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

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.


Introduction
In this paper, we consider the following elliptic boundary value problem with rapidly oscillatory coefficients: where Ω ⊂ R 2 is a smooth-bounded domain,   () : R 2 → R is symmetric and satisfies (1)           2 ≤   ()     ≤  −1         2 , ∀ ∈ R 2 , ∃ ∈ (0, 1] , (2)   ( +   ) =   () , ∀ ∈ R 2 , ∃  ∈  2 , 1 ≤ ,  ≤ , (3)             1 (R 2 ) ≤ , ∃ > 0, where  = /,  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][2][3][4] about the numerical methods of (1).A large amount of examples and applications can also be found in the classical books [5][6][7][8], where the formal asymptotic expansions for the limit solution are deduced when  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 that   is the solution of (1) and ûℎ 0 ,ℎ 1 ,ℎ is the finite element solution [9].For all , 1 <  < +∞, there exists a constant  such that where  0 is the homogenization solution of (1), and dist(, Ω) is the distance between the point  and the boundary Ω.
Unfortunately, the needed CPU time of the method presented in [9] is ( 1− ℎ − ).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   ().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  = { | 0 <   < 1,  = 1, 2}, and the capital letter  (with or without subscripts) denotes a positive constant, which is independent of the small parameter  and the mesh size ℎ (with or without subscripts).

An Improved Multiscale Finite Element Method
Firstly, let us simply recall the homogenization method described in [5].

Homogenization Method.
Let   ()( = 1, 2) be a 1periodic function, which satisfies Then, the matrix â = (â  ) 2 × 2 can be obtained by The first-order approximation of   () can be written as where  0 () satisfies the homogenization problem The boundary corrector term of the homogenization method   is defined by In the next two subsections, we will compute numerically the first-order approximation ũ and the boundary corrector term   , respectively, and furthermore give the multiscale finite element solution of (1).

Finite Element Approximation of ũ.
Let T ℎ 0 be a quasiuniform triangular partition of  with the mesh size ℎ 0 . ℎ 0 denotes the conforming  1 finite element spaces with respect to T ℎ 0 , and  ℎ 0 0 =  ℎ 0 ∩  1 0 ().The finite element scheme of (4) is to find Then, the numerical approximation âℎ 0  of â can be calculated by Let T ℎ 1 be a quasiuniform triangular partition of Ω with the mesh size ℎ 1 and satisfy min where    is the area of the triangular element   . ℎ 1  denotes the corresponding conforming  1 finite element spaces, and The finite element approximation  ℎ 0 ,ℎ 1 0 of the homogenization problem (7) is to find Furthermore, we turn to the computation of  ℎ 0 ,ℎ 1 0 ()/   and ũ().Let Σ ℎ 1 be the set of all nodal points of the mesh T ℎ 1 .Define  ℎ 0 ,ℎ 1  ( = 1, 2) by the following: Therefore, we have a numerical approximation ũℎ 0 ,ℎ 1 () of ũ() which is defined by (13)

Finite Element Approximation of 𝜃 𝜀 (𝑥).
Let  be a positive integer satisfying Then, the domain Ω can be divided by Ω = ⋃  =0 Ω  and Let T ℎ be the regular triangular partition of Ω with the mesh size ℎ and satisfy where  and  are independent of  and   ,    denotes the area of   , and   ,   are the length of two edges of   .Let Σ ℎ be the set of all nodal points in T ℎ , Σ ℎ  = Σ ℎ ∩ Ω, and let  ℎ be the conforming  1 finite element spaces with respect to T ℎ ; we define Then, the finite element approximation of   is to find 2.4.Multiscale Finite Element Approximation of   .For any  ∈  ℎ (Ω), we define the linear operator  ℎ by Then, the multiscale finite element approximation ûℎ 0 ,ℎ 1 ,ℎ () of   () can be defined by 3. Superconvergence Result of ûℎ 0 ,ℎ 1 ,ℎ Firstly, we have the following assumption.
Assuming that (C1) holds, then there exists , which is independent of  and  such that From Lemma 2, one can easily deduce.
Considering the proof of (23),    can be divided into where v ,1 satisfies and v ,2 satisfies In view of Lemma 2, we have      v Following the same line of [5] (1992, Theorem 1.2, pages 124-128), we have Combining ( 37) with (38), we can conclude the result of this lemma.
Assuming that T ℎ is defined as ( 16), and let  ℎ  and    be the linear finite element approximation and the linear interpolation of   with respect to T ℎ , respectively.Then, we have the following.

𝜀
is the linear interpolation of θℎ 0 ,ℎ Firstly, considering the first item of the right-hand side of (48) and assuming that  ℎ 0 ,ℎ 1  () satisfies the problem we have Using the same method of Lemma 2, we have Then, we have Combining ( 52) with (54), we have Next, considering the second item of the right-hand side of (48), θ,ℎ 0 ,ℎ Similarly, we have (64) Combining the above lemmas, we can conclude the following result.(66)

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 that  * denotes the finite element solution of (1) in a fine mesh, and ûℎ 0 ,ℎ 1 ,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:  (68) In Table 1, the numerical results of the multiscale method for  0 and  1 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 ∇  () can effectively be computed for problem (1) by using the above method, even if dist(, Ω) is very small.If we only need to get a good numerical solution for problem (1) in Sobolev space  1 (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.