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

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 EQ FE schemes under a reasonable regularity of the exact solution u ∈ H5/2 Ω , which seem to be never discovered in the previous literature. The optimal L2-norm error estimate is also derived for EQ FE. At last, some numerical results are provided to verify the theoretical analysis.


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 O h 3/2 .The authors of 3 first obtained the error bound O h 3/2−ε for any ε > 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 O h .The authors of 7 obtained the same error estimate with order O h 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 9-11 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 O h 3/4 | log h| 1/4 if the displacement field is of H 2 regularity and also showed that if stronger but reasonable regularity is available u ∈ W 2,p , p > 2 , the above result can be improved to optimal order O h .The authors of 13 applied a class of Crouzeix-Raviart-type FEs to Signorini problem and obtained O h 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 O h 3/2 when the exact solution u ∈ H 5/2 Ω .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 Ω ⊂ R 2 is a bounded convex polygonal domain; K * is defined as follows: Ω ∇u∇v μuv dx dy, μ is a positive constant, f, v Ω fv dx dy, j v Γ d ψ v ds, and 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 u ∈ H 5/2 Ω .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 O h 3/2 is derived under the broken energy norm.In Section 3, the nonconforming EQ rot FE see 26 approximation is used, and the same superclose result is obtained under the energy norm; the optimal error estimate of L 2 -norm is also derived when u ∈ H 2 Ω .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.

The Equivalent Form and Conforming FE Scheme
It has been shown in 21, 22 that 1.1 is equivalent to 2.1 and 2.1 has the unique solution u in K * .It can be verified that ϕ t satisfies the following two properties: for all a, b ∈ R 1 , Let T h be a rectangular partition with a maximum size h in x, y plane, K ∈ T h a general element; V 1 h and V 2 h are the conforming bilinear FE space and the nonconforming EQ rot FE space.We denote by Π 1 h and Π 2 h the associated interpolation operators on V 1 h and V 2 h , respectively.In the meantime, we denote K i h by a convex set associated with K * in V i h i 1, 2 as follows: where F is an edge of K.The following two lemmas will play an important role in the FE analysis, which can be found in 14, 24 , respectively. where The corresponding conforming FE approximation version of 2.1 reads as

2.6
Theorem 2.3.Let u ∈ H 5/2 Ω be the exact solution of 1.1 and u h ∈ K 1 h the bilinear FE solution of 2.6 , then there holds here and later, c is a generic positive constant, which is independent of h, K, and u.
Proof.Subtracting 2.1 from 2.6 , then taking v v h in it, one can get Let ξ Π 1 h u − u h and η u − Π 1 h u.Taking v h ξ in the above equation, there yields By the definition of a v, v , we have

2.10
Noticing 2.3 , we have From 2.2 and Lemma 2.1, I 1 can be estimated as

2.12
Applying the interpolation theory and Lemma 2.2, we get

2.13
The desired result follows directly from the combination of 2.12 and 2.13 .

The Nonconforming FE Scheme
The corresponding nonconforming FE approximation scheme of 2.1 reads as

3.1
where a h u, v K K ∇u∇v μuv dx dy.First, we introduce the following Lemma 3.1, which can be found in 25 .
By using the similar technique in 26 , one now states and proves the following important conclusion.

Lemma 3.2. For all
x 0 h x , y 0 h y , and Z 4 x 0 −h x , y 0 h y be the four vertices of K, F i Z i Z i 1 i 1, 2, 3, 4, mod 4 .We define operators P 0 and P 0i as M i M.

3.5
By the definition of P 01 , we get

3.6
Noticing that v h − P 01 v h | F 1 equals v h − P 03 v h | F 3 and ∂v h /∂x is only dependent on x, we can derive that x, y 0 h y − ∂u ∂y x, y 0 − h y v h − P 01 v h dx By using the same technique as 14, 15 , M can be estimated as 3.8 Thus the desired result follows.
Theorem 3.3.Let u ∈ H 5/2 Ω be the exact solution of 1.1 and u h ∈ K 2 h the nonconforming FE solution of 3.1 .Then one has 3.9 Proof.Subtracting 2.1 from 3.1 gives For convenience, we still denote ξ

3.11
By Lemma 3.1, we can derive that

3.12
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 u ∈ H 2 Ω instead of u ∈ H 5/2 Ω , the consistency error can be estimated as which can be found in 26 .Then we can derive the following optimal error estimate:

3.14
Now we start to give the L 2 -norm estimate through a duality argument.
Theorem 3.5.Let u ∈ K 2 Ω and u h ∈ V 2 h be the solutions of 1.1 and 3.1 , respectively, there holds 3.15 Proof.Let ω ∈ H 2 Ω be the solution of the following auxiliary elliptic problem: 3.17

3.18
where h w ds, and J 3 K ∂K ∂w/∂n u − u h ds K ∂K ∂u/∂n w −Π 2 h w ds.These three terms can be estimated one by one as follows.By 3.14 , 3.17 , and the interpolation theory, J 1 can be estimated as

3.19
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 J 1 , J 2 , and J 3 .
Remark 3.6.As to the L 2 -norm error estimate of bilinear FE scheme, the readers may refer to 21, 22 .

The Global Superconvergence Result
In order to obtain the global superconvergence, we combine the four neighbouring elements K 1 , K 2 , K 3 , K 4 ∈ T h into one new rectangular element K 0 , whose four edges are L 1 , L 2 , L 3 , and L 4 .T 2h represents the corresponding new partition.For the conforming FE scheme, we construct the postprocessing operator Π 1 2h u| K 0 : C K 0 → P 2 K 0 as follows: in which Z j is the four vertices and four mid point of edges of K 0 .For the nonconforming FE scheme, we construct the postprocessing Π 2 2h operator as

4.2
It is easy to validate that the interpolation operator is well posed and has the following properties 23 :

Numerical Results
In this section, we will present an example to confirm the correctness of our theoretical analysis.In 1.1 , we choose Ω 0, 1 × 0, 1 with boundary ∂Ω The right hand term f 1.Since there may be no exact solution to the above problem, we use the conforming FE solution on a sufficient refined mesh h 1/256 as the reference solution.Then we compare the conforming and nonconforming FE solutions see Figure 1 on the coarser meshes h 1/2, 1/4, 1/8, 1/16, 1/32, 1/64 with the reference one in Tables 1 and 2.
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.

By 3 .
16 and Lemma 3.1, we can derive that

Figure 1 :
Figure 1: The conforming FE solution a and the nonconforming FE solution b on the 64 × 64 mesh.

Table 1 :
The error estimates for conforming FE scheme.

Table 2 :
The error estimates for nonconforming FE scheme.