# Superconvergence Analysis of a Multiscale Finite Element Method for Elliptic Problems with Rapidly Oscillating Coefficients

Authors: Xiaofei Guan; Xiaoling Wang; Cheng Wang; Xian Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A new multiscale finite element method is presented for solving the elliptic equations with rapidly oscillating coefficients. The proposed method is based on asymptotic analysis and  careful numerical treatments for the boundary corrector terms by virtue of the recovery technique. Under the assumption that the oscillating coefficient is periodic, some superconvergence results are derived, which seem to be never discovered in the previous literature. Finally, some numerical experiments are carried out to demonstrate the efficiency and accuracy of this method, and it is seen that they agree very well with the analytical result.


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## Body

## 1. Introduction

In this paper, we consider the following elliptic boundary value problem with rapidly oscillatory coefficients:(1)Lεuε≡∂∂xi(aij(xε)∂uε∂xj)=f(x),inΩ,uε=g(x),on∂Ω,
where Ω⊂ℜ2 is a smooth-bounded domain, aij(ξ):ℜ2→ℜ is symmetric and satisfies
(2)(1)λ|ξ|2≤aij(ξ)ξiξj≤λ-1|ξ|2,∀ξ∈ℜ2,∃λ∈(0,1],(2)aij(ξ+ξ′)=aij(ξ),∀ξ∈ℜ2,∃ξ′∈Z2,1≤i,j≤n,(3)∥aij∥H1(ℜ2)≤C,∃C>0,
where ξ=x/ε, ε is a small scale parameter. This kind of equation has widely been applied in many areas, such as the behavior of flow in porous media or the thermal and mechanical behavior of composite material structure. In practice, the oscillatory coefficients may span many scales to a great extent. In such cases, the direct accurate numerical computation of the solution becomes difficult because it would require a very fine mesh, and it can easily exceed the limit of today’s computer resources because of the requirement of tremendous amount of computer memory and CPU time. Meanwhile, it is desirable to have a numerical method that can solve this equation on a large-scale mesh with capturing the effect of small scales details. Thus, various methods of upscaling or homogenization have been developed.Based on the homogenization method, there are many discussions [1–4] about the numerical methods of (1). A large amount of examples and applications can also be found in the classical books [5–8], where the formal asymptotic expansions for the limit solution are deduced when ε is small enough. In these books, the first-order approximation of these expansions is justified by proving sharp error estimates, from which a general method that allowed us to treat some structures with rapidly oscillatory coefficients is also developed. However, the general method cannot effectively compute the boundary corrector on boundary layer. It should be noted that the boundary corrector is the important source of error estimates. In [9], He and Cui present a novel finite element method to solve (1) which can effectively compute the boundary corrector even if the boundary layer is very small. The crucial idea is to combine the numerical approximation of the first-order terms of asymptotic expansions with the numerical approximation of the boundary corrector from different meshes exploiting the need for different levels of resolution. The following result (Theorem  2.13 in [9]) can be obtained.Lemma 1.
Assume thatuε is the solution of (1) and u^h0,h1,h is the finite element solution [9]. For all p,1<p<+∞, there exists a constant C such that
(3)∥∇(uε-u^h0,h1,h)dist(·,∂Ω)+ε∥L2(Ω)≤C[(h1+h0+h+ε)|lnε|1/2+ε(2p-1)/2p]×(∥u0∥W2,∞(Ω)+∥u0∥H3(Ω)),
where u0 is the homogenization solution of (1), and dist(x,∂Ω) is the distance between the point x and the boundary ∂Ω.Unfortunately, the needed CPU time of the method presented in [9] is O(ε1-nh-n). In this paper, a high-effective finite element method to compute boundary corrector by virtue of the recovery technique is proposed, and some superconvergence results for the multiscale finite element approximation of (1) are obtained. The rest of this paper is organized as follows. In the next section, we present a multiscale finite element method to compute uε(x). Its convergence analysis are shown in Section 3. Finally, some numerical results conforming our analytical estimates are given in Section 4.Notation.  Before closing this section, we would like to fix some notations. First, the Einstein summation is used. Let Q={ξ∣0<ξi<1,i=1,2}, and the capital letter C (with or without subscripts) denotes a positive constant, which is independent of the small parameter ε and the mesh size h (with or without subscripts).


## 2. An Improved Multiscale Finite Element Method

Firstly, let us simply recall the homogenization method described in [5].


### 2.1. Homogenization Method

LetNk(ξ)(k=1,2) be a 1-periodic function, which satisfies
(4)∂∂ξi(aij(ξ)∂Nk∂ξj)=-∂∂ξiaik,inℜ2,∫Q‍Nk(ξ)dξ=0.
Then, the matrix a^=(a^ij)2×2 can be obtained by
(5)a^ij=∫Q‍(aij+aik∂Nj∂ξk)dξ.The first-order approximation ofuε(x) can be written as
(6)u~(x)=u0(x)+εNk(ξ)∂u0(x)∂xk,
where u0(x) satisfies the homogenization problem
(7)L0u0(x):=∂∂xi(a^ij∂u0∂xj)=f(x),inΩ,u0(x)=g(x),on∂Ω.The boundary corrector term of the homogenization methodθε is defined by
(8)Lεθε=0,inΩ,θε=-εNk∂u0∂xk,on∂Ω.In the next two subsections, we will compute numerically the first-order approximationu~ and the boundary corrector term θε, respectively, and furthermore give the multiscale finite element solution of (1).


### 2.2. Finite Element Approximation ofu~

Let𝒯h0 be a quasiuniform triangular partition of Q with the mesh size h0. Sh0 denotes the conforming P1 finite element spaces with respect to 𝒯h0, and S0h0=Sh0∩H01(Q). The finite element scheme of (4) is to find Nkh0∈Sh0 such that
(9)∫Q‍aij(ξ)∂Nkh0(ξ)∂ξj∂v(ξ)∂ξidξ=-∫Q‍aik(ξ)∂v(ξ)∂ξidξ,∀v∈S0h0(Q),Nkh0(ξ)isa1-periodicfunction.
Then, the numerical approximation a^ijh0 of a^ij can be calculated by
(10)a^ijh0=∫Q‍(aij(ξ)+aik(ξ)∂Njh0(ξ)∂ξk)dξ.Let𝒯h1 be a quasiuniform triangular partition of Ω with the mesh size h1 and satisfy
(11)minek∈𝒯h1Sek≥Ch12,
where Sek is the area of the triangular element ek. Sgh1 denotes the corresponding conforming P1 finite element spaces, and S0h1=Sh1∩H01(Q).The finite element approximationu0h0,h1 of the homogenization problem (7) is to find u0h0,h1∈Sgh1 such that
(12)∫Ω‍a^ijh0∂u0h0,h1∂xi∂vh∂xjdx=∫Ω‍f(x)vh(x)dx,∀vh∈S0h1.Furthermore, we turn to the computation of∂u0h0,h1(x)/∂xk and u~(x). Let Σh1 be the set of all nodal points of the mesh 𝒯h1. Define ukh0,h1(k=1,2) by the following: (A1)
for all x∈Σh1,  the value of ukh0,h1 at the nodal point x is the average of∂u0h0,h1(x)/∂xk in all elements including x,(A2)
ukh0,h1(x) is a piecewise linear function in every element.Therefore, we have a numerical approximationu~h0,h1(x) of u~(x) which is defined by
(13)u~h0,h1(x)=u0h0,h1(x)+εNkh0(ξ)ukh0,h1(x).


### 2.3. Finite Element Approximation ofθε(x)

Letm be a positive integer satisfying
(14)2m-1ε<maxx∈Ωdist(x,∂Ω)≤2mε.
Then, the domain Ω can be divided by Ω=⋃i=0mΩi and
(15)Ωi={x∈Ω∣dist(x,∂Ω)≤ε},ifi=0,Ωi={x∈Ω∣2i-1ε<dist(x,∂Ω)≤2iε},if1≤i≤m.Let𝒯h be the regular triangular partition of Ω with the mesh size h and satisfy
(16)(B1)2i/2λεh≤Sei≤2i/2λ-1εh,∀ei∈𝒯h,ei⊂Ωi∪Ωi+1,(17)(B2)|li-lj|≤Chli,∀ei∈𝒯h,
where λ and C are independent of i and ei, Sei denotes the area of ei, and li,lj are the length of two edges of ei. Let Σh be the set of all nodal points in 𝒯h, ΣBh=Σh∩∂Ω, and let  Sh be the conforming P1 finite element spaces with respect to 𝒯h; we define
(18)S1h={v∈Sh∣v(x)=-εNkh0(xε)ukh0,h1(x),∀x∈ΣBh},S0h={v∈Sh∣v(x)=0,∀x∈ΣBh}.Then, the finite element approximation ofθε is to find θ~εh0,h1,h∈S1h such that
(19)∫Ω‍aij(ξ)∂θ~εh0,h1,h∂xi∂v∂xjdx=0,∀v∈S0h.


### 2.4. Multiscale Finite Element Approximation ofuε

For anyv∈Sh(Ω), we define the linear operator Rh by
(20)∂Rhv(x)∂xiisapiecewisefunctionon𝒯h,∂Rhv(x)∂xi=∂v(x)∂xi,ifxisthemiddlepointofelementsof𝒯h.
Then, the multiscale finite element approximation u^h0,h1,h(x) of uε(x) can be defined by
(21)u^h0,h1,h(x)=u~h0,h1(x)+Rhθ~εh0,h1,h(x).


## 2.1. Homogenization Method

LetNk(ξ)(k=1,2) be a 1-periodic function, which satisfies
(4)∂∂ξi(aij(ξ)∂Nk∂ξj)=-∂∂ξiaik,inℜ2,∫Q‍Nk(ξ)dξ=0.
Then, the matrix a^=(a^ij)2×2 can be obtained by
(5)a^ij=∫Q‍(aij+aik∂Nj∂ξk)dξ.The first-order approximation ofuε(x) can be written as
(6)u~(x)=u0(x)+εNk(ξ)∂u0(x)∂xk,
where u0(x) satisfies the homogenization problem
(7)L0u0(x):=∂∂xi(a^ij∂u0∂xj)=f(x),inΩ,u0(x)=g(x),on∂Ω.The boundary corrector term of the homogenization methodθε is defined by
(8)Lεθε=0,inΩ,θε=-εNk∂u0∂xk,on∂Ω.In the next two subsections, we will compute numerically the first-order approximationu~ and the boundary corrector term θε, respectively, and furthermore give the multiscale finite element solution of (1).


## 2.2. Finite Element Approximation ofu~

Let𝒯h0 be a quasiuniform triangular partition of Q with the mesh size h0. Sh0 denotes the conforming P1 finite element spaces with respect to 𝒯h0, and S0h0=Sh0∩H01(Q). The finite element scheme of (4) is to find Nkh0∈Sh0 such that
(9)∫Q‍aij(ξ)∂Nkh0(ξ)∂ξj∂v(ξ)∂ξidξ=-∫Q‍aik(ξ)∂v(ξ)∂ξidξ,∀v∈S0h0(Q),Nkh0(ξ)isa1-periodicfunction.
Then, the numerical approximation a^ijh0 of a^ij can be calculated by
(10)a^ijh0=∫Q‍(aij(ξ)+aik(ξ)∂Njh0(ξ)∂ξk)dξ.Let𝒯h1 be a quasiuniform triangular partition of Ω with the mesh size h1 and satisfy
(11)minek∈𝒯h1Sek≥Ch12,
where Sek is the area of the triangular element ek. Sgh1 denotes the corresponding conforming P1 finite element spaces, and S0h1=Sh1∩H01(Q).The finite element approximationu0h0,h1 of the homogenization problem (7) is to find u0h0,h1∈Sgh1 such that
(12)∫Ω‍a^ijh0∂u0h0,h1∂xi∂vh∂xjdx=∫Ω‍f(x)vh(x)dx,∀vh∈S0h1.Furthermore, we turn to the computation of∂u0h0,h1(x)/∂xk and u~(x). Let Σh1 be the set of all nodal points of the mesh 𝒯h1. Define ukh0,h1(k=1,2) by the following: (A1)
for all x∈Σh1,  the value of ukh0,h1 at the nodal point x is the average of∂u0h0,h1(x)/∂xk in all elements including x,(A2)
ukh0,h1(x) is a piecewise linear function in every element.Therefore, we have a numerical approximationu~h0,h1(x) of u~(x) which is defined by
(13)u~h0,h1(x)=u0h0,h1(x)+εNkh0(ξ)ukh0,h1(x).


## 2.3. Finite Element Approximation ofθε(x)

Letm be a positive integer satisfying
(14)2m-1ε<maxx∈Ωdist(x,∂Ω)≤2mε.
Then, the domain Ω can be divided by Ω=⋃i=0mΩi and
(15)Ωi={x∈Ω∣dist(x,∂Ω)≤ε},ifi=0,Ωi={x∈Ω∣2i-1ε<dist(x,∂Ω)≤2iε},if1≤i≤m.Let𝒯h be the regular triangular partition of Ω with the mesh size h and satisfy
(16)(B1)2i/2λεh≤Sei≤2i/2λ-1εh,∀ei∈𝒯h,ei⊂Ωi∪Ωi+1,(17)(B2)|li-lj|≤Chli,∀ei∈𝒯h,
where λ and C are independent of i and ei, Sei denotes the area of ei, and li,lj are the length of two edges of ei. Let Σh be the set of all nodal points in 𝒯h, ΣBh=Σh∩∂Ω, and let  Sh be the conforming P1 finite element spaces with respect to 𝒯h; we define
(18)S1h={v∈Sh∣v(x)=-εNkh0(xε)ukh0,h1(x),∀x∈ΣBh},S0h={v∈Sh∣v(x)=0,∀x∈ΣBh}.Then, the finite element approximation ofθε is to find θ~εh0,h1,h∈S1h such that
(19)∫Ω‍aij(ξ)∂θ~εh0,h1,h∂xi∂v∂xjdx=0,∀v∈S0h.


## 2.4. Multiscale Finite Element Approximation ofuε

For anyv∈Sh(Ω), we define the linear operator Rh by
(20)∂Rhv(x)∂xiisapiecewisefunctionon𝒯h,∂Rhv(x)∂xi=∂v(x)∂xi,ifxisthemiddlepointofelementsof𝒯h.
Then, the multiscale finite element approximation u^h0,h1,h(x) of uε(x) can be defined by
(21)u^h0,h1,h(x)=u~h0,h1(x)+Rhθ~εh0,h1,h(x).


## 3. Superconvergence Result ofu^h0,h1,h

Firstly, we have the following assumption.Assumption C1.  The functions aij∈W1,∞(Q)∩H2(Q), and the homogenization solution u0∈W2,∞(Ω)∩H4(Ω).Then, we introduce the following lemma.Lemma 2 (see [1, 5]).
LetΩr={x∈Ω∣dist(x,∂Ω)≥r}. Assuming that (C1) holds, then there exists C, which is independent of ε and r such that
(22)∥∇θε∥L2(Ωr)≤Cr-1/2ε∥u0∥H2(Ω),∥θε∥H2(Ω)≤Cε-1/2∥u0∥H3(Ω),∥θε∥H1(Ωr)≤Cr-1/2ε∥u0∥W1,∞(∂Ω),∥θε∥H2(Ωr)≤Cr-1/2∥u0∥W1,∞(∂Ω).From Lemma2, one can easily deduce.Lemma 3.
Assuming that (C1) holds, then there existsC, which is independent of ε such that
(23)∥θε∥H3(Ω)≤Cε-3/2∥u0∥H4(Ω),(24)∥θε∥H3(Ωr)≤Cr-1/2ε-1(∥u0∥W2,∞(∂Ω)+∥u0∥H4(Ω)).Proof.
Fork=1,2, we define
(25)vkε(x)=∂θε(x)∂xk.
Then, the upper bound of ∥vkε∥H2(Ω) and ∥vkε∥H1(Ωr) can be estimated, respectively.
Using the result from (8) and ξ=x/ε, we have
(26)Lεvkε=-ε-1∂∂xi(∂aij(ξ)∂ξk∂θε∂xj),x∈Ω,vkε(x)=-ε∂(Nl(ξ)(∂u0(x)/∂xl))∂xk,x∈∂Ω.
Then, we estimate ∥vkε∥H2(Ω). Obviously, vkε can be divided into
(27)vkε=vk,1ε+vk,2ε,
where vk,1ε satisfies
(28)Lεvk,1ε=-ε-1∂∂xi(∂aij(ξ)∂ξk∂θε∂xj),x∈Ω,vk,1ε=0,x∈∂Ω,
and vk,2ε satisfies
(29)Lεvk,2ε=0,x∈Ω,vk,2ε=-ε∂(Nl(x/ε)(∂u0/∂xl))∂xk,x∈∂Ω.
Using the result from Lemma 2 and (28), we have
(30)∥vk,1ε∥H2(Ω)≤Cε-1∥θε∥H2(Ω)≤Cε-3/2∥u0∥H3(Ω).
Letvk,2,k′ε=∂vk,2ε/∂xk′, and using the result from (29), we have
(31)Lεvk,2,k′ε=0,x∈Ω,vk,2,k′ε=-ε∂2(Nl(x/ε)(∂u0/∂xl))∂xk∂xk′,x∈∂Ω.
Following the same line of [5] (1992, Theorem  1.2, pages 124–128), we have
(32)∥vk,2,k′∥H1(Ω)≤Cε-3/2∥u0∥H4(Ω)≤Cε-3/2∥u0∥H4(Ω)
which indicates
(33)∥vk,2ε∥H2(Ω)≤Cε-3/2∥u0∥H4(Ω).
Combining (30) with (33), we can derive (24) immediately.
Considering the proof of (23), vkε can be divided into
(34)vkε=v^k,1ε+v^k,2ε,
where v^k,1ε satisfies
(35)Lεv^k,1ε=-ε-1∂∂xi(∂aij(ξ)∂ξk∂θε∂xj),x∈Ωr,v^k,1ε=0,x∈∂Ωr,
and v^k,2ε satisfies
(36)Lεv^k,2ε=0,x∈Ωr,v^k,2ε=∂θε∂xk,x∈∂Ωr.
In view of Lemma2, we have
(37)∥v^k,1ε∥H2(Ωr)≤Cε-1∥θε∥H2(Ωr)≤Cε-1r-1/2∥u0∥W2,∞(∂Ω).
Following the same line of [5] (1992, Theorem  1.2, pages 124–128), we have
(38)∥v^k,2ε∥H2(Ωr)≤Cε-1r-1/2∥θ∥W1,∞(Ωr/2)+Cr-1/2∥θ∥W2,∞(Ωr/2)≤Cε-1r-1/2r-1ε(∥u0∥W2,∞(∂Ω)+∥u0∥H4(Ω))+Cr-1/2r-1(∥u0∥W2,∞(∂Ω)+∥u0∥H4(Ω))≤Cε-1r-1/2(∥u0∥W2,∞(∂Ω)+∥u0∥H4(Ω)).
Combining (37) with (38), we can conclude the result of this lemma.Assuming that𝒯h is defined as (16), and let θεh and θεI be the linear finite element approximation and the linear interpolation of θε with respect to 𝒯h, respectively. Then, we have the following.Lemma 4.
Assuming that (C1) holds, then there existsC such that
(39)∥∇(θεI-θεh)dist(·,∂Ω)+ε∥L2(Ω)≤Cε-1/2h2(∥u0∥W1,∞(∂Ω)+∥u0∥H4(Ω)).Proof.
Assuming thatm is defined as (14) and Ω=⋃i=k+1mΩi (k<m,k∈ℕ). Considering aij∈W1,∞(Ω) and using the result from Lemma 2, we have
(40)(∥∇(θεI-θεh)∥L2(Ω))2≤C|∫Ω‍aij(xε)∂(θε,rh-θεI)∂xi∂(θεh-θεI)∂xjdx|≤C|∫Ω‍aij(xε)∂(θε-θεI)∂xi∂(θε,rh-θεI)∂xjdx|≤∑i=1k|∫Ωi‍aij(xε)∂(θε-θεI)∂xi∂(θε,rh-θεI)∂xjdx|≤C∑i=1k‍(2i/2εh)2(ε-1∥θε∥H2(Ωi)∥∇(θεI-θε,rh)∥L2(Ωi)≤C∑i=1k‍(2i/2εh)2+∥θε∥H3(Ωi)∥∇(θεI-θεr,h)∥L2(Ωi))≤C∑i=1k‍2iε2h2(ε-1(2iε)-1/2(∥u0∥W1,∞(∂Ω)+∥u0∥H4(Ω))≤C∑i=1k‍2iε2h2×∥∇(θεI-θε,rh)∥L2(Ω-Ωr))≤C2k/2ε1/2h2(∥u0∥W1,∞(∂Ω)+∥u0∥H4(Ω))×∥∇(θεI-θε,rh)∥L2(Ω)≤Ch2(∥u0∥W1,∞(∂Ω)+∥u0∥H4(Ω))×∥∇(θεI-θε,rh)∥L2(Ω-Ωr),
which indicates that
(41)∥∇(θεI-θεh)∥L2(Ω)≤Ch2(∥u0∥W1,∞(∂Ω)+∥u0∥H4(Ω)).
Then  Lemma 4 can be easily derived.Furthermore, we can obtain the following lemma.Lemma 5.
Assuming that (C1) holds, then there existsC such that
(42)∥∇(θεI-θεh)dist(·,∂Ω)+ε∥L2(Ω)≤C|lnε|h2(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω)).Proof.
Assume thatΩrj is defined as Lemma 4. Considering θεh=θε,r0h (∀x∈Ωrj), and we divide θεI-θεh into
(43)θεI-θεh=(θεI-θε,rjh)+∑i=0j-1‍(θε,rih-θε,ri+1h).
Then,
(44)∥θεI-θεh∥H1(Ω-Ωrj)≤∥θεI-θε,rj+1h∥H1(Ω-Ωrj)+∑i=1j-1‍∥θε,riI-θε,ri+1h∥H1(Ω-Ωrj)≤crj1/2h2(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω))+∑i=1j-1‍∥θε,riI-θε,ri+1h∥H1(Ω-Ωrj)≤Crj1/2h2(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω))+Crjjh2(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω))≤Crj1/2h2(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω)).
Using the result from (44), we have
(45)∥∇(θεI-θεh)dist(·,∂Ω)+ε∥L2(Ω)≤C∑j=1m‍∥∇(θεI-θεh)dist(·,∂Ω)+ε∥L2(Ωrj-Ωrj-1)≤∑j=1m‍Crj-1/2∥∇(θεI-θεh)∥L2(Ωrj-Ωrj-1)≤∑j=1m‍Crj-1/2∥∇(θεI-θεh)∥L2(Ω-Ωrj-1)≤∑j=1m‍Crj-1/2rj1/2h2(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω))≤Cmh2(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω))≤C|lnε|h2(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω)).Based on the previous lemmas, the estimate of∥|∇(θεI-θ~εh0,h1,h)|/dist(·,∂Ω)+ε∥L2(Ω) can be given as follows.Lemma 6.
Assuming that (C1) holds, then there existsC such that
(46)∥|∇(θεI-θ~εh0,h1,h)|dist(·,∂Ω)+ε∥L2(Ω)≤C(h0+h1+h2)|lnε|×(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω)).Proof.
Assuming thatθ~εh0,h1 satisfies
(47)Lεθ~εh0,h1(x)=0,x∈Ω,θ~εh0,h1(x)=-εNkh0(xε)ukh0,h1(x),x∈∂Ω,
and θ~εI,h0,h1 is the linear interpolation of θ~εh0,h1 on 𝒯h.
Then, we divideθεI-θ~εh0,h1,h into
(48)θεI-θ~εh0,h1,h=(θεI-θ~εI,h0,h1)+(θ~εI,h0,h1-θ~εh0,h1,h).
Firstly, considering the first item of the right-hand side of (48) and assuming that θεh0,h1(x) satisfies the problem
(49)Lεθεh0,h1(x)=0,x∈Ω,θεh0,h1(x)=-εNkh0(xε)ukh0,h1(x),x∈∂Ω,
we have
(50)∥∇(θεI-θ~εI,h0,h1)∥L2(Ω)≤C∥∇(θε-θ~εh0,h1)∥L2(Ω)≤C(∥∇(θε-θεh0,h1)∥L2(Ω)+∥∇(θ~εh0,h1-θεh0,h1)∥L2(Ω)).
Using the same method of Lemma2, we have
(51)∥θε-θεh0,h1∥L∞(Ω)≤Cε(h0+h1),(52)∥∇(θε-θεh0,h1)∥L2(Ω)≤Cε1/2(h0+h1).
Then, we have(53)|εNkh0(xε)ukh0,h1(x)-εNk(xε)∂u0∂xk|L∞(Ω)≤Cε(h0+h1)∥u0∥W2,∞(Ω),(54)∥∇(θ~εh0,h1-θεh0,h1)∥L2(Ω)≤Cε1/2(h0+h1)∥u0∥W2,∞(Ω).
Combining (52) with (54), we have
(55)∥∇(θεI-θ~εI,h0,h1)∥L2(Ω)≤Cε1/2(h0+h1)∥u0∥W2,∞(Ω).
Next, considering the second item of the right-hand side of (48),  θ~εI,h0,h1-θ~εh0,h1,h can be divided into
(56)θ~εI,h0,h1-θ~εh,h0,h1=(θ~εI,h0,h1-θεI)+(θεI-θεh)+(θεh-θ~εh0,h1,h).
Similarly, we have(57)|∇(θ~εI,h0,h1-θεI)|L2(Ω)≤Cε1/2(h0+h1)∥u0∥W2,∞(Ω),|∇(θεI-θεh)|L2(Ω)≤Cε1/2(h0+h1)∥u0∥W2,∞(Ω),(58)|∇(θεh-θ~εh0,h1,h)|L2(Ω)≤Cε1/2(h0+h1)∥u0∥W2,∞(Ω).
Combining Lemma 6 with (55)–(58), we have (46).Next, using the extrapolation technique [10], we are in a position to estimate
(59)θε(x)-Rh4θ~εh0,h1,h/2(x)-θ~εh0,h1,h(x)3
instead of
(60)θε(x)-Rhθ~εh0,h1,h(x).Lemma 7.
Assuming that (C1) holds, then there existsC such that
(61)∥∇(θε-Rh((4θ~εh0,h1,h/2-θ~εh0,h1,h)/3))dist(·,∂Ω)+ε∥L2(Ω)≤C(h0+h1+h2)|lnε|(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω)).Proof.
Let𝒯h and θεI be defined as above. Assuming that θεI,h/2 is the linear interpolation of θε on 𝒯h/2, we have
(62)∥∇(θε-Rh4θ~εh0,h1,h/2-θ~εh0,h1,h3)∥L2(Ω)≤∥∇(θε-Rh4θεI,h/2-θεI3)∥L2(Ω)+∥∇Rh(4θεI,h/2-θεI3-4θ~εh0,h1,h/2-θ~εh0,h1,h3)∥L2(Ω)≤Cε1/2h2|lnε|(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω))+Cε1/2(h0+h1+h2)(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω))≤Cε1/2(h0+h1+h2|lnε|)(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω)).
Then, (61) can be easily derived.Next, we turn to estimate∥∇wε/dist(·,∂Ω)+ε∥L2(Ω).Lemma 8.
Assuming that (C1) holds, then there existsC such that
(63)∥∇wεdist(·,∂Ω)+ε∥L2(Ω)≤Cε|lnε|∥u0∥W2,∞(Ω).Proof.
Following the same line of [5] and ωε=uε-u~-θε, there exists C such that
(64)∥∇wεdist(·,∂Ω)+ε∥L2(Ω)≤C∥∇wε∥L2|lnε|(Ω)×∥1dist(·,∂Ω)+ε∥L4|lnε|/(2|lnε|-1)(Ω)≤Cε|lnε|1/2∥u0∥W2,2|lnε|(Ω)|lnε|1/2≤Cε|lnε|∥u0∥W2,∞(Ω).
Finally, noting the definitions of u~, u~h0,h1, θε, Rhθ~εh0,h1,h, and ωε, and combining Lemma 3 with Lemmas 7-8 and Lemma  2.4 in [9], we have
(65)∥∇(uε-u^h0,h1,h)dist(·,∂Ω)+ε∥L2(Ω)≤∥∇(u~-u~h0,h1)dist(·,∂Ω)+ε∥L2(Ω)+∥∇(θε-Rhθ~εh0,h1,h)dist(·,∂Ω)+ε∥L2(Ω)+∥∇ωεdist(·,∂Ω)+ε∥L2(Ω)≤C(h1+h0+ε+h2)|lnε|×(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω)).Combining the above lemmas, we can conclude the following result.Theorem 9.
Assuming that (C1) holds, then there existsC such that
(66)∥∇(uε-u^h0,h1,h)dist(·,∂Ω)+ε∥L2(Ω)≤C[(h1+h0+ε+h2)|lnε|]×(∥u0∥W2,∞(Ω)+∥u0∥H4(Ω)).


## 4. Numerical Example

In this section, some numerical results will be shown. In order to show the numerical accuracy of the method presented in this paper, the exact solution of problem (1) should firstly be obtained. However, it is very difficult to find them out. Then, the exact solution will be replaced by the finite element solution in a fine mesh with the mesh size 1/256.It should not be confused thatu* denotes the finite element solution of (1) in a fine mesh, and u^h0,h1,h, obtained by the multiscale finite element scheme presented in the above section, is the multiscale finite element solution of problem (1). Some numerical results will be presented by solving the following model problem:
(67)a11=0.3+2x1(1-x1),a12=a21=x1(1-x1)x2(1-x2),a22=0.1+2x2(1-x2),f(x)=ex1+x2,g(x)=2sin(x1)+4cos(x2),Ω={x∣(x1-0.5)2+(x1-0.5)2<0.25}.
Moreover, let
(68)e0=∥∇(u*-u~h0,h1)/dist(·,∂Ω)+ε∥L2(Ω)∥∇u*/dist(·,∂Ω)+ε∥L2(Ω),e1=∥∇(u*-u^h0,h1,h)/dist(·,∂Ω)+ε∥L2(Ω)∥∇u*/dist(·,∂Ω)+ε∥L2(Ω).
In Table 1, the numerical results of the multiscale method for e0 and e1 are given. It can be seen that the improvement obtained in the final approximation by considering the numerical approximation for the boundary corrector, and the numerical result agree well with the theoretical result from Theorem 9.Table 1
Comparison of computational results withε=1/30.
h
0
↓
h
1
↓
h
↓
e
0
e
1
1
/
8
1
/
8
1
/
8
0.1856
0.0426
1
/
32
1
/
32
1
/
16
0.1629
0.0178
1
/
128
1
/
128
1
/
32
0.1573
0.0043According to Table2, it can be seen that ∇uε(x) can effectively be computed for problem (1) by using the above method, even if dist(x,Ω) is very small. If we only need to get a good numerical solution for problem (1) in Sobolev space H1(Ω), the boundary corrector needs not to be computed. However, the boundary corrector is a very important part of error estimate in the real applications. It can be concluded that this method is an exceedingly important and effective finite element algorithm.Table 2
Comparison of error order.
∥
∇
(
u
*
-
u
~
h
0
,
h
1
)
dist
(
·
,
∂
Ω
)
+
ε
∥
L
2
(
Ω
)
O (1)
∥
∇
(
u
*
-
u
^
h
0
,
h
1
,
h
)
dist
(
·
,
∂
Ω
)
+
ε
∥
L
2
(
Ω
)
O ((h1+h0+ε+h2)|lnε|)



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*Source: 764165-2013-02-28.xml*