Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 764165, 8 pages
http://dx.doi.org/10.1155/2013/764165
Research Article
Superconvergence Analysis of a Multiscale
Finite Element Method for Elliptic Problems with
Rapidly Oscillating Coefficients
Xiaofei Guan,1 Xiaoling Wang,2 Cheng Wang,1 and Xian Liu3
1
Department of Mathematics, Tongji University, Shanghai 200092, China
Department of Fundamental Subject, Tianjin Institute of Urban Construction, Tianjin 300384, China
3
Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China
2
Correspondence should be addressed to Cheng Wang; wangcheng@tongji.edu.cn
Received 29 October 2012; Accepted 17 January 2013
Academic Editor: Song Cen
Copyright © 2013 Xiaofei Guan et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
1. Introduction
In this paper, we consider the following elliptic boundary
value problem with rapidly oscillatory coefficients:
𝐿 𝜀 𝑢𝜀 ≡
𝜕
𝑥 𝜕𝑢𝜀
(𝑎𝑖𝑗 ( )
) = 𝑓 (𝑥) ,
𝜕𝑥𝑖
𝜀 𝜕𝑥𝑗
𝑢𝜀 = 𝑔 (𝑥) ,
in Ω,
(1)
on 𝜕Ω,
where Ω ⊂ R2 is a smooth-bounded domain, 𝑎𝑖𝑗 (𝜉) : R2 →
R is symmetric and satisfies
󵄨 󵄨2
󵄨 󵄨2
(1) 𝜆󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ≤ 𝑎𝑖𝑗 (𝜉) 𝜉𝑖 𝜉𝑗 ≤ 𝜆−1 󵄨󵄨󵄨𝜉󵄨󵄨󵄨 ,
(2) 𝑎𝑖𝑗 (𝜉 + 𝜉󸀠 ) = 𝑎𝑖𝑗 (𝜉) ,
∀𝜉 ∈ R2 , ∃𝜆 ∈ (0, 1] ,
∀𝜉 ∈ R2 , ∃𝜉󸀠 ∈ 𝑍2 ,
1 ≤ 𝑖, 𝑗 ≤ 𝑛,
󵄩 󵄩
(3) 󵄩󵄩󵄩󵄩𝑎𝑖𝑗 󵄩󵄩󵄩󵄩𝐻1 (R2 ) ≤ 𝐶,
∃𝐶 > 0,
(2)
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
2
Journal of Applied Mathematics
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 that 𝑢𝜀 is the solution of (1) and 𝑢̂ℎ0 ,ℎ1 ,ℎ is
the finite element solution [9]. For all 𝑝, 1 < 𝑝 < +∞, there
exists a constant 𝐶 such that
󵄩
󵄩󵄩
󵄩󵄩 ∇ (𝑢𝜀 − 𝑢̂ℎ0 ,ℎ1 ,ℎ ) 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩 2
󵄩𝐿 (Ω)
󵄩
Then, the matrix 𝑎̂ = (̂
𝑎𝑖𝑗 )2 × 2 can be obtained by
𝑎̂𝑖𝑗 = ∫ (𝑎𝑖𝑗 + 𝑎𝑖𝑘
𝑄
𝑢̃ (𝑥) = 𝑢0 (𝑥) + 𝜀𝑁𝑘 (𝜉)
𝐿 0 𝑢0 (𝑥) :=
𝜕𝑢0
𝜕
(̂
𝑎𝑖𝑗
) = 𝑓 (𝑥) , in Ω,
𝜕𝑥𝑖
𝜕𝑥𝑗
𝜃𝜀 = −𝜀𝑁𝑘
Unfortunately, the needed CPU time of the method presented in [9] is 𝑂(𝜀1−𝑛 ℎ−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).
(7)
on 𝜕Ω.
𝜕𝑢0
,
𝜕𝑥𝑘
(8)
on 𝜕Ω.
In the next two subsections, we will compute numerically
the first-order approximation 𝑢̃ and the boundary corrector
term 𝜃𝜀 , respectively, and furthermore give the multiscale
finite element solution of (1).
2.2. Finite Element Approximation of 𝑢̃. Let Tℎ0 be a quasiuniform triangular partition of 𝑄 with the mesh size ℎ0 . 𝑆ℎ0
denotes the conforming 𝑃1 finite element spaces with respect
ℎ
to Tℎ0 , and 𝑆00 = 𝑆ℎ0 ∩ 𝐻01 (𝑄). The finite element scheme of
ℎ
(4) is to find 𝑁𝑘 0 ∈ 𝑆ℎ0 such that
ℎ
∫ 𝑎𝑖𝑗 (𝜉)
𝑄
𝜕𝑁𝑘 0 (𝜉) 𝜕𝑣 (𝜉)
𝑑𝜉
𝜕𝜉𝑗
𝜕𝜉𝑖
= − ∫ 𝑎𝑖𝑘 (𝜉)
𝑄
𝜕𝑣 (𝜉)
𝑑𝜉,
𝜕𝜉𝑖
ℎ
∀𝑣 ∈ 𝑆00 (𝑄) ,
(9)
ℎ
𝑁𝑘 0 (𝜉) is a 1-periodic function.
ℎ
Then, the numerical approximation 𝑎̂𝑖𝑗0 of 𝑎̂𝑖𝑗 can be calculated by
2. An Improved Multiscale Finite
Element Method
Firstly, let us simply recall the homogenization method described in [5].
2.1. Homogenization Method. Let 𝑁𝑘 (𝜉) (𝑘 = 1, 2) be a 1periodic function, which satisfies
𝑄
(6)
𝐿 𝜀 𝜃𝜀 = 0, in Ω,
where 𝑢0 is the homogenization solution of (1), and dist(𝑥, 𝜕Ω)
is the distance between the point 𝑥 and the boundary 𝜕Ω.
∫ 𝑁𝑘 (𝜉) 𝑑𝜉 = 0.
𝜕𝑢0 (𝑥)
,
𝜕𝑥𝑘
The boundary corrector term of the homogenization
method 𝜃𝜀 is defined by
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻3 (Ω) ) ,
𝜕𝑁𝑘
𝜕
𝜕
(𝑎𝑖𝑗 (𝜉)
) = − 𝑎𝑖𝑘 ,
𝜕𝜉i
𝜕𝜉𝑗
𝜕𝜉𝑖
(5)
where 𝑢0 (𝑥) satisfies the homogenization problem
𝑢 (𝑥) = 𝑔 (𝑥) ,
(3)
) 𝑑𝜉.
𝜕𝜉𝑘
The first-order approximation of 𝑢𝜀 (𝑥) can be written as
0
≤ 𝐶 [(ℎ1 + ℎ0 + ℎ + 𝜀) |ln 𝜀|1/2 + 𝜀(2𝑝−1)/2𝑝 ]
𝜕𝑁𝑗
ℎ
ℎ
𝑎̂𝑖𝑗0 = ∫ (𝑎𝑖𝑗 (𝜉) + 𝑎𝑖𝑘 (𝜉)
𝜕𝑁𝑗 0 (𝜉)
𝑄
min 𝑆𝑒𝑘 ≥ 𝐶ℎ12 ,
in R ,
(4)
) 𝑑𝜉.
(10)
Let Tℎ1 be a quasiuniform triangular partition of Ω with
the mesh size ℎ1 and satisfy
𝑒𝑘 ∈Tℎ1
2
𝜕𝜉𝑘
(11)
where 𝑆𝑒𝑘 is the area of the triangular element 𝑒𝑘 . 𝑆𝑔ℎ1 denotes
the corresponding conforming 𝑃1 finite element spaces, and
ℎ
𝑆01 = 𝑆ℎ1 ∩ 𝐻01 (𝑄).
Journal of Applied Mathematics
3
ℎ ,ℎ
The finite element approximation 𝑢00 1 of the homogeℎ ,ℎ
nization problem (7) is to find 𝑢00 1 ∈ 𝑆𝑔ℎ1 such that
∫
Ω
ℎ
𝑎̂𝑖𝑗0
ℎ ,ℎ
𝜕𝑢00 1 𝜕𝑣ℎ
𝑑𝑥 = ∫ 𝑓 (𝑥) 𝑣ℎ (𝑥) 𝑑𝑥,
𝜕𝑥𝑖 𝜕𝑥𝑗
Ω
∀𝑣ℎ ∈
ℎ ,ℎ
Furthermore, we turn to the computation of 𝜕𝑢00 1 (𝑥)/
𝜕𝑥𝑘 and 𝑢̃(𝑥). Let Σℎ1 be the set of all nodal points of the mesh
ℎ ,ℎ
Tℎ1 . Define 𝑢𝑘0 1 (𝑘 = 1, 2) by the following:
ℎ ,ℎ
𝑢𝑘0 1
(𝐴 1 ) for all 𝑥 ∈ Σ , the value of
at the nodal point
ℎ0 ,ℎ1
𝑥 is the average of 𝜕𝑢0 (𝑥)/𝜕𝑥𝑘 in all elements
including 𝑥,
ℎ ,ℎ
(𝐴 2 ) 𝑢𝑘0 1 (𝑥)
is a piecewise linear function in every ele-
ment.
Therefore, we have a numerical approximation 𝑢̃
of 𝑢̃(𝑥) which is defined by
𝑢̃
ℎ0 ,ℎ1
(𝑥) =
ℎ ,ℎ
𝑢00 1
(𝑥) +
ℎ
𝜀𝑁𝑘 0
ℎ ,ℎ
(𝜉) 𝑢𝑘0 1
ℎ0 ,ℎ1
(𝑥) .
(𝑥)
2𝑚−1 𝜀 < max dist (𝑥, 𝜕Ω) ≤ 2𝑚 𝜀.
𝑥∈Ω
(13)
(14)
Then, the domain Ω can be divided by Ω = ⋃𝑚
𝑖=0 Ω𝑖 and
if 𝑖 = 0,
Ω𝑖 = {𝑥 ∈ Ω | 2𝑖−1 𝜀 < dist (𝑥, 𝜕Ω) ≤ 2𝑖 𝜀} ,
if 1 ≤ 𝑖 ≤ 𝑚.
(15)
Let Tℎ be the regular triangular partition of Ω with the
mesh size ℎ and satisfy
(𝐵1 ) 2𝑖/2 𝜆𝜀ℎ ≤ √𝑆𝑒𝑖 ≤ 2𝑖/2 𝜆−1 𝜀ℎ,
∀𝑒𝑖 ∈ Tℎ ,
(16)
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 , Σℎ𝐵 = Σℎ ∩ 𝜕Ω, and let 𝑆ℎ be the
conforming 𝑃1 finite element spaces with respect to Tℎ ; we
define
𝑥 ℎ ,ℎ
ℎ
𝑆1ℎ = {𝑣 ∈ 𝑆ℎ | 𝑣 (𝑥) = −𝜀𝑁𝑘 0 ( ) 𝑢𝑘0 1 (𝑥) , ∀𝑥 ∈ Σℎ𝐵 } ,
𝜀
𝑆0ℎ = {𝑣 ∈ 𝑆ℎ | 𝑣 (𝑥) = 0, ∀𝑥 ∈ Σℎ𝐵 } .
(18)
Then, the finite element approximation of 𝜃𝜀 is to find
𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ ∈ 𝑆1ℎ such that
𝜕𝜃̃ℎ0 ,ℎ1 ,ℎ 𝜕𝑣
𝑑𝑥 = 0,
∫ 𝑎𝑖𝑗 (𝜉) 𝜀
𝜕𝑥𝑖 𝜕𝑥𝑗
Ω
∀𝑣 ∈ 𝑆0ℎ .
𝜕𝑅ℎ 𝑣 (𝑥)
𝜕𝑥𝑖
=
𝜕𝑣 (𝑥)
,
𝜕𝑥𝑖
if 𝑥 is the middle point of elements of Tℎ .
(20)
Then, the multiscale finite element approximation 𝑢̂ℎ0 ,ℎ1 ,ℎ (𝑥)
of 𝑢𝜀 (𝑥) can be defined by
𝑢̂ℎ0 ,ℎ1 ,ℎ (𝑥) = 𝑢̃ℎ0 ,ℎ1 (𝑥) + 𝑅ℎ 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ (𝑥) .
(21)
3. Superconvergence Result of 𝑢̂ℎ0 ,ℎ1 ,ℎ
Firstly, we have the following assumption.
2.3. Finite Element Approximation of 𝜃𝜀 (𝑥). Let 𝑚 be a
positive integer satisfying
Ω𝑖 = {𝑥 ∈ Ω | dist (𝑥, 𝜕Ω) ≤ 𝜀} ,
𝜕𝑅ℎ 𝑣 (𝑥)
is a piecewise function on Tℎ ,
𝜕𝑥𝑖
ℎ
𝑆01 .
(12)
ℎ1
2.4. Multiscale Finite Element Approximation of 𝑢𝜀 . For any
𝑣 ∈ 𝑆ℎ (Ω), we define the linear operator 𝑅ℎ by
(19)
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]). Let Ω𝑟 = {𝑥 ∈ Ω | dist(𝑥, 𝜕Ω) ≥
𝑟}. Assuming that (C1) holds, then there exists 𝐶, which is
independent of 𝜀 and 𝑟 such that
󵄩󵄩 󵄩󵄩
−1/2 󵄩 0 󵄩
󵄩󵄩∇𝜃𝜀 󵄩󵄩𝐿2 (Ω𝑟 ) ≤ 𝐶𝑟 𝜀󵄩󵄩󵄩󵄩𝑢 󵄩󵄩󵄩󵄩𝐻2 (Ω) ,
󵄩󵄩 󵄩󵄩
−1/2 󵄩 0 󵄩
󵄩󵄩𝜃𝜀 󵄩󵄩𝐻2 (Ω) ≤ 𝐶𝜀 󵄩󵄩󵄩󵄩𝑢 󵄩󵄩󵄩󵄩𝐻3 (Ω) ,
(22)
−1/2 󵄩
󵄩󵄩𝑢0 󵄩󵄩󵄩
󵄩󵄩󵄩𝜃𝜀 󵄩󵄩󵄩 1
≤
𝐶𝑟
𝜀
,
󵄩
󵄩
1,∞
󵄩 󵄩𝐻 (Ω𝑟 )
󵄩 󵄩𝑊 (𝜕Ω)
󵄩
−1/2 󵄩 0 󵄩
󵄩
󵄩󵄩󵄩𝜃𝜀 󵄩󵄩󵄩 2
󵄩 󵄩𝐻 (Ω𝑟 ) ≤ 𝐶𝑟 󵄩󵄩󵄩𝑢 󵄩󵄩󵄩𝑊1,∞ (𝜕Ω) .
From Lemma 2, one can easily deduce.
Lemma 3. Assuming that (C1) holds, then there exists 𝐶, which
is independent of 𝜀 such that
󵄩󵄩 󵄩󵄩
−3/2 󵄩
󵄩 0 󵄩󵄩
(23)
󵄩󵄩𝜃𝜀 󵄩󵄩𝐻3 (Ω) ≤ 𝐶𝜀 󵄩󵄩󵄩𝑢 󵄩󵄩󵄩𝐻4 (Ω) ,
󵄩 0󵄩
󵄩󵄩 󵄩󵄩
−1/2 −1 󵄩 0 󵄩
󵄩󵄩𝜃𝜀 󵄩󵄩𝐻3 (Ω𝑟 ) ≤ 𝐶𝑟 𝜀 (󵄩󵄩󵄩󵄩𝑢 󵄩󵄩󵄩󵄩𝑊2,∞ (𝜕Ω) + 󵄩󵄩󵄩󵄩𝑢 󵄩󵄩󵄩󵄩𝐻4 (Ω) ) . (24)
Proof. For 𝑘 = 1, 2, we define
𝑣𝑘𝜀 (𝑥) =
𝜕𝜃𝜀 (𝑥)
.
𝜕𝑥𝑘
(25)
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
𝐿 𝜀 𝑣𝑘𝜀 = −𝜀−1
𝑣𝑘𝜀 (𝑥) = −𝜀
𝜕 𝜕𝑎𝑖𝑗 (𝜉) 𝜕𝜃𝜀
(
),
𝜕𝑥𝑖
𝜕𝜉𝑘 𝜕𝑥𝑗
𝜕 (𝑁𝑙 (𝜉) (𝜕𝑢0 (𝑥) /𝜕𝑥𝑙 ))
𝜕𝑥𝑘
𝑥 ∈ Ω,
(26)
,
𝑥 ∈ 𝜕Ω.
4
Journal of Applied Mathematics
Then, we estimate ‖𝑣𝑘𝜀 ‖𝐻2 (Ω) . Obviously, 𝑣𝑘𝜀 can be divided into
𝜀
𝜀
𝑣𝑘𝜀 = 𝑣𝑘,1
+ 𝑣𝑘,2
,
(27)
𝜀
where 𝑣𝑘,1
satisfies
𝜀
𝐿 𝜀 𝑣𝑘,1
= −𝜀−1
𝜕 𝜕𝑎𝑖𝑗 (𝜉) 𝜕𝜃𝜀
(
),
𝜕𝑥𝑖
𝜕𝜉𝑘 𝜕𝑥𝑗
𝜀
𝑣𝑘,1
= 0,
𝑥 ∈ Ω,
Following the same line of [5] (1992, Theorem 1.2, pages
124–128), we have
󵄩󵄩 𝜀 󵄩󵄩
−1 −1/2
−1/2
󵄩󵄩𝑣̂𝑘,2 󵄩󵄩 2
󵄩 󵄩𝐻 (Ω𝑟 ) ≤ 𝐶𝜀 𝑟 ‖𝜃‖𝑊1,∞ (Ω𝑟/2 ) + 𝐶𝑟 ‖𝜃‖𝑊2,∞ (Ω𝑟/2 )
󵄩 󵄩
󵄩 󵄩
≤ 𝐶𝜀−1 𝑟−1/2 𝑟−1 𝜀 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (𝜕Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
(28)
󵄩 󵄩
󵄩 󵄩
+ 𝐶𝑟−1/2 𝑟−1 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (𝜕Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
𝑥 ∈ 𝜕Ω,
󵄩 󵄩
󵄩 󵄩
≤ 𝐶𝜀−1 𝑟−1/2 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (𝜕Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) ) .
𝜀
and 𝑣𝑘,2
satisfies
𝜀
= 0,
𝐿 𝜀 𝑣𝑘,2
𝜀
𝑣𝑘,2
= −𝜀
𝑥 ∈ Ω,
𝜕 (𝑁𝑙 (𝑥/𝜀) (𝜕𝑢0 /𝜕𝑥𝑙 ))
𝜕𝑥𝑘
,
(29)
Combining (37) with (38), we can conclude the result of this
lemma.
(30)
Assuming that Tℎ 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.
𝑥 ∈ 𝜕Ω.
Using the result from Lemma 2 and (28), we have
󵄩󵄩 𝜀 󵄩󵄩
󵄩 󵄩
󵄩󵄩𝑣𝑘,1 󵄩󵄩 2 ≤ 𝐶𝜀−1 󵄩󵄩󵄩󵄩𝜃𝜀 󵄩󵄩󵄩󵄩𝐻2 (Ω) ≤ 𝐶𝜀−3/2 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩 3 .
󵄩 󵄩𝐻 (Ω)
󵄩 󵄩𝐻 (Ω)
𝜀
𝜀
Let 𝑣𝑘,2,𝑘
󸀠 = 𝜕𝑣𝑘,2 /𝜕𝑥𝑘󸀠 , and using the result from (29), we
have
𝜀
𝐿 𝜀 𝑣𝑘,2,𝑘
󸀠 = 0,
𝜀
𝑣𝑘,2,𝑘
󸀠
= −𝜀
𝑥 ∈ Ω,
𝜕2 (𝑁𝑙 (𝑥/𝜀) (𝜕𝑢0 /𝜕𝑥𝑙 ))
𝜕𝑥𝑘 𝜕𝑥𝑘󸀠
(31)
,
𝑥 ∈ 𝜕Ω.
Following the same line of [5] (1992, Theorem 1.2, pages 124–
128), we have
󵄩󵄩
󵄩
−3/2 󵄩 0 󵄩
−3/2 󵄩 0 󵄩
󵄩󵄩𝑣𝑘,2,𝑘󸀠 󵄩󵄩󵄩𝐻1 (Ω) ≤ 𝐶𝜀 󵄩󵄩󵄩󵄩𝑢 󵄩󵄩󵄩󵄩𝐻4 (Ω) ≤ 𝐶𝜀 󵄩󵄩󵄩󵄩𝑢 󵄩󵄩󵄩󵄩𝐻4 (Ω) (32)
which indicates
󵄩 󵄩
󵄩󵄩 𝜀 󵄩󵄩
󵄩󵄩𝑣𝑘,2 󵄩󵄩 2 ≤ 𝐶𝜀−3/2 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩 4 .
󵄩 󵄩𝐻 (Ω)
󵄩 󵄩𝐻 (Ω)
(33)
Combining (30) with (33), we can derive (24) immediately.
Considering the proof of (23), 𝑣𝑘𝜀 can be divided into
𝜀
𝜀
+ 𝑣̂𝑘,2
,
𝑣𝑘𝜀 = 𝑣̂𝑘,1
(34)
𝜀
where 𝑣̂𝑘,1
satisfies
𝜀
𝐿 𝜀 𝑣̂𝑘,1
= −𝜀−1
𝜕 𝜕𝑎𝑖𝑗 (𝜉) 𝜕𝜃𝜀
(
),
𝜕𝑥𝑖
𝜕𝜉𝑘 𝜕𝑥𝑗
𝜀
𝑣̂𝑘,1
= 0,
𝑥 ∈ Ω𝑟 ,
(35)
𝑥 ∈ 𝜕Ω𝑟 ,
𝜀
and 𝑣̂𝑘,2
satisfies
𝜀
= 0,
𝐿 𝜀 𝑣̂𝑘,2
𝜀
𝑣̂𝑘,2
𝜕𝜃
= 𝜀,
𝜕𝑥𝑘
In view of Lemma 2, we have
󵄩󵄩 𝜀 󵄩󵄩
−1 󵄩 󵄩
󵄩󵄩󵄩𝑣̂𝑘,1 󵄩󵄩󵄩𝐻2 (Ω𝑟 ) ≤ 𝐶𝜀 󵄩󵄩󵄩𝜃𝜀 󵄩󵄩󵄩𝐻2 (Ω𝑟 )
󵄩 󵄩
≤ 𝐶𝜀−1 𝑟−1/2 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (𝜕Ω) .
Lemma 4. Assuming that (C1) holds, then there exists 𝐶 such
that
󵄩
󵄩󵄩
󵄩󵄩 ∇ (𝜃𝜀𝐼 − 𝜃𝜀ℎ ) 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩 dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩 2
󵄩𝐿 (Ω)
󵄩
󵄩
󵄩
󵄩 󵄩
≤ 𝐶𝜀−1/2 ℎ2 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊1,∞ (𝜕Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) ) .
(39)
Proof. Assuming that 𝑚 is defined as (14) and Ω = ⋃𝑚
𝑖=𝑘+1 Ω𝑖
(𝑘 < 𝑚, 𝑘 ∈ N). Considering 𝑎𝑖𝑗 ∈ 𝑊1,∞ (Ω) and using the
result from Lemma 2, we have
2
󵄩
󵄩
(󵄩󵄩󵄩󵄩∇ (𝜃𝜀𝐼 − 𝜃𝜀ℎ )󵄩󵄩󵄩󵄩𝐿2 (Ω) )
󵄨󵄨
󵄨󵄨
ℎ
𝐼
ℎ
𝐼
󵄨󵄨
𝑥 𝜕 (𝜃𝜀,𝑟 − 𝜃𝜀 ) 𝜕 (𝜃𝜀 − 𝜃𝜀 ) 󵄨󵄨󵄨
󵄨
󵄨
≤ 𝐶 󵄨󵄨∫ 𝑎𝑖𝑗 ( )
𝑑𝑥󵄨󵄨󵄨
𝜀
𝜕𝑥𝑖
𝜕𝑥𝑗
󵄨󵄨 Ω
󵄨󵄨
󵄨
󵄨
󵄨󵄨
𝐼
ℎ
𝐼
󵄨󵄨󵄨
𝑥 𝜕 (𝜃𝜀 − 𝜃𝜀 ) 𝜕 (𝜃𝜀,𝑟 − 𝜃𝜀 ) 󵄨󵄨󵄨
󵄨
≤ 𝐶 󵄨󵄨󵄨󵄨∫ 𝑎𝑖𝑗 ( )
𝑑𝑥󵄨󵄨󵄨
𝜀
𝜕𝑥𝑖
𝜕𝑥𝑗
󵄨󵄨 Ω
󵄨󵄨
󵄨
󵄨
󵄨
󵄨󵄨
𝐼
ℎ
𝐼
󵄨
𝑘 󵄨
𝑥 𝜕 (𝜃𝜀 − 𝜃𝜀 ) 𝜕 (𝜃𝜀,𝑟 − 𝜃𝜀 ) 󵄨󵄨󵄨
󵄨
≤ ∑ 󵄨󵄨󵄨󵄨∫ 𝑎𝑖𝑗 ( )
𝑑𝑥󵄨󵄨󵄨
𝜀
𝜕𝑥𝑖
𝜕𝑥𝑗
󵄨󵄨
𝑖=1 󵄨󵄨󵄨 Ω𝑖
󵄨
𝑘
2
󵄩
󵄩 󵄩
ℎ 󵄩
≤ 𝐶∑(2𝑖/2 𝜀ℎ) (𝜀−1 󵄩󵄩󵄩𝜃𝜀 󵄩󵄩󵄩𝐻2 (Ω𝑖 ) 󵄩󵄩󵄩󵄩∇ (𝜃𝜀𝐼 − 𝜃𝜀,𝑟
)󵄩󵄩󵄩󵄩𝐿2 (Ω )
𝑖
𝑥 ∈ Ω𝑟 ,
𝑥 ∈ 𝜕Ω𝑟 .
(38)
(36)
𝑖=1
󵄩
󵄩
󵄩 󵄩
+󵄩󵄩󵄩𝜃𝜀 󵄩󵄩󵄩𝐻3 (Ω𝑖 ) 󵄩󵄩󵄩󵄩∇ (𝜃𝜀𝐼 − 𝜃𝜀𝑟,ℎ )󵄩󵄩󵄩󵄩𝐿2 (Ω ) )
𝑖
𝑘
≤ 𝐶∑2𝑖 𝜀2 ℎ2 (𝜀−1 (2𝑖 𝜀)
𝑖=1
(37)
−1/2
󵄩 󵄩
󵄩 󵄩
(󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊1,∞ (𝜕Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
󵄩
ℎ 󵄩
×󵄩󵄩󵄩󵄩∇ (𝜃𝜀𝐼 − 𝜃𝜀,𝑟
)󵄩󵄩󵄩󵄩𝐿2 (Ω−Ω ) )
𝑟
Journal of Applied Mathematics
5
󵄩 󵄩
󵄩 󵄩
≤ 𝐶2𝑘/2 𝜀1/2 ℎ2 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊1,∞ (𝜕Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
Using the result from (44), we have
󵄩
ℎ 󵄩
× 󵄩󵄩󵄩󵄩∇ (𝜃𝜀𝐼 − 𝜃𝜀,𝑟
)󵄩󵄩󵄩󵄩𝐿2 (Ω)
󵄩 󵄩
󵄩 󵄩
≤ 𝐶ℎ2 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊1,∞ (𝜕Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
󵄩
ℎ 󵄩
× 󵄩󵄩󵄩󵄩∇ (𝜃𝜀𝐼 − 𝜃𝜀,𝑟
)󵄩󵄩󵄩󵄩𝐿2 (Ω−Ω ) ,
𝑟
(40)
󵄩
󵄩󵄩
󵄩󵄩 ∇ (𝜃𝜀𝐼 − 𝜃𝜀ℎ ) 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩 2
󵄩𝐿 (Ω)
󵄩
󵄩
󵄩
𝑚󵄩
󵄩󵄩 ∇ (𝜃𝜀𝐼 − 𝜃𝜀ℎ ) 󵄩󵄩󵄩
󵄩󵄩
󵄩
≤ 𝐶∑ 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 2
󵄩
√
dist
𝜕Ω)
+
𝜀
(⋅,
𝑗=1󵄩
󵄩𝐿 (Ω𝑟𝑗 −Ω𝑟𝑗−1 )
󵄩
𝑚
󵄩
󵄩
≤ ∑ 𝐶𝑟𝑗−1/2 󵄩󵄩󵄩󵄩∇ (𝜃𝜀𝐼 − 𝜃𝜀ℎ )󵄩󵄩󵄩󵄩𝐿2 (Ω
which indicates that
𝑟𝑗 −Ω𝑟𝑗−1 )
𝑗=1
󵄩󵄩 0 󵄩󵄩
󵄩󵄩
󵄩
󵄩 󵄩
󵄩 󵄩
󵄩󵄩∇ (𝜃𝜀𝐼 − 𝜃𝜀ℎ )󵄩󵄩󵄩 2 ≤ 𝐶ℎ2 (󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩 1,∞
󵄩
󵄩𝐿 (Ω)
󵄩 󵄩𝑊 (𝜕Ω) + 󵄩󵄩𝑢 󵄩󵄩𝐻4 (Ω) ) . (41)
Then Lemma 4 can be easily derived.
≤
𝑚
󵄩
∑ 𝐶𝑟𝑗−1/2 󵄩󵄩󵄩󵄩∇ (𝜃𝜀𝐼
𝑗=1
−
󵄩
𝜃𝜀ℎ )󵄩󵄩󵄩󵄩𝐿2 (Ω−Ω
(45)
𝑟𝑗−1 )
𝑚
󵄩 󵄩
󵄩 󵄩
≤ ∑ 𝐶𝑟𝑗−1/2 𝑟𝑗1/2 ℎ2 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
𝑗=1
Furthermore, we can obtain the following lemma.
Lemma 5. Assuming that (C1) holds, then there exists 𝐶 such
that
󵄩
󵄩󵄩
󵄩󵄩 ∇ (𝜃𝜀𝐼 − 𝜃𝜀ℎ ) 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩 2
󵄩𝐿 (Ω)
󵄩
󵄩 󵄩
󵄩 󵄩
≤ 𝐶 |ln 𝜀| ℎ2 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) ) .
󵄩 󵄩
󵄩 󵄩
≤ 𝐶𝑚ℎ2 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
󵄩 󵄩
󵄩 󵄩
≤ 𝐶 |ln 𝜀| ℎ2 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) ) .
(42)
Based on the previous lemmas, the estimate of
‖|∇(𝜃𝜀𝐼 − 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ )|/√dist(⋅, 𝜕Ω) + 𝜀‖𝐿2 (Ω) can be given as
follows.
Proof. Assume that Ω𝑟𝑗 is defined as Lemma 4. Considering
Lemma 6. Assuming that (C1) holds, then there exists 𝐶 such
that
󵄨󵄩
󵄩󵄩 󵄨󵄨
󵄩󵄩 󵄨󵄨∇ (𝜃𝜀𝐼 − 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ )󵄨󵄨󵄨 󵄩󵄩󵄩
󵄨 󵄩󵄩
󵄩󵄩 󵄨
≤ 𝐶 (ℎ0 + ℎ1 + ℎ2 ) |ln 𝜀|
󵄩
󵄩󵄩
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩 2
󵄩𝐿 (Ω)
󵄩
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) ) .
(46)
𝜃𝜀ℎ
=
ℎ
𝜃𝜀,𝑟
0
(∀𝑥 ∈ Ω𝑟𝑗 ), and we divide
𝜃𝜀𝐼
−
𝜃𝜀ℎ
=
(𝜃𝜀𝐼
−
ℎ
𝜃𝜀,𝑟
)
𝑗
𝜃𝜀𝐼
−
𝜃𝜀ℎ
into
𝑗−1
ℎ
ℎ
+ ∑ (𝜃𝜀,𝑟
− 𝜃𝜀,𝑟
).
𝑖
𝑖+1
(43)
𝑖=0
Then,
󵄩󵄩
󵄩󵄩 𝐼
󵄩󵄩 𝐼
󵄩
ℎ
󵄩
󵄩
󵄩󵄩𝜃𝜀 − 𝜃𝜀ℎ 󵄩󵄩󵄩 1
󵄩
󵄩𝐻 (Ω−Ω𝑟𝑗 ) ≤ 󵄩󵄩󵄩𝜃𝜀 − 𝜃𝜀,𝑟𝑗+1 󵄩󵄩󵄩𝐻1 (Ω−Ω𝑟𝑗 )
Proof. Assuming that 𝜃̃𝜀ℎ0 ,ℎ1 satisfies
𝑗−1
󵄩 𝐼
󵄩󵄩
ℎ
󵄩󵄩
+ ∑ 󵄩󵄩󵄩󵄩𝜃𝜀,𝑟
− 𝜃𝜀,𝑟
𝑖
𝑖+1 󵄩𝐻1 (Ω−Ω𝑟 )
𝑗
𝑖=1
󵄩 󵄩
󵄩 󵄩
≤ 𝑐𝑟𝑗1/2 ℎ2 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
𝑗−1
󵄩 𝐼
ℎ
󵄩󵄩󵄩
+ ∑ 󵄩󵄩󵄩󵄩𝜃𝜀,𝑟
− 𝜃𝜀,𝑟
󵄩
𝑖
𝑖+1 󵄩𝐻1 (Ω−Ω𝑟 )
𝑖=1
𝑗
󵄩 󵄩
󵄩 󵄩
≤
(󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
󵄩 󵄩
󵄩 󵄩
+ 𝐶𝑟𝑗 𝑗ℎ2 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
󵄩 󵄩
󵄩 󵄩
≤ 𝐶𝑟𝑗1/2 ℎ2 (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) ) .
(44)
𝐶𝑟𝑗1/2 ℎ2
𝐿 𝜀 𝜃̃𝜀ℎ0 ,ℎ1 (𝑥) = 0,
𝑥 ∈ Ω,
𝑥 ℎ ,ℎ
ℎ
𝜃̃𝜀ℎ0 ,ℎ1 (𝑥) = −𝜀𝑁𝑘 0 ( ) 𝑢𝑘0 1 (𝑥) ,
𝜀
𝑥 ∈ 𝜕Ω,
(47)
and 𝜃̃𝜀𝐼,ℎ0 ,ℎ1 is the linear interpolation of 𝜃̃𝜀ℎ0 ,ℎ1 on Tℎ .
Then, we divide 𝜃𝜀𝐼 − 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ into
𝜃𝜀𝐼 − 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ = (𝜃𝜀𝐼 − 𝜃̃𝜀𝐼,ℎ0 ,ℎ1 ) + (𝜃̃𝜀𝐼,ℎ0 ,ℎ1 − 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ ) .
(48)
Firstly, considering the first item of the right-hand side of
(48) and assuming that 𝜃𝜀ℎ0 ,ℎ1 (𝑥) satisfies the problem
𝐿 𝜀 𝜃𝜀ℎ0 ,ℎ1 (𝑥) = 0,
𝑥 ∈ Ω,
𝑥 ℎ ,ℎ
ℎ
𝜃𝜀ℎ0 ,ℎ1 (𝑥) = −𝜀𝑁𝑘 0 ( ) 𝑢𝑘0 1 (𝑥) ,
𝜀
𝑥 ∈ 𝜕Ω,
(49)
6
Journal of Applied Mathematics
we have
󵄩
󵄩󵄩
󵄩󵄩∇ (𝜃𝜀𝐼 − 𝜃̃𝜀𝐼,ℎ0 ,ℎ1 )󵄩󵄩󵄩 2
󵄩𝐿 (Ω)
󵄩
󵄩
󵄩
≤ 𝐶󵄩󵄩󵄩󵄩∇ (𝜃𝜀 − 𝜃̃𝜀ℎ0 ,ℎ1 )󵄩󵄩󵄩󵄩𝐿2 (Ω)
󵄩
󵄩
󵄩
󵄩
≤ 𝐶 (󵄩󵄩󵄩󵄩∇ (𝜃𝜀 − 𝜃𝜀ℎ0 ,ℎ1 )󵄩󵄩󵄩󵄩𝐿2 (Ω) + 󵄩󵄩󵄩󵄩∇ (𝜃̃𝜀ℎ0 ,ℎ1 − 𝜃𝜀ℎ0 ,ℎ1 )󵄩󵄩󵄩󵄩𝐿2 (Ω) ) .
(50)
Using the same method of Lemma 2, we have
󵄩󵄩
󵄩
󵄩󵄩𝜃𝜀 − 𝜃𝜀ℎ0 ,ℎ1 󵄩󵄩󵄩 ∞ ≤ 𝐶𝜀 (ℎ0 + ℎ1 ) ,
󵄩
󵄩𝐿 (Ω)
󵄩󵄩
󵄩
󵄩󵄩∇ (𝜃𝜀 − 𝜃𝜀ℎ0 ,ℎ1 )󵄩󵄩󵄩 2 ≤ 𝐶𝜀1/2 (ℎ0 + ℎ1 ) .
󵄩
󵄩𝐿 (Ω)
(51)
(52)
Then, we have
󵄨󵄨
󵄨
𝑥 𝜕𝑢0 󵄨󵄨󵄨
󵄨󵄨 ℎ0 𝑥 ℎ0 ,ℎ1
󵄨󵄨𝜀𝑁𝑘 ( ) 𝑢𝑘 (𝑥) − 𝜀𝑁𝑘 ( )
󵄨󵄨
󵄨󵄨
𝜀
𝜀 𝜕𝑥𝑘 󵄨󵄨󵄨𝐿∞ (Ω)
󵄨
(53)
󵄩󵄩 0 󵄩󵄩
≤ 𝐶𝜀 (ℎ0 + ℎ1 ) 󵄩󵄩󵄩𝑢 󵄩󵄩󵄩𝑊2,∞ (Ω) ,
󵄩󵄩 ̃ℎ0 ,ℎ1
󵄩
󵄩 󵄩
󵄩󵄩∇ (𝜃𝜀 − 𝜃𝜀ℎ0 ,ℎ1 )󵄩󵄩󵄩 2 ≤ 𝐶𝜀1/2 (ℎ0 + ℎ1 ) 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩 2,∞ .
󵄩
󵄩𝐿 (Ω)
󵄩 󵄩𝑊 (Ω)
(54)
Combining (52) with (54), we have
󵄩󵄩
󵄩 󵄩
󵄩
󵄩󵄩∇ (𝜃𝜀𝐼 − 𝜃̃𝜀𝐼,ℎ0 ,ℎ1 )󵄩󵄩󵄩 2 ≤ 𝐶𝜀1/2 (ℎ0 + ℎ1 ) 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩 2,∞ . (55)
󵄩
󵄩 󵄩𝑊 (Ω)
󵄩𝐿 (Ω)
Next, considering the second item of the right-hand side
of (48), 𝜃̃𝜀𝐼,ℎ0 ,ℎ1 − 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ can be divided into
𝜃̃𝜀𝐼,ℎ0 ,ℎ1 − 𝜃̃𝜀ℎ,ℎ0 ,ℎ1 = (𝜃̃𝜀𝐼,ℎ0 ,ℎ1 − 𝜃𝜀𝐼 ) + (𝜃𝜀𝐼 − 𝜃𝜀ℎ )
+ (𝜃𝜀ℎ − 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ ) .
(56)
󵄨󵄨 ̃𝐼,ℎ0 ,ℎ1
󵄨
󵄩 󵄩
󵄨󵄨∇ (𝜃𝜀
− 𝜃𝜀𝐼 )󵄨󵄨󵄨󵄨𝐿2 (Ω) ≤ 𝐶𝜀1/2 (ℎ0 + ℎ1 ) 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) ,
󵄨
(57)
󵄨󵄨
󵄨
󵄩 󵄩
󵄨󵄨∇ (𝜃𝜀𝐼 − 𝜃𝜀ℎ )󵄨󵄨󵄨 2 ≤ 𝐶𝜀1/2 (ℎ0 + ℎ1 ) 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩 2,∞ ,
󵄨
󵄨𝐿 (Ω)
󵄩 󵄩𝑊 (Ω)
󵄨󵄨
󵄨
󵄩 󵄩
󵄨󵄨∇ (𝜃𝜀ℎ − 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ )󵄨󵄨󵄨 2 ≤ 𝐶𝜀1/2 (ℎ0 + ℎ1 ) 󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩 2,∞ . (58)
󵄨
󵄨𝐿 (Ω)
󵄩 󵄩𝑊 (Ω)
Combining Lemma 6 with (55)–(58), we have (46).
Next, using the extrapolation technique [10], we are in a
position to estimate
4𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ/2 (𝑥) − 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ (𝑥)
3
(59)
instead of
𝜃𝜀 (𝑥) − 𝑅ℎ 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ (𝑥) .
󵄩
󵄩󵄩
󵄩󵄩 ∇ (𝜃𝜀 − 𝑅ℎ ((4𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ/2 − 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ ) /3)) 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩 2
󵄩󵄩
√dist (⋅, 𝜕Ω) + 𝜀
󵄩𝐿 (Ω)
󵄩
(61)
󵄩
󵄩
󵄩
󵄩
≤ 𝐶 (ℎ0 + ℎ1 + ℎ2 ) |ln 𝜀| (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) ) .
Proof. Let Tℎ and 𝜃𝜀𝐼 be defined as above. Assuming that 𝜃𝜀𝐼,ℎ/2
is the linear interpolation of 𝜃𝜀 on Tℎ/2 , we have
󵄩󵄩
̃ℎ0 ,ℎ1 ,ℎ/2 − 𝜃̃ℎ0 ,ℎ1 ,ℎ 󵄩󵄩󵄩󵄩
󵄩󵄩
𝜀
󵄩󵄩∇ (𝜃𝜀 − 𝑅ℎ 4𝜃𝜀
)󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩 2
3
󵄩󵄩
󵄩𝐿 (Ω)
󵄩
󵄩󵄩󵄩
4𝜃𝐼,ℎ/2 − 𝜃𝜀𝐼 󵄩󵄩󵄩󵄩
󵄩
≤ 󵄩󵄩󵄩∇ (𝜃𝜀 − 𝑅ℎ 𝜀
)󵄩󵄩
󵄩󵄩
󵄩󵄩 2
3
󵄩
󵄩𝐿 (Ω)
󵄩󵄩
󵄩
󵄩󵄩
4𝜃𝜀𝐼,ℎ/2 − 𝜃𝜀𝐼 4𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ/2 − 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ 󵄩󵄩󵄩󵄩
󵄩
+ 󵄩󵄩∇𝑅ℎ (
−
)󵄩󵄩
󵄩󵄩
󵄩󵄩 2
3
3
󵄩
󵄩𝐿 (Ω)
󵄩 󵄩
󵄩 󵄩
≤ 𝐶𝜀1/2 ℎ2 |ln 𝜀| (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
󵄩 󵄩
󵄩 󵄩
+ 𝐶𝜀1/2 (ℎ0 + ℎ1 + ℎ2 ) (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) )
󵄩 󵄩
󵄩 󵄩
≤ 𝐶𝜀1/2 (ℎ0 + ℎ1 + ℎ2 |ln 𝜀|) (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) ) .
(62)
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 exists 𝐶 such
that
Similarly, we have
𝜃𝜀 (𝑥) − 𝑅ℎ
Lemma 7. Assuming that (C1) holds, then there exists 𝐶 such
that
(60)
󵄩󵄩
󵄩󵄩
∇𝑤𝜀
󵄩󵄩
󵄩󵄩
󵄩󵄩 0 󵄩󵄩
󵄩
󵄩󵄩
󵄩 󵄩
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩 2 ≤ 𝐶𝜀 |ln 𝜀| 󵄩󵄩𝑢 󵄩󵄩𝑊2,∞ (Ω) .
󵄩𝐿 (Ω)
󵄩
(63)
Proof. Following the same line of [5] and 𝜔𝜀 = 𝑢𝜀 − 𝑢̃ − 𝜃𝜀 ,
there exists 𝐶 such that
󵄩󵄩
󵄩󵄩
∇𝑤𝜀
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩 2
󵄩
󵄩𝐿 (Ω)
󵄩󵄩
󵄩󵄩
1
󵄩
󵄩󵄩
󵄩
󵄩
󵄩󵄩
≤ 𝐶󵄩󵄩󵄩∇𝑤𝜀 󵄩󵄩󵄩𝐿2| ln 𝜀| (Ω) × 󵄩󵄩󵄩󵄩
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩𝐿4| ln 𝜀|/(2| ln 𝜀|−1) (Ω)
󵄩 󵄩
≤ 𝐶𝜀|ln 𝜀|1/2 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,2| ln 𝜀| (Ω) |ln 𝜀|1/2
󵄩 󵄩
≤ 𝐶𝜀 |ln 𝜀| 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) .
(64)
Journal of Applied Mathematics
7
Table 1: Comparison of computational results with 𝜀 = 1/30.
ℎ0 ↓
1/8
1/32
1/128
ℎ1 ↓
1/8
1/32
1/128
ℎ↓
1/8
1/16
1/32
𝑒0
0.1856
0.1629
0.1573
𝑒1
0.0426
0.0178
0.0043
Finally, noting the definitions of 𝑢̃, 𝑢̃ℎ0 ,ℎ1 , 𝜃𝜀 , 𝑅ℎ 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ , and 𝜔𝜀 ,
and combining Lemma 3 with Lemmas 7-8 and Lemma 2.4
in [9], we have
󵄩
󵄩󵄩󵄩 ∇ (̃
󵄩󵄩󵄩 ∇ (𝑢𝜀 − 𝑢̂ℎ0 ,ℎ1 ,ℎ ) 󵄩󵄩󵄩
𝑢 − 𝑢̃ℎ0 ,ℎ1 ) 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩
≤
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩
󵄩󵄩𝐿2 (Ω) 󵄩󵄩
󵄩󵄩𝐿2 (Ω)
󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩 ∇ (𝜃𝜀 − 𝑅ℎ 𝜃̃𝜀ℎ0 ,ℎ1 ,ℎ ) 󵄩󵄩󵄩
󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩
󵄩
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩 2
󵄩𝐿 (Ω)
󵄩
󵄩󵄩
󵄩󵄩
𝜀
∇𝜔
󵄩
󵄩󵄩
󵄩󵄩
+ 󵄩󵄩󵄩󵄩
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩𝐿2 (Ω)
≤ 𝐶 (ℎ1 + ℎ0 + 𝜀 + ℎ2 ) |ln 𝜀|
󵄩
󵄩
× (󵄩󵄩󵄩󵄩𝑢0 ‖𝑊2,∞ (Ω) +󵄩󵄩󵄩󵄩 𝑢0 ‖𝐻4 (Ω) ) .
(65)
Table 2: Comparison of error order.
󵄩
󵄩󵄩
∗
ℎ0 ,ℎ1 󵄩
) 󵄩󵄩
󵄩󵄩 ∇(𝑢 − 𝑢̃
󵄩
󵄩󵄩
𝑂 (1)
󵄩󵄩 √dist(⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩 2
󵄩𝐿 (Ω)
󵄩
󵄩󵄩
∗
ℎ0 ,ℎ1 ,ℎ 󵄩
) 󵄩󵄩󵄩
󵄩󵄩󵄩 ∇(𝑢 − 𝑢̂
󵄩󵄩
𝑂 ((ℎ1 + ℎ0 + 𝜀 + ℎ2 )| ln 𝜀|)
󵄩󵄩
󵄩
󵄩󵄩 √dist(⋅, 𝜕Ω) + 𝜀 󵄩󵄩𝐿2 (Ω)
𝑓 (𝑥) = 𝑒𝑥1 +𝑥2 ,
𝑔 (𝑥) = 2 sin (𝑥1 ) + 4 cos (𝑥2 ) ,
2
2
Ω = {𝑥 | (𝑥1 − 0.5) + (𝑥1 − 0.5) < 0.25} .
(67)
Moreover, let
󵄩󵄩
󵄩
󵄩󵄩∇ (𝑢∗ − 𝑢̃ℎ0 ,ℎ1 ) /√dist (⋅, 𝜕Ω) + 𝜀󵄩󵄩󵄩 2
󵄩
󵄩𝐿 (Ω)
𝑒0 =
,
󵄩󵄩 ∗
󵄩
󵄩󵄩∇𝑢 /√dist (⋅, 𝜕Ω) + 𝜀󵄩󵄩󵄩 2
󵄩
󵄩𝐿 (Ω)
󵄩󵄩
󵄩
󵄩󵄩∇ (𝑢∗ − 𝑢̂ℎ0 ,ℎ1 ,ℎ ) /√dist (⋅, 𝜕Ω) + 𝜀󵄩󵄩󵄩 2
󵄩
󵄩𝐿 (Ω)
.
𝑒1 =
󵄩󵄩 ∗
󵄩
󵄩󵄩∇𝑢 /√dist (⋅, 𝜕Ω) + 𝜀󵄩󵄩󵄩 2
󵄩
󵄩𝐿 (Ω)
(68)
Theorem 9. Assuming that (C1) holds, then there exists 𝐶 such
that
󵄩󵄩
𝜀
ℎ0 ,ℎ1 ,ℎ 󵄩
) 󵄩󵄩󵄩󵄩
󵄩󵄩󵄩 ∇ (𝑢 − 𝑢̂
󵄩
󵄩󵄩
≤ 𝐶 [(ℎ1 + ℎ0 + 𝜀 + ℎ2 ) |ln 𝜀|]
󵄩󵄩 √dist (⋅, 𝜕Ω) + 𝜀 󵄩󵄩󵄩
󵄩󵄩𝐿2 (Ω)
󵄩󵄩
󵄩 󵄩
󵄩 󵄩
× (󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝑊2,∞ (Ω) + 󵄩󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩󵄩𝐻4 (Ω) ) .
(66)
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.
According to Table 2, 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.
4. Numerical Example
Acknowledgments
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 that 𝑢∗ denotes the finite
element solution of (1) in a fine mesh, and 𝑢̂ℎ0 ,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:
The authors would like to thank the referees for their valuable
suggestions and corrections, which contribute significantly to
the improvement of the paper. This research was financially
supported by the National Basic Research Program of China
(973 Program: 2011CB013800), and the National Natural
Science Foundation of China, with Grant nos. 11126132,
50838004, 50908167, and 11101311. The financial support of
research is gratefully acknowledged.
Combining the above lemmas, we can conclude the following result.
𝑎11 = 0.3 + 2𝑥1 (1 − 𝑥1 ) ,
𝑎12 = 𝑎21 = 𝑥1 (1 − 𝑥1 ) 𝑥2 (1 − 𝑥2 ) ,
𝑎22 = 0.1 + 2𝑥2 (1 − 𝑥2 ) ,
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