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.
In this paper, we consider the following elliptic boundary value problem with rapidly oscillatory coefficients: where is a smooth-bounded domain, is symmetric and satisfies 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 [1–4] 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 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 , 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 ) can be obtained.
Lemma 1. Assume that is the solution of (1) and is the finite element solution . For all , there exists a constant such that where is the homogenization solution of (1), and is the distance between the point and the boundary .
Unfortunately, the needed CPU time of the method presented in  is . 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 , and the capital letter (with or without subscripts) denotes a positive constant, which is independent of the small parameter and the mesh size (with or without subscripts).
2. An Improved Multiscale Finite Element Method
Firstly, let us simply recall the homogenization method described in .
2.1. Homogenization Method
Let be a 1-periodic function, which satisfies Then, the matrix can be obtained by
The first-order approximation of can be written as where 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 and the boundary corrector term , respectively, and furthermore give the multiscale finite element solution of (1).
2.2. Finite Element Approximation of
Let be a quasiuniform triangular partition of with the mesh size . denotes the conforming finite element spaces with respect to , and . The finite element scheme of (4) is to find such that Then, the numerical approximation of can be calculated by
Let be a quasiuniform triangular partition of with the mesh size and satisfy where is the area of the triangular element . denotes the corresponding conforming finite element spaces, and .
The finite element approximation of the homogenization problem (7) is to find such that
Furthermore, we turn to the computation of and . Let be the set of all nodal points of the mesh . Define by the following: () for all , the value of at the nodal point is the average of in all elements including ,() is a piecewise linear function in every element.
Therefore, we have a numerical approximation of which is defined by
2.3. Finite Element Approximation of
Let be a positive integer satisfying Then, the domain can be divided by and
Let be the regular triangular partition of with the mesh size and satisfy where and are independent of and , denotes the area of , and are the length of two edges of . Let be the set of all nodal points in , , and let be the conforming finite element spaces with respect to ; we define
Then, the finite element approximation of is to find such that
2.4. Multiscale Finite Element Approximation of
For any , we define the linear operator by Then, the multiscale finite element approximation of can be defined by
3. Superconvergence Result of
Firstly, we have the following assumption.
Assumption C1. The functions , and the homogenization solution .
Then, we introduce the following lemma.
From Lemma 2, one can easily deduce.
Lemma 3. Assuming that (C1) holds, then there exists , which is independent of such that
Proof. For , we define
Then, the upper bound of and can be estimated, respectively.
Using the result from (8) and , we have Then, we estimate . Obviously, can be divided into where satisfies and satisfies Using the result from Lemma 2 and (28), we have
Let , and using the result from (29), we have Following the same line of  (1992, Theorem 1.2, pages 124–128), we have which indicates Combining (30) with (33), we can derive (24) immediately.
Considering the proof of (23), can be divided into where satisfies and satisfies
In view of Lemma 2, we have
Following the same line of  (1992, Theorem 1.2, pages 124–128), we have Combining (37) with (38), we can conclude the result of this lemma.
Assuming that is defined as (16), and let and be the linear finite element approximation and the linear interpolation of with respect to , respectively. Then, we have the following.
Lemma 4. Assuming that (C1) holds, then there exists such that
Furthermore, we can obtain the following lemma.
Lemma 5. Assuming that (C1) holds, then there exists such that
Based on the previous lemmas, the estimate of can be given as follows.
Lemma 6. Assuming that (C1) holds, then there exists such that
Proof. Assuming that satisfies
and is the linear interpolation of on .
Then, we divide into
Firstly, considering the first item of the right-hand side of (48) and assuming that 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), can be divided into
Similarly, we have Combining Lemma 6 with (55)–(58), we have (46).
Next, using the extrapolation technique , we are in a position to estimate instead of
Lemma 7. Assuming that (C1) holds, then there exists such that
Proof. Let and be defined as above. Assuming that is the linear interpolation of on , we have Then, (61) can be easily derived.
Next, we turn to estimate .
Lemma 8. Assuming that (C1) holds, then there exists such that
Combining the above lemmas, we can conclude the following result.
Theorem 9. Assuming that (C1) holds, then there exists such that
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 that denotes the finite element solution of (1) in a fine mesh, and , 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: Moreover, let In Table 1, the numerical results of the multiscale method for and 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 is very small. If we only need to get a good numerical solution for problem (1) in Sobolev space , 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.
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.
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