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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 156095, 12 pages
http://dx.doi.org/10.1155/2012/156095
Research Article

Superconvergence Analysis of Finite Element Method for a Second-Type Variational Inequality

1Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China
2Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China
3Department of Mathematics, Tongji University, Shanghai 200092, China

Received 10 May 2012; Accepted 14 October 2012

Academic Editor: Song Cen

Copyright © 2012 Dongyang Shi 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.

Abstract

This paper studies the finite element (FE) approximation to a second-type variational inequality. The supe rclose and superconvergence results are obtained for conforming bilinear FE and nonconforming FE schemes under a reasonable regularity of the exact solution , which seem to be never discovered in the previous literature. The optimal -norm error estimate is also derived for FE. At last, some numerical results are provided to verify the theoretical analysis.

1. Introduction

Variational inequality (VI) theory has been playing an important role in the obstacle problem, contact problem, elasticity problem, and so on [1]. FE methods for solving VI problems have attracted more and more attentions. For example, as regards to the first type-VI case, the authors of [2] used piecewise quadratic FE to approximate the obstacle problem and suggested the error order between the FE solution and the exact solution should be . The authors of [3] first obtained the error bound (for any 0) for the above FE when the obstacle vanished. Then through a detailed analysis, the authors of [4] obtained the same error bound as the ones of [3] under the hypothesis that the free boundary has finite length. Later, the authors of [5] obtained the same error bound as the ones of [3] for the same element without the hypothesis of finite length of the free boundary. Furthermore, [6] investigated the Wilson’s element approximation to the obstacle problem and derived the error bound with order . The authors of [7] obtained the same error estimate with order on anisotropic meshes by making the full use of the bilinear part of the Wilson element, which relaxed the interpolation restriction and simplified the proofs of [5, 6]. Recently, the authors of [8] proposed a class of nonconforming FE methods for the parabolic obstacle VI problem with moving grids and obtained the optimal error estimates on anisotropic meshes. On the other hand, some studies [911] have been devoted to FE approximation to Signorini problem which arises in contact problems and obtained different error estimates under different assumptions. The authors of [12] derived the convergence result of if the displacement field is of regularity and also showed that if stronger but reasonable regularity is available , the above result can be improved to optimal order . The authors of [13] applied a class of Crouzeix-Raviart-type FEs to Signorini problem and obtained order estimate on anisotropic meshes. The authors of [14] used the bilinear FE to approximate the frictionless Signorini problem by virtue of the information on the contact zone and derived a superconvergence rate of when the exact solution . The authors of [15] presented the nonconforming Carey FE approximation to the problem of [14] and obtained the same convergence and superconvergence results are also obtained.

For the second type case, the authors of [16] proposed a Galerkin FE schemes for deriving a posteriori error estimates for a friction problem and a model flow of Bingham fluid. The authors of [17] considered the FE approximation to the plate contact problem and obtained some error estimates by employing the technique of mesh dependent norm.

In this paper, we will consider the following second type-VI problem [18, 19]: where is a bounded convex polygonal domain; is defined as follows: in which and , . is a positive constant, , and and and are positive constants. (1.1) may describe many practical engineering problems and attracts many scholars’ interests. For instance, the authors of [20] obtained the error estimate of energy norm for linear FE; the authors of [21] got the error estimate in energy norm by improving the result of [20] for ; the authors of [22] derived the optimal error estimate of norm and error estimate of energy norm when . But all the above studies mentioned above only paid attention to the convergence analysis for the conforming FE with no consideration on the superconvergence property, although it is surely an interesting and useful phenomenon in scientific computing of industrial problems [23].

In this paper, as a first attempt, we try to investigate the superconvergence of conforming and nonconforming FE schemes for problem (1.1) with a reasonable assumption of . The rest of this paper is organized as follows. In the next section, we give the equivalent form of (1.1) and the conforming bilinear FE (see [14]) approximation of (1.1). Moreover, superclose result of is derived under the broken energy norm. In Section 3, the nonconforming FE (see [26]) approximation is used, and the same superclose result is obtained under the energy norm; the optimal error estimate of -norm is also derived when . In Section 4, we construct a postprocessing interpolation operator to obtain the superconvergence properties. In Section 5, we present some numerical results to verify the theoretical analysis.

2. The Equivalent Form and Conforming FE Scheme

It has been shown in [21, 22] that (1.1) is equivalent to and (2.1) has the unique solution in . It can be verified that satisfies the following two properties: for all , Let be a rectangular partition with a maximum size in plane, a general element; and are the conforming bilinear FE space and the nonconforming FE space. We denote by and the associated interpolation operators on and , respectively. In the meantime, we denote by a convex set associated with in ( as follows: where is an edge of . The following two lemmas will play an important role in the FE analysis, which can be found in [14, 24], respectively.

Lemma 2.1. For all , there holds .

Lemma 2.2. Let , then for , there holds where .

The corresponding conforming FE approximation version of (2.1) reads as

Theorem 2.3. Let be the exact solution of (1.1) and the bilinear FE solution of (2.6), then there holds here and later, is a generic positive constant, which is independent of , , and .

Proof. Subtracting (2.1) from (2.6), then taking in it, one can get Let and . Taking in the above equation, there yields By the definition of , we have Noticing (2.3), we have ; thus in which .
From (2.2) and Lemma 2.1, can be estimated as
Applying the interpolation theory and Lemma 2.2, we get
The desired result follows directly from the combination of (2.12) and (2.13).

3. The Nonconforming FE Scheme

The corresponding nonconforming FE approximation scheme of (2.1) reads as where .

First, we introduce the following Lemma 3.1, which can be found in [25].

Lemma 3.1 (see [25]). If , one has

By using the similar technique in [26], one now states and proves the following important conclusion.

Lemma 3.2. For all , there holds where .

Proof. Let , and be the four vertices of , . We define operators and as respectively, where and denote the measures of and , respectively.
It can be checked that
By the definition of , we get
Noticing that equals and is only dependent on , we can derive that
Similarly, . By using the same technique as [14, 15], can be estimated as
Thus the desired result follows.

Theorem 3.3. Let be the exact solution of (1.1) and the nonconforming FE solution of (3.1). Then one has

Proof. Subtracting (2.1) from (3.1) gives
For convenience, we still denote and . Taking in (3.10) yields
By Lemma 3.1, we can derive that
Noticing Lemma 3.2 and using the analysis technique of Theorem 2.3, one can immediately get the desired result.

Remark 3.4. As a by-product, if we assume instead of , the consistency error can be estimated as which can be found in [26]. Then we can derive the following optimal error estimate:
Now we start to give the -norm estimate through a duality argument.

Theorem 3.5. Let and be the solutions of (1.1) and (3.1), respectively, there holds

Proof. Let be the solution of the following auxiliary elliptic problem: in which , then
By (3.16) and Lemma 3.1, we can derive that where . These three terms can be estimated one by one as follows. By (3.14), (3.17), and the interpolation theory, can be estimated as
By the trace theorem, (3.17), and Lemma 2.1, one gets By (3.13), (3.14), and (3.17), we have
The desired result follows the combination of the above estimates of , , and .

Remark 3.6. As to the -norm error estimate of bilinear FE scheme, the readers may refer to [21, 22].

4. The Global Superconvergence Result

In order to obtain the global superconvergence, we combine the four neighbouring elements into one new rectangular element , whose four edges are , and . represents the corresponding new partition. For the conforming FE scheme, we construct the postprocessing operator as follows: in which is the four vertices and four mid point of edges of . For the nonconforming FE scheme, we construct the postprocessing operator as

It is easy to validate that the interpolation operator is well posed and has the following properties [23]:

Theorem 4.1. If is the exact solution of (1.1), is the conforming or nonconforming FE solution. The following superconvergence result holds.

Proof. By (4.3), one gets
Noticing , the proof is completed.

5. Numerical Results

In this section, we will present an example to confirm the correctness of our theoretical analysis. In (1.1), we choose with boundary . The right hand term . Since there may be no exact solution to the above problem, we use the conforming FE solution on a sufficient refined mesh as the reference solution. Then we compare the conforming and nonconforming FE solutions (see Figure 1) on the coarser meshes with the reference one in Tables 1 and 2.

tab1
Table 1: The error estimates for conforming FE scheme.
tab2
Table 2: The error estimates for nonconforming FE scheme.
fig1
Figure 1: The conforming FE solution (a) and the nonconforming FE solution (b) on the mesh.

From the above tables, we can see that the conforming and nonconforming FE solutions both converge. At the same time, the superconvergence results in our experiments are a little better than the theoretical ones. We may explain this phenomenon with some special properties of this nonconforming FE that we have not discovered.

Acknowledgments

The first author was supported by the National Natural Science Foundation of China under Grant 10971203. The third author was supported by the National Natural Science Foundation of China under Grant 11126132. The authors would like to thank the referees for their valuable suggestions and corrections, which contribute significantly to the improvement of the paper.

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