}^{m})$ satisfying

$\int_{\Omega}\nabla\varphi\cdot\widehat{A}\nabla\psi\,\mathrm{d}x=\left\langle
F,\psi\right\rangle\qquad\text{for all\quad}\psi\in
H_{0}^{1}(\Omega;\mathbb{R}^{m})$

For $m=1$ or $d=2$, if the shift estimate

(4.27) $\left\|\,\varphi\,\right\|_{H^{2}(\Omega)}\leq
C\left\|\,F\,\right\|_{L^{2}(\Omega)}$

holds true, then

(4.28) $\left\|\,u-u_{h}\,\right\|_{L^{2}(\Omega)}\leq
C(\varepsilon+h^{2}+\varepsilon/h)\left(\,\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\Omega)}+\left\|\,f\,\right\|_{L^{2}(\Omega)}\,\right).$

For $m\geq 2$ and $d=3$, if the shift estimate

(4.29) $\left\|\,\varphi\,\right\|_{W^{2,3}(\Omega)}\leq
C\left\|\,F\,\right\|_{L^{3}(\Omega)}$

holds true, then

(4.30) $\left\|\,u-u_{h}\,\right\|_{L^{3/2}(\Omega)}\leq
C(\varepsilon+h^{2}+\varepsilon/h)\left(\,\left\|\,\nabla
u_{0}\,\right\|_{W^{1,3}(\Omega)}+\left\|\,f\,\right\|_{L^{3}(\Omega)}\,\right).$

Except the resonance error $\varepsilon/h$, the other two items in the above
error estimates are optimal. For scalar elliptic equation and elliptic system
in two dimension, we obtain the L2 error estimate.

###### Proof.

For any $g\in L^{2}(\Omega;\mathbb{R}^{m})$, we find $v^{\varepsilon}\in
H_{0}^{1}(\Omega;\mathbb{R}^{m})$ such that

(4.31) $\int_{\Omega}\nabla w\cdot(A(x/\varepsilon))^{t}\nabla
v^{\varepsilon}\,\mathrm{d}x=\int_{\Omega}g\cdot w\,\mathrm{d}x\qquad\text{for
all\quad}w\in H_{0}^{1}(\Omega;\mathbb{R}^{m}).$

Let $v_{h}$ be the MsFEM approximation of $v^{\varepsilon}$ defined by

(4.32) $a_{h}(w,v_{h})=\int_{\Omega}g\cdot w\,\mathrm{d}x\qquad\text{for
all\quad}w\in V_{h}^{0}.$

It follows from (4.31) and (4.32) that

$\displaystyle\int_{\Omega}g\cdot(u^{\varepsilon}-u_{h})\,\mathrm{d}x$
$\displaystyle=a(u^{\varepsilon},v^{\varepsilon})-a_{h}(u_{h},v_{h})$
$\displaystyle=a_{h}(u^{\varepsilon}-u_{h},v^{\varepsilon}-v_{h})+a_{h}(u^{\varepsilon}-u_{h},v_{h})+a_{h}(u_{h},v^{\varepsilon}-v_{h})$
$\displaystyle=a_{h}(u^{\varepsilon}-u_{h},v^{\varepsilon}-v_{h})$
$\displaystyle\quad+\bigl{[}a_{h}(u^{\varepsilon},v_{h})-\left\langle
f,v_{h}\right\rangle+a_{h}(u_{h},v^{\varepsilon})-\left\langle
g,u_{h}\right\rangle\bigr{]}.$

For $m=1$ or $d=2$, using the energy error estimate (4.1) and the regularity
assumption (4.27), we obtain

$\displaystyle\left\lvert
a_{h}(u^{\varepsilon}-u_{h},v^{\varepsilon}-v_{h})\right\rvert$
$\displaystyle\leq\Lambda\left\|\,u^{\varepsilon}-u_{h}\,\right\|_{h}\left\|\,v^{\varepsilon}-v_{h}\,\right\|_{h}$
$\displaystyle\leq
C(\varepsilon+h^{2}+\varepsilon^{2}/h^{2})\left(\,\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\Omega)}+\left\|\,f\,\right\|_{L^{2}(\Omega)}\,\right)\left\|\,g\,\right\|_{L^{2}(\Omega)}.$

Using (4.21) and (4.27), we bound the consistency error functional as

$\displaystyle\quad\left\lvert a_{h}(u^{\varepsilon},v_{h})-\left\langle
f,v_{h}\right\rangle+a_{h}(u_{h},v^{\varepsilon})-\left\langle
g,u_{h}\right\rangle\right\rvert$ $\displaystyle\leq
C(\varepsilon+\varepsilon/h)\left(\,\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\Omega)}+\left\|\,f\,\right\|_{L^{2}(\Omega)}\,\right)\left\|\,g\,\right\|_{L^{2}(\Omega)}.$

A combination of the above three estimates yields (4.28).

For $m\geq 2$ and $d=3$, $\chi$ is unbounded. Replacing (4.27), (4.1) and
(4.21) by (4.29), (4.2) and (4.22), respectively, and proceeding along the
same line that leads to (4.28), we obtain (4.30). ∎

###  4.3. Error estimates for MsFEM without oversampling

We visit the error estimates of MsFEM without oversampling [19]. The
multiscale basis function is $\phi^{\beta}=\\{\phi_{i}^{\beta}\\}_{i=1}^{d+1}$
is constructed as (2.4) with $S(\tau)$ replaced by $\tau$.

$V_{h}:=\text{Span}\\{\phi_{i}\quad\text{for all
nodes\;}x_{i}\text{\;of\;}\mathcal{T}_{h}\\},$

and $V_{h}^{0}{:}=\\{\,v\in
V_{h}\,\mid\,v=0\quad\text{on~{}}\partial\Omega\,\\}.$ The approximation
problem reads as: Find $u_{h}\in V_{h}^{0}$ such that

(4.33) $a(u_{h},v)=\left\langle f,v\right\rangle\qquad\text{for all\quad}v\in
V_{h}^{0}.$

Under the same assumptions of Theorem 4.1 except that $A$ is not necessarily
symmetric when $m\geq 2$, we prove the energy error estimate for MsFEM without
oversampling.

######  Theorem 4.10.

Assume that $A$ is $1-$periodic and satisfies (2.1). Let $\Omega$ be a bounded
Lipschitz domain in $\mathbb{R}^{d}$. Let $u^{\varepsilon}$ and $u_{h}$ be the
solutions of (2.3) and (4.33), respectively.

For $m=1$ or $d=2$, if $u_{0}\in H^{2}(\Omega;\mathbb{R}^{m})$, then

(4.34) $\left\|\,\nabla(u^{\varepsilon}-u_{h})\,\right\|_{L^{2}(\Omega)}\leq
C\left(\,(\sqrt{\varepsilon}+h)\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\Omega)}+\sqrt{\varepsilon/h}\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\Omega)}\,\right),$

where $C$ depends on
$\lambda,\Lambda,\left\|\,\chi\,\right\|_{L^{\infty}},\Omega$ and the mesh
parameters $\sigma_{0}$ and $\sigma_{1}$.

For $m\geq 2$ and $d=3$, if $u_{0}\in W^{2,3}(\Omega;\mathbb{R}^{m})$, then

(4.35) $\left\|\,\nabla(u^{\varepsilon}-u_{h})\,\right\|_{L^{2}(\Omega)}\leq
C\left(\,(\sqrt{\varepsilon}+h)\left\|\,\nabla
u_{0}\,\right\|_{W^{1,3}(\Omega)}+\sqrt{\varepsilon/h}\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\Omega)}\,\right),$

where $C$ depends on $\lambda,\Lambda,\Omega$ and the mesh parameters
$\sigma_{0}$ and $\sigma_{1}$.

As a direct consequence of the above theorem, we obtain the Ld/(d-1) error
estimate for MsFEM without oversampling. The proof follows the same line that
leads to Theorem 4.9, we omit the proof.

######  Corollary 4.11.

Under the same assumption of Theorem 4.9 except that $A$ is not necessarily
symmetric for $m\geq 2$. Let $u^{\varepsilon}$ and $u_{h}$ be the solutions of
(2.3) and (4.33), respectively. For $m=1$ or $d=2$, there holds

$\left\|\,u-u_{h}\,\right\|_{L^{2}(\Omega)}\leq
C(\varepsilon+h^{2}+\varepsilon/h)\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\Omega)}.$

For $m\geq 2$ and $d=3$, there holds

$\left\|\,u-u_{h}\,\right\|_{L^{3/2}(\Omega)}\leq
C(\varepsilon+h^{2}+\varepsilon/h)\left\|\,\nabla
u_{0}\,\right\|_{W^{1,3}(\Omega)}.$

The proof of Theorem 4.10 relies on Theorem 3.1 and Lemma 4.2. We only sketch
the main steps because the details are the same with the line leading to
Theorem 4.1. Proof of Theorem 4.10 Using the fact that MsFEM without
oversampling is conforming, i.e., $V_{h}^{0}\subset
H_{0}^{1}(\Omega;\mathbb{R}^{m})$, we obtain the following Cea’s lemma:

(4.36)
$\left\|\,\nabla(u^{\varepsilon}-u_{h})\,\right\|_{L^{2}(\Omega)}\leq(1+\Lambda/\lambda)\inf_{v\in
V_{h}^{0}}\left\|\,\nabla(u^{\varepsilon}-v)\,\right\|_{L^{2}(\Omega)},$

Define MsFEM interpolant $\widetilde{u}(x)$ as (4.13). Using triangle
inequality, we obtain

$\left\|\,\nabla(u^{\varepsilon}-\widetilde{u})\,\right\|_{L^{2}(\Omega)}\leq\left\|\,\nabla(u^{\varepsilon}-u_{1}^{\varepsilon})\,\right\|_{L^{2}(\Omega)}+\left\|\,\nabla(\widetilde{u}-\widetilde{u}_{1}^{\varepsilon})\,\right\|_{L^{2}(\Omega)}+\left\|\,\nabla(u_{1}^{\varepsilon}-\widetilde{u}_{1}^{\varepsilon})\,\right\|_{L^{2}(\Omega)}.$

The estimate of
$\left\|\,\nabla(u^{\varepsilon}-u_{1}^{\varepsilon})\,\right\|_{L^{2}(\Omega)}$
follows from Theorem 3.1, and the estimate of
$\left\|\,\nabla(u_{1}^{\varepsilon}-\widetilde{u}_{1}^{\varepsilon})\,\right\|_{L^{2}(\Omega)}\leq$
is the same with the corresponding terms in Lemma 4.5. Note that
$\widetilde{u}_{1}^{\varepsilon}$ is the first order approximation of
$\widetilde{u}$ over $\tau$. For $m=1$ or $d=2$, using (3.8), we get

$\displaystyle\left\|\,\nabla(\widetilde{u}-\widetilde{u}_{1}^{\varepsilon})\,\right\|_{L^{2}(\tau)}$
$\displaystyle\leq C\sqrt{\varepsilon/h_{\tau}}\left\|\,\nabla\pi
u_{0}\,\right\|_{L^{2}(\tau)}$ $\displaystyle\leq
C\left(\,\sqrt{\varepsilon/h_{\tau}}\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\tau)}+\sqrt{\varepsilon h_{\tau}}\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\tau)}\,\right).$

Summing up the above estimate for all $\tau\in\mathcal{T}_{h}$, and using the
inverse assumption of $\mathcal{T}_{h}$, we obtain

(4.37)
$\left\|\,\nabla(\widetilde{u}-\widetilde{u}_{1}^{\varepsilon})\,\right\|_{L^{2}(\Omega)}\leq
C\left(\,\sqrt{\varepsilon/h}\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\Omega)}+\sqrt{\varepsilon h}\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\Omega)}\,\right).$

For $m\geq 2$ and $d=3$, using (3.7) and the fact that $\nabla\pi u_{0}$ is a
piecewise constant vector over $\tau$, we get

$\left\|\,\nabla(\widetilde{u}-\widetilde{u}_{1}^{\varepsilon})\,\right\|_{L^{2}(\tau)}\leq
C\sqrt{\varepsilon/h_{\tau}}\left\lvert\tau\right\rvert^{1/6}\left\|\,\nabla\pi
u_{0}\,\right\|_{L^{3}(\tau)}=C\sqrt{\varepsilon/h_{\tau}}\left\|\,\nabla\pi
u_{0}\,\right\|_{L^{2}(\tau)}.$

Proceeding along the same line that leads to (4.37), we obtain

$\left\|\,\nabla(\widetilde{u}-\widetilde{u}_{1}^{\varepsilon})\,\right\|_{L^{2}(\Omega)}\leq
C\left(\,\sqrt{\varepsilon/h}\left\|\,\nabla
u_{0}\,\right\|_{L^{2}(\Omega)}+\sqrt{\varepsilon h}\left\|\,\nabla
u_{0}\,\right\|_{H^{1}(\Omega)}\,\right).$

A combination of all the above estimates completes the proof.∎

##  5\. Conclusion

Under suitable regularity assumptions on the homogenized solution, we proved
the optimal energy error estimates for MsFEM with or without oversampling
applying to elliptic system with bounded and measurable periodic coefficients.
The present work may be extended to elliptic system with locally periodic
coefficients, i.e., $A^{\varepsilon}=A(x,x/\varepsilon)$ with the aid of a new
local multiplier estimate. The extension to elliptic system for the
coefficients with stratified structure is also very interesting. We believe
the machineries developed in the present work may be useful to analyze other
version MsFEM such as the mixed MsFEM [8], Crouzeix-Raviart MsFEM [23], or
other type of over-sampling technique [16]. We shall leave these for further
pursuit.

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