A two-order and two-scale computation method for nonselfadjoint

Appl. Math. Mech. -Engl. Ed. 30(12), 1579–1588 (2009) DOI: 10.1007/s10483-009-1209-z c Shanghai University and Springer-Verlag 2009

Applied Mathematics and Mechanics (English Edition)

A two-order and two-scale computation method for nonselfadjoint elliptic problems with rapidly oscillatory coefficients ∗ Fang SU (苏芳)1 , Jun-zhi CUI (崔俊芝)2 , Zhan XU (徐湛)3 (1. Department of Mathematics and Systems Science, National University of Defense Technology, Changsha 410073, P. R. China; 2. Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China; 3. School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, P. R. China) (Communicated by Xing-ming GUO)

Abstract The purpose of this paper is to solve nonselfadjoint elliptic problems with rapidly oscillatory coefficients. A two-order and two-scale approximate solution expression for nonselfadjoint elliptic problems is considered, and the error estimation of the twoorder and two-scale approximate solution is derived. The numerical result shows that the presented approximation solution is effective. Key words nonselfadjoint elliptic problems, rapidly oscillatory coefficients, two-order and two-scale finite element method Chinese Library Classification O242.1 2000 Mathematics Subject Classification

65F10, 35P15

Introduction Porous or periodic media are often encountered in modern engineering[1-4] . The flow and transport through such kind of media can be represented by the boundary value problems of second-order nonselfadjoint elliptic equations with rapidly oscillatory coefficients. One of the main difficulties for numerically solving these kinds of problems is the too large computing scale. To reduce the large computing scale, the homogenization method was proposed theoretically by Lions, Babuska, and others[1,5-6] . The main idea of this method is to obtain an average field equation by properly constructing a local smoothing operator. As a result, one can solve numerically the homogenized equation in a coarse mesh. Bensoussan et al.[1] gave the homogenized equation and the first-order correction for the second-order elliptic equation with a periodic or quasi-periodic structure. Oleinik et al.[3] extended the homogenization method to the mixed ∗ Received May 10, 2009 / Revised Oct. 4, 2009 Project supported by the National Natural Science Foundation of China (No. 10590353) and the Science Research Project of National University of Defense Technology (No. JC09-02-05) Corresponding author Fang SU, Ph. D., E-mail: [email protected]

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Fang SU, Jun-zhi CUI, and Zhan XU

problem with a perforated domain for Dirichlet boundary conditions on the outer boundary part and Neumann conditions on the surface of cavities. Since the homogenization method and the first-order correction are not enough to describe the local fluctuation of some physical fields in some cases (refer to Section 3 in this paper), it is necessary to find the higher-order asymptotic expansion for considered solutions. Cui and Cao[7-11] proposed a higher-order asymptotic expansion for the Dirichlet problem of the elastic systems with rapidly oscillating coefficients over a domain formed by entirely basic configurations without cavity. Feng and Cui[12] proposed the higher-order asymptotic expansion for the problem under the conditions of coupled thermoelasticity for the structure of composite materials with a small periodic configuration. Li et al.[13-15] extended the higher-order method to predict the heat transfer performance of composite materials with random grain distributions. However, the methods above only consider selfadjoint elliptic problems. Chen and Cui[16] studied nonselfadjoint elliptic problems with rapidly oscillatory coefficients. In [16], the homogenized solutions and one-order and two-scale approximate solutions were considered. When the coefficients are distinct with O(102 ) upwards, the errors between approximate solutions and original solutions obviously increase. But the higher-order asymptotic expansion cannot be obtained by usual multiscale methods. Now, we introduce a two-order and two-scale approximate solution by constructing two correct terms to the auxiliary functions for eliminating disturbance. The remainder of this paper is organized as follows. In Section 1, we introduce the problem and some notations. In Section 2, a two-order and two-scale approximate solution is obtained, and the error estimation is presented. In the last section, some numerical results are reported, which strongly support the theoretical assertion of this paper.

1

Governing equations and the new method

2 ¯  Assume the bounded domain Ω ∈ R consists of entirely basic configurations, i.e., Ω = ¯ where  is a small positive number, (z + Q), z∈I 

I  = {z ∈ Z2 |(z + Q) ⊂ Ω}, Q = {ξ|0 < ξi < 1, i = 1, 2}. Consider the following problem:    L u = −div(a ∇u + α u ) + β  · ∇u + γ  u + λu = f u = 0 on ∂Ω,

in Ω, (1)

where f is a sufficiently smooth function, a = (aij (x)) is a bounded symmetric positive definite matrix valued functions with period , α = (αi (x)) and β  = (βi (x)) are vector valued functions with period , γ  (x) is a function with period , and λ > 0 is a sufficiently large real number such that the operator L is coercive, i.e., (L v, v) ≥ Cv21 ,

∀v ∈ H01 (Ω).

Let ξ = x/, a = (aij (ξ)) = (aij (x)), α = (αi (ξ)) = (αi (x)), β = (βi (ξ)) = (βi (x)), and γ(ξ) = γ  (x), and then aij (ξ), αi (ξ), βi (ξ), and γ(ξ) are periodic functions with period 1. Assume aij (ξ), αi (ξ) ∈ H 1 (Q), βi (ξ), γ(ξ) ∈ L∞ (Q). First, we introduce periodic functions M0 (ξ) and Nα1 (ξ) (α1 = 1, 2), which are defined by the following equations: ⎧ ∂αi (ξ) ∂M0 (ξ) ⎪ ∂  ⎨ aij (ξ) =− in Q, ∂ξi ∂ξj ∂ξi (2) ⎪ ⎩ M (ξ) = 0 on ∂Q, 0

A two-order and two-scale computation method for nonselfadjoint elliptic problems

⎧  ⎪ ⎨ ∂ aij (ξ) ∂Nα1 (ξ) = − ∂aiα1 (ξ) ∂ξi ∂ξj ∂ξi ⎪ ⎩ N (ξ) = 0 on ∂Q.

in Q,

1581

(3)

α1

Then, we define a constant matrix a0 = (a0ij ) by

 1 ∂Nj (ξ) 0 aij (ξ) + aik (ξ) dξ. aij = |Q| Q ∂ξk The constant vectors α0 = (α0i ) and β 0 = (βi0 ) are defined by

 1 ∂M0 (ξ) 0 αi (ξ) + aik (ξ) dξ, αi = |Q| Q ∂ξk

 1 ∂Ni (ξ) βi0 = βi (ξ) + βk (ξ) dξ. |Q| Q ∂ξk Also, we define a constant γ 0 by γ0 =

1 |Q|

 ∂M0 (ξ) γ(ξ) + βk (ξ) dξ. ∂ξk Q

Now, we introduce the following homogenization equation of (1):  − div(a0 ∇u0 + α0 u0 ) + β 0 · ∇u0 + γ 0 u0 + λu0 = f

in Ω,

0

u = 0 on ∂Ω.

(4)

For convenience, we present the notations gij , pi , qi , and r defined by ∂Nj − a0ij , ∂ξk ∂M0 pi = αi + aik − α0i , ∂ξk ∂Ni qi = βi + βk − βi0 , ∂ξk ∂M0 r = γ + βk − γ0. ∂ξk gij = aij + aik

From the definition of L , we know  ∂ 1 ∂  ∂ 1 ∂  ∂ 1 ∂ L = − aij − αi + + + ∂xi  ∂ξi ∂xj  ∂ξj ∂xi  ∂ξi  ∂ 1 ∂ +γ+λ + βi + ∂xi  ∂ξi = −2 L1 + −1 L2 + L3 , where the variables x and ξ are independent, and ∂ ∂  aij , L1 = − ∂ξi ∂ξj ∂  ∂ ∂ ∂ ∂ ∂  aij − aij − L2 = − αi + βi , ∂xi ∂ξj ∂ξi ∂xj ∂ξi ∂ξi ∂ ∂ ∂ ∂  aij − αi + βi + γ + λ. L3 = − ∂xi ∂xj ∂xi ∂xi

(5)

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2

Fang SU, Jun-zhi CUI, and Zhan XU

Two-order and two-scale approximate solution and error estimation

Following [16], we define the two-order and two-scale approximate expression of the solution u to the problem (1) by u2 = u0 (x) + u1 (x, ξ) + 2 u2 (x, ξ)  ∂u0 = u0 (x) +  M0 (ξ)u0 + Nα1 (ξ) ∂xα1  0 ∂u ∂u0 ∂ 2 u0 , + 2 C0 (ξ)u0 + Mα1 (ξ) + Dα1 (ξ) + Nα1 α2 (ξ) ∂xα1 ∂xα1 ∂xα1 ∂xα2

(6)

where u0 (x), M0 (ξ), and Nα1 (ξ) are defined by (4), (2), and (3), respectively. In this section, we first give the definitions of C0 (ξ), Mα1 (ξ), Dα1 (ξ), and Nα1 α2 (ξ) : ⎧  ⎪ ⎨ ∂ aij (ξ) ∂C0 (ξ) = −γ 0 + γ(ξ) + βi (ξ) ∂M0 (ξ) − ∂ (αi (ξ)M0 (ξ)) in Q, ∂ξi ∂ξj ∂ξi ∂ξi ⎪ ⎩ C0 (ξ) = 0 on ∂Q, ⎧  ⎪ ⎨ ∂ aij (ξ) ∂Mα1 (ξ) = α0α − αα1 (ξ) − aα1 j (ξ) ∂M0 (ξ) − ∂ (aiα1 (ξ)M0 (ξ)) in Q, 1 ∂ξi ∂ξj ∂ξj ∂ξi ⎪ ⎩ Mα1 (ξ) = 0 on ∂Q, ⎧  ⎪ ⎨ ∂ aij (ξ) ∂Dα1 (ξ) = −βα0 + βα1 (ξ) + βi (ξ) ∂Nα1 (ξ) − ∂ (αi (ξ)Nα1 (ξ)) in Q, 1 ∂ξi ∂ξj ∂ξi ∂ξi ⎪ ⎩ Dα1 (ξ) = 0 on ∂Q, ⎧  ⎪ ⎨ ∂ aij (ξ) ∂Nα1 α2 (ξ) = a0α α − aα1 α2 (ξ) − aα2 j (ξ) ∂Nα1 (ξ) − ∂ (aiα2 (ξ)Nα1 (ξ)) in Q, 1 2 ∂ξi ∂ξj ∂ξj ∂ξi ⎪ ⎩ Nα1 α2 (ξ) = 0 on ∂Q. We have given all definitions of the auxiliary functions of the two-order and two-scale approximate solutions. Then, we estimate the error of the approximate solution u2 (x). To this purpose, we give the following results. 4 (Ω) ∩ H01 (Ω), and then u − u2 satisfies the variational Lemma 2.1 Assume u0 ∈ W∞ equation (L (u − u2 ), v) = (f ∗ , v), ∀v ∈ H01 (Ω), (7) where f ∗ = gij +

∂ 2 u0 ∂u0 ∂u0 ∂ ∂ 2 u0 + pi − qi − ru0 + (aij Nk ) ∂xi ∂xj ∂xi ∂xi ∂ξi ∂xj ∂xk

∂ ∂ ∂u0 ∂u0 ∂ (aij M0 ) + (αi Nk ) + u0 (αi M0 ) ∂ξi ∂xj ∂xk ∂ξi ∂ξi

+u0

+

∂  ∂Mα1 ∂u0 ∂  ∂C0 aij + aij ∂ξi ∂ξj ∂ξi ∂ξj ∂xα1

∂  ∂Dα1 ∂u0 ∂  ∂Nα1 α2 ∂ 2 u0 aij aij + ∂ξi ∂ξj ∂xα1 ∂ξi ∂ξj ∂xα1 ∂xα2

+

 ∂u0 ∂xi

aij

∂u0 ∂ ∂C0 ∂ ∂C0 + (aij C0 ) + u0 (αi C0 ) − βi u0 ∂ξj ∂xj ∂ξi ∂ξi ∂ξi

A two-order and two-scale computation method for nonselfadjoint elliptic problems

1583

 ∂ 2 u0 ∂ 2 u0 ∂ ∂u0 ∂ ∂Mα1 ∂u0 ∂Mα1 + aij + (aij Mα1 ) + (αi Mα1 ) − βi ∂xα1 ∂xi ∂ξj ∂xα1 ∂xj ∂ξi ∂xα1 ∂ξi ∂xα1 ∂ξi +

 ∂ 2 u0 ∂ 2 u0 ∂ ∂u0 ∂ ∂Dα1 ∂u0 ∂Dα1 aij + (aij Dα1 ) + (αi Dα1 ) − βi ∂xα1 ∂xi ∂ξj ∂xα1 ∂xj ∂ξi ∂xα1 ∂ξi ∂xα1 ∂ξi 

+

∂Nα1 α2 ∂ ∂ 3 u0 ∂ 3 u0 aij + (aij Nα1 α2 ) ∂xα1 ∂xα2 ∂xi ∂ξj ∂xα1 ∂xα2 ∂xj ∂ξi

∂ 2 u0 ∂ ∂ 2 u0 ∂Nα1 α2 (αi Nα1 α2 ) − βi ∂xα1 ∂xα2 ∂ξi ∂xα1 ∂xα2 ∂ξi  2 0 ∂u0 ∂ u +M0 aij + (αi − βi ) − (γ + λ)u0 ∂xi ∂xj ∂xi  ∂ 3 u0 ∂ 2 u0 ∂u0 +Nk aij + (αi − βi ) − (γ + λ) ∂xi ∂xj ∂xk ∂xi ∂xk ∂xk  ∂ 2 u0 ∂u0 +2 C0 aij + (αi − βi ) − (γ + λ)u0 ∂xi ∂xj ∂xi  ∂ 3 u0 ∂ 2 u0 ∂u0 +2 Mk aij + (αi − βi ) − (γ + λ) ∂xi ∂xj ∂xk ∂xi ∂xk ∂xk  3 0 2 0 ∂ u ∂ u ∂u0 +2 Dk aij + (αi − βi ) − (γ + λ) ∂xi ∂xj ∂xk ∂xi ∂xk ∂xk  4 0 3 0 ∂ u ∂ u ∂ 2 u0 . +2 Nks aij + (αi − βi ) − (γ + λ) ∂xi ∂xj ∂xk ∂xs ∂xi ∂xk ∂xs ∂xk ∂xs +

Lemma 2.2 |M0 |2 ≤ C−2 ;

M0 0 ≤ C1 |M0 |1 ≤ C2 , Nα1 0 ≤ C1 |Nα1 |1 ≤ C2 , C0 0 ≤ C1 |C0 |1 ≤ C2 ,

|Nα1 |2 ≤ C−2 ;

|C0 |2 ≤ C−2 ;

Mα1 0 ≤ C1 |Mα1 |1 ≤ C2 ,

|Mα1 |2 ≤ C−2 ;

Dα1 0 ≤ C1 |Dα1 |1 ≤ C2 ,

|Dα1 |2 ≤ C−2 ;

Nα1 α2 0 ≤ C1 |Nα1 α2 |1 ≤ C2 , Lemma 2.3

|Nα1 α2 |2 ≤ C−2 .

For any v ∈ H 1 (Ω), the following inequalities hold :



gij vdx ≤ C1 |v|1 ,

Ω



pi vdx ≤ C1 |v|1 ,

Ω



qi vdx ≤ C1 |v|1 ,

Ω



rvdx ≤ C1 |v|1 .

Ω

The proof of Lemmas 2.1, 2.2, and 2.3 can be found in [16]. Now, we prove the main result of this section.

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Theorem 2.1

Fang SU, Jun-zhi CUI, and Zhan XU 4 Assuming u0 (x) ∈ W∞ (Ω) ∩ H01 (Ω), it holds 4 (Ω) , u (x) − u2 (x)1 ≤ Cu0 W∞

(8)

where C > 0 has nothing with , u , and u0 . Proof Since the operator L is coercive and u − u2 ∈ H01 (Ω), from (7), it follows that Cu − u2 21 ≤ (L (u − u2 ), u − u2 ) = (f ∗ , u − u2 ).

(9)

Using integration by parts and some theorems in [17], we deduce that (f ∗ , u − u2 )





∂ 2 u0 ∂u0 



gij (u − u2 )dx + pi (u − u2 )dx ≤ ∂x ∂x ∂x i j i Ω Ω



0 ∂u



+ qi (u − u2 )dx + ru0 (u − u2 )dx ∂xi Ω Ω



∂ 2 u0 ∂ ∂ ∂u0 



  +  aij Nk ( (u − u2 ))dx +  aij M0 ( (u − u2 ))dx ∂xi ∂xj ∂xk ∂xi ∂xj Ω Ω





0 ∂ ∂u ∂



αi Nk ( (u − u2 ))dx +  αi M0 (u0 (u − u2 ))dx +  ∂x ∂x ∂x i k i Ω Ω





∂C0 ∂ ∂Mα1 ∂ ∂u0 



+  aij (u0 (u − u2 ))dx +  aij ( (u − u2 ))dx ∂ξ ∂x ∂ξ ∂x ∂x j i j i α Ω Ω 1



0 ∂D ∂ ∂N ∂ ∂u ∂ 2 u0



α1 α1 α2 +  aij ( (u − u2 ))dx +  aij ( (u − u2 ))dx ∂ξj ∂xi ∂xα1 ∂ξj ∂xi ∂xα1 ∂xα2 Ω Ω

 ∂u0 ∂C0 ∂ ∂C0  ∂u0 ∂

(u − u2 )dx aij + (aij C0 ) + u0 (αi C0 ) − βi u0 +  ∂ξj ∂xj ∂ξi ∂ξi ∂ξi Ω ∂xi

 ∂ 2 u0 ∂Mα1 ∂ 2 u0 ∂

+  aij + (aij Mα1 ) ∂ξj ∂xα1 ∂xj ∂ξi Ω ∂xα1 ∂xi

∂u0 ∂ ∂u0 ∂Mα1 

+ (u − u2 )dx (αi Mα1 ) − βi ∂xα1 ∂ξi ∂xα1 ∂ξi

 ∂ 2 u0 ∂ 2 u0 ∂ ∂Dα1

+  aij + (aij Dα1 ) ∂ξj ∂xα1 ∂xj ∂ξi Ω ∂xα1 ∂xi

∂u0 ∂ ∂u0 ∂Dα1 

+ (u − u2 )dx (αi Dα1 ) − βi ∂xα1 ∂ξi ∂xα1 ∂ξi

 ∂ 3 u0 ∂Nα1 α2 ∂ ∂ 3 u0

+  aij + (aij Nα1 α2 ) ∂ξj ∂xα1 ∂xα2 ∂xj ∂ξi Ω ∂xα1 ∂xα2 ∂xi

∂ 2 u0 ∂ ∂ 2 u0 ∂Nα1 α2 

+ (u − u2 )dx (αi Nα1 α2 ) − βi ∂xα1 ∂xα2 ∂ξi ∂xα1 ∂xα2 ∂ξi

 2 0 ∂u0 ∂ u

M0 aij + (αi − βi ) − (γ + λ)u0 (u − u2 )dx +  ∂x ∂x ∂x i j i Ω

 3 0 ∂ u ∂ 2 u0 ∂u0 

(u − u2 )dx Nα1 aij + (αi − βi ) − (γ + λ) +  ∂x ∂x ∂x ∂x ∂x ∂x α1 i j α1 i α1 Ω

A two-order and two-scale computation method for nonselfadjoint elliptic problems

1585



 ∂u0 ∂ 2 u0

+ 2 C0 aij + (αi − βi ) − (γ + λ)u0 (u − u2 )dx ∂xi ∂xj ∂xi Ω

 ∂ 3 u0 ∂ 2 u0 ∂u0 

+ 2 (u − u2 )dx Mα1 aij + (αi − βi ) − (γ + λ) ∂xα1 ∂xi ∂xj ∂xα1 ∂xi ∂xα1 Ω

 ∂ 3 u0 ∂ 2 u0 ∂u0 

2 + (u − u2 )dx Dα1 aij + (αi − βi ) − (γ + λ) ∂xα1 ∂xi ∂xj ∂xα1 ∂xi ∂xα1 Ω

 ∂ 4 u0 ∂ 3 u0 2 + Nα1 α2 aij + (αi − βi ) ∂xα1 ∂xα2 ∂xi ∂xj ∂xα1 ∂xα2 ∂xi Ω

∂ 2 u0 

(u − u2 )dx − (γ + λ) ∂xα1 ∂xα2   0   0   2 |u − u |1 + Nk 0 u W 3 u − u 1 + M0 0 u W 2 u − u 1 ≤ C(u0 W∞ 2 2 2 ∞ ∞   0   2 u − u 1 + Mα 1 u W 3 u − u 1 + C0 1 u0 W∞ 2 2 1 ∞   0   3 u − u 1 + Nα α 1 u W 4 u − u 1 ) + Dα1 1 u0 W∞ 1 2 2 2 ∞   4 u − u 1 . ≤ Cu0 W∞ 2

From Lemmas 2.2 and 2.3, combining the above inequality with (9) gives 4 . u − u2 1 ≤ Cu0 W∞

3

Numerical results

Consider the Dirichlet boundary value problem (1), where Ω is shown in Fig. 1, the periodicity cell Q is shown in Fig. 2, and  = 18 . We consider two cases as follows: Case 1 : Case 2 :

1 , 8 1 = , 8

=

1 δij , 1 000 1 δij , = 500

aij0 = δij , aij1 =

f (x) = 105 (x1 x2 (1 − x1 )(1 − x2 ))3 .

aij0 = δij , aij1

f (x) = 10 sin(πx1 ) sin(πx2 ).

y 1

1 2/3

(1) aijhk (0) aijhk

1/3

0

1

Fig. 1

0

x

Domain Ω

Fig. 2

Here, δij =

 1 0

1/3

2/3

1

Unit cell Q = [0, 1]2

if i = j, if i =

j.

Since it is difficult to find the analytic solution to (1), we have to replace u (x) with its finite element (FE) solution on a very refined mesh. Now, we implement the quadrangle partition for

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Fang SU, Jun-zhi CUI, and Zhan XU

Ω such that the discontinuities of the coefficients aij coincide with the sides of the quadrangle. The numbers of the quadrangles and nodes are shown in Table 1. Various relative errors between different order solutions and the FE solution on a refined mesh are shown in Table 2. Here, e0 = urm − u0 , e1 = urm − u1 , e2 = urm − u2 , u0 (x) is the FE solution of the homogenized equation, urm (x) is the FE solution of (1) on a refined mesh, and u1 (x), u2 (x) are the first-order and the second-order multiscale FE solutions. Tu2 is the computation time of the second-order multiscale FE solution, and Turm is the computation time of the FE solution on a very refined mesh. Here, u2 = u0 (x) + u1 (x, ξ) + 2 u2 (x, ξ)   ∂u0 ∂u0 + 2 C0 (ξ)u0 + Mα1 (ξ) = u0 (x) +  M0 (ξ)u0 + Nα1 (ξ) ∂xα1 ∂xα1 0 2 0 ∂u ∂ u (α1 , α2 = 1, 2). + Dα1 (ξ) + Nα1 α2 (ξ) ∂xα1 ∂xα1 ∂xα2

(10)

Figures 3 and 4 show the interpolation figures of u0 , u1 , u2 , and urm on the refined mesh for these two cases, respectively. From these figures, we can see that the two-order and two-scale Table 1

Comparison of the numbers of elements and nodes

Original equation,  =

1 8

Unit cell

Homogenized equation

Element number

9 216

576

2 304

Node number

37 249

2 401

9 409

Table 2

Comparison of the computing results of L2 norm, H 1 norm, and time

e0 L2 urm L2

e1 L2 urm L2

e2 L2 urm L2

e0 H 1 urm H 1

e1 H 1 urm H 1

e2 H 1 urm H 1

Tu2 /s

Turm /s

Case 1

0.330 89

0.330 13

0.035 69

0.330 93

0.330 16

0.035 31

19.671

57.516 0

Case 2

0.309 45

0.308 62

0.033 27

0.309 48

0.308 64

0.032 91

16.031

47.187 0

Fig. 3

Interpolation figures on the refined mesh in Case 1

A two-order and two-scale computation method for nonselfadjoint elliptic problems

Fig. 4

1587

Interpolation figures on the refined mesh in Case 2

approximate solution is in good agreement with the FE solution on a refined mesh. However, the homogenization solution and the first-order approximate solution have less effect approaching the refined-mesh FE solution. From Tables 1 and 2, we can see that the mesh partition numbers of the two-order and two-scale approximate solution are much less than that of the refined FE solution, and the computation time is also much less than that of the refined FE solution. This means that the approximate solution we present can greatly save the computer memory and CPU time, which is very important in engineering computations. All these information shows that the two-order and two-scale solution is effective to approximate nonselfadjoint elliptic problems with rapidly oscillatory coefficients.

References [1] Bensoussan, A., Lions, J., and Papanicolaou, G. Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978) [2] Hornung, U. Homogenization and Porous Media, Springer-Verlag, New York (1997) [3] Oleinik, O. A., Shamaev, A. S., and Yosifian, G. A. Mathematical Problems in Elasticity and Homogenization, North-Holland (1992) [4] Zhikov, V. V., Kozlov, S. M., and Oleinik, O. A. Homogenization of Differential Operators and Interal Functionals, Springer-Verlag, Berlin (1994) [5] Bourget, J. F. and Iria-Laboria. Numerical experiments to the homogenization method for operators with periodic coefficients. Lecture Notes in Mathematics 705, 330–356 (1977) [6] Cioranescu, D. and Donato, P. An Introduction to Homogenization, Oxford University Press, Oxford (1999) [7] Cui, J. Z. and Cao, L. Q. Two-scale asymptotic analysis methods for a class of elliptic boundary value problems with small periodic coefficients. Math. Numer. Sinica 21(1), 19–28 (1999) [8] Cao, L. Q., Cui, J. Z., and Luo, J. L. Multiscale asymptotic expansion and a post-processing algorithm for second-order elliptic problems with highly oscillatory coefficients over general convex domains. J. Comp. Appl. Math. 157(1), 1–29 (2003)

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[9] Cao, L. Q. and Cui, J. Z. Finite element computation for elastic structures of composite materials formed by entirely basic configuration. J. Num. Math. Appl. 20, 25–37 (1998) [10] Cao, L. Q., Cui, J. Z., and Zhu, D. C. Multiscale asymptotic analysis and numerical simulation for the second order Helmholtz with rapidly oscillating coefficients over general convex domains. SIAM J. Numer. Anal. 40(2), 543–577 (2003) [11] Cao, L. Q. and Cui, J. Z. Homogenization method for the quasi-periodic structures of composite materials. Math. Numer. Sinica 21(3), 331–344 (1999) [12] Feng, Y. P. and Cui, J. Z. Multi-scale analysis and FE computation for the structure of composite materials with small periodic configurarion under condition of coupled thermoelasticity. Int. J. Numer. Meth. Engng. 60(11), 1879–1910 (2004) [13] Li, Y. Y. and Cui, J. Z. Two-scale analysis method for predicting heat transfer performance of composite materials with random grain distribution. Science in China Series A: Mathematics 47(1), 101–110 (2004) [14] Cui, J. Z. and Yu, X. G. A two-scale method for identifying mechanical parameters of composite materials with periodic configuration. Acta Mechanica Sinica 22(6), 581–594 (2006) [15] Yu, X. G. and Cui, J. Z. The prediction on mechanical properties of 4-step braided composites via two-scale method. Compos. Sci. Technol. 67(3-4), 471–480 (2007) [16] Chen, J. R. and Cui, J. Z. Two-scale finite element method for nonselfadjoint elliptic problems with rapidly oscillatory coefficients. Appl. Math. Comp. 150(2), 585–601 (2004) [17] Adams, R. A. Sobolev Space, Academic Press, New York (1975)