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

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 EQ^rot FE schemes under a reasonable regularity of the exact solution 𝑢∈𝐻^5/2(Ω), which seem to be never discovered in the previous literature. The optimal 𝐿²-norm error estimate is also derived for EQ^rot 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 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|logℎ|^1/4) if the displacement field is of 𝐻² 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]: find𝑢∈𝐾^∗𝑎()(𝑣)(𝑢)≥,suchthat𝑢,𝑣−𝑢+𝑗−𝑗𝑓,𝑣−𝑢,∀𝑣∈𝐾^∗, (1.1) where Ω⊂𝑅² is a bounded convex polygonal domain; 𝐾^∗ is defined as follows: 𝐾^∗=𝑣∈𝐻¹(Ω)∣𝑣=0,onΓ−Γ_𝑑;𝑣≥0,𝜕𝑣𝜕𝑛≥0,𝑣𝜕𝑣𝜕𝑛=0,onΓ_𝑑=Γ⁰_𝑑∪Γ⁺_𝑑, (1.2) in which Γ=𝜕Ω,Γ_𝑑⊂Γ and Γ⁰_𝑑={𝑥∈Γ_𝑑∣𝑣(𝑥)=0}, Γ⁺_𝑑={𝑥∈Γ_𝑑∣𝑣(𝑥)>0}. 𝑎(𝑢,𝑣)=∫_Ω(∇𝑢∇𝑣+𝜇𝑢𝑣)𝑑𝑥𝑑𝑦,𝜇 is a positive constant, (𝑓,𝑣)=∫_Ω𝑓𝑣𝑑𝑥𝑑𝑦,𝑗(𝑣)=∫_Γ𝑑𝜓(𝑣)𝑑𝑠, and 𝜓(𝑡)=^𝑡₀𝜑(𝜏)(𝜏)=𝜏𝑑𝜏,𝜑𝑔,𝜏≥𝑘𝑔,𝑘,|𝜏|≤𝑘𝑔,−𝑔𝜏≤−𝑘𝑔, (1.3) and 𝑔 and 𝑘 are positive constants. (1.1) may describe many practical engineering problems and attracts many scholars’ interests. For instance, the authors of [20] obtained the 𝑂(ℎ^1/2−𝜀) error estimate of energy norm for linear FE; the authors of [21] got the 𝑂(ℎ^1/2) error estimate in energy norm by improving the result of [20] for 𝑢∈𝐻^3/2(Ω); the authors of [22] derived the optimal 𝑂(ℎ²) error estimate of 𝐿² norm and 𝑂(ℎ) error estimate of energy norm when 𝑢∈𝐻²(Ω). 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 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 𝐿²-norm is also derived when 𝑢∈𝐻²(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.

2. The Equivalent Form and Conforming FE Scheme

It has been shown in [21, 22] that (1.1) is equivalent to find𝑢∈𝐾^∗𝑎()+,suchthat𝑢,𝑣_Γ𝑑𝜑(𝑢)𝑣𝑑𝑠=𝑓,𝑣,∀𝑣∈𝐾^∗, (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∈𝑅¹, 𝜑(𝑎)(𝑏≤1−𝜑)𝑘||,𝜑(𝑎)(𝑏))𝑎−𝑏−𝜑(𝑎−𝑏≥0. (2.2)(2.3) Let 𝑇_ℎ be a rectangular partition with a maximum size ℎ in (𝑥,𝑦) plane, 𝐾∈𝑇_ℎ a general element;𝑉¹_ℎ and 𝑉²_ℎ are the conforming bilinear FE space and the nonconforming EQ^rot FE space. We denote by Π¹_ℎ and Π²_ℎ the associated interpolation operators on 𝑉¹_ℎ and 𝑉²_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: 𝐾¹_ℎ=𝑣_ℎ∈𝑉¹_ℎ∣𝑣_ℎ=0onΓ−Γ_𝑑,𝐾²_ℎ=𝑣_ℎ∈𝑉²_ℎ∣_𝐹𝑣_ℎ𝑑𝑠=0,𝐹⊂Γ−Γ_𝑑,_𝐹𝑣_ℎ𝑑𝑠≥0,𝐹⊂Γ_𝑑, (2.4) 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.

Lemma 2.1. For all 𝑢∈𝐻²(Ω),𝐹⊂𝜕𝐾, there holds ‖𝑢−Π^𝑖_ℎ𝑢‖_0,𝐹≤𝐶ℎ^3/2|𝑢|_2,𝐾.

Lemma 2.2. Let 𝑢∈𝐻^5/2(Ω), then for 𝑣_ℎ∈𝐾¹_ℎ, there holds ∇𝑢−Π¹_ℎ𝑢,𝑣_ℎ=𝑂ℎ^3/2|𝑢|_5/2𝑣_ℎ1, (2.5) where |𝑢|_5/2=∑_|𝛼|=2∬_Ω|𝑢^(𝛼)(𝜗)−𝑢^(𝛼)(𝜃)|²/|𝜗−𝜃|³𝑑𝜗𝑑𝜃.

The corresponding conforming FE approximation version of (2.1) reads as find𝑢∈𝐾¹_ℎ𝑎𝑢,suchthat_ℎ,𝑣_ℎ+_Γ𝑑𝜑𝑢_ℎ𝑣_ℎ𝑑𝑠=𝑓,𝑣_ℎ,∀𝑣_ℎ∈𝐾¹_ℎ. (2.6)

Theorem 2.3. Let 𝑢∈𝐻^5/2(Ω) be the exact solution of (1.1) and 𝑢_ℎ∈𝐾¹_ℎ the bilinear FE solution of (2.6), then there holds Π¹_ℎ𝑢−𝑢_ℎ1≤𝑐ℎ^3/2|𝑢|_5/2, (2.7) 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 𝑎𝑢−𝑢_ℎ,𝑣_ℎ+_Γ𝑑𝜑(𝑢)−𝜑𝑢_ℎ𝑣_ℎ𝑑𝑠=0. (2.8) Let 𝜉=Π¹_ℎ𝑢−𝑢_ℎ and 𝜂=𝑢−Π¹_ℎ𝑢. Taking 𝑣_ℎ=𝜉 in the above equation, there yields 𝑎𝑢−𝑢_ℎ,𝜉+_Γ𝑑𝜑(𝑢)−𝜑𝑢_ℎ𝜉𝑑𝑠=0. (2.9) By the definition of 𝑎(𝑣,𝑣), we have 𝜉²₁≤𝑎(𝜉,𝜉)=𝑎𝑢−𝑢_ℎ,𝜉−𝑎𝜂,𝜉=−_Γ𝑑𝜑(𝑢)−𝜑𝑢_ℎ𝜉𝑑𝑠−𝑎𝜂,𝜉=−_Γ𝑑𝜑(𝑢)−𝜑Π¹_ℎ𝑢𝜉𝑑𝑠−_Γ𝑑𝜑Π¹_ℎ𝑢−𝜑𝑢_ℎ𝜉𝑑𝑠−∇𝜂,∇𝜉−𝜇𝜂,𝜉. (2.10) Noticing (2.3), we have −∫_Γ𝑑(𝜑(Π¹_ℎ𝑢)−𝜑(𝑢_ℎ))𝜉𝑑𝑠≤0; thus 𝜉²₁≤𝐼₁+𝐼₂, (2.11) in which 𝐼₁=−∫_Γ𝑑(𝜑(𝑢)−𝜑(Π¹_ℎ𝑢))𝜉𝑑𝑠,𝐼₂=−(∇𝜂,∇𝜉)−𝜇(𝜂,𝜉).
From (2.2) and Lemma 2.1, 𝐼₁ can be estimated as 𝐼₁≤𝑐𝑘_Γ𝑑𝜂𝜉𝑑𝑠≤𝜂_0,Γ𝑑𝜉_0,Γ𝑑≤𝑐ℎ^3/2|𝑢|₂𝜉₁. (2.12)
Applying the interpolation theory and Lemma 2.2, we get 𝐼₂≤𝑐ℎ^3/2|𝑢|_5/2𝜉₁. (2.13)
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 find𝑢∈𝐾²_ℎ𝑎,suchthat_ℎ𝑢_ℎ,𝑣_ℎ+_Γ𝑑𝜑𝑢_ℎ𝑣_ℎ𝑑𝑠=𝑓,𝑣_ℎ,∀𝑣_ℎ∈𝐾²_ℎ, (3.1) where 𝑎_ℎ(𝑢,𝑣)=∑_𝐾∫_𝐾(∇𝑢∇𝑣+𝜇𝑢𝑣)𝑑𝑥𝑑𝑦.

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

Lemma 3.1 (see [25]). If 𝑢∈𝐻²(Ω),𝑣_ℎ∈𝐾²_ℎ, one has ∇𝑢−Π²_ℎ𝑢,∇𝑣_ℎ=0. (3.2)

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

Lemma 3.2. For all 𝑢∈𝐻^5/2(Ω),𝑣_ℎ∈𝐾²_ℎ, there holds _𝐾𝜕𝐾𝜕𝑢𝑣𝜕𝑛^ℎ𝑑𝑠≤𝑐ℎ^3/2|𝑢|_5/2𝑣_ℎℎ, (3.3) where ‖𝑣_ℎ‖_ℎ=(∑_{𝐾∈𝑇ℎ}|𝑣_ℎ|²_1,𝐾)^1/2.

Proof. Let 𝑍₁=(𝑥₀−ℎ_𝑥,𝑦₀−ℎ_𝑦),𝑍₂=(𝑥₀+ℎ_𝑥,𝑦₀−ℎ_𝑦),𝑍₃=(𝑥₀+ℎ_𝑥,𝑦₀+ℎ_𝑦), and 𝑍₄=(𝑥₀−ℎ_𝑥,𝑦₀+ℎ_𝑦) be the four vertices of 𝐾, 𝐹_𝑖=𝑍_𝑖𝑍_𝑖+1(𝑖=1,2,3,4,mod4). We define operators 𝑃₀ and 𝑃_0𝑖 as 𝑃₀1𝑣=|𝐾|_𝐾𝑣𝑑𝑥,𝑃_0𝑖1𝜔=𝐹_𝑖𝐹𝑖𝜔𝑑𝑠, (3.4) respectively, where |𝐾| and |𝐹_𝑖| denote the measures of 𝐾 and 𝐹_𝑖, respectively.
It can be checked that _𝐾𝜕𝐾𝜕𝑢𝑣𝜕𝑛^ℎ𝑑𝑠=_𝐾−_𝐹1𝜕𝑢𝑣𝜕𝑦^ℎ−𝑃₀₁𝑣_ℎ𝑑𝑥+_𝐹2𝜕𝑢𝑣𝜕𝑥^ℎ−𝑃₀₂𝑣_ℎ+𝑑𝑦_𝐹3𝜕𝑢𝑣𝜕𝑦^ℎ−𝑃₀₃𝑣_ℎ𝑑𝑥−_𝐹4𝜕𝑢𝑣𝜕𝑥^ℎ−𝑃₀₄𝑣_ℎ𝑑𝑦+_{𝐹⊂Γ𝑑𝐹}𝜕𝑢𝑣𝜕𝑛^ℎ𝑑𝑠≐_𝐾⁴_𝑖=1𝑀_𝑖+𝑀. (3.5)
By the definition of 𝑃₀₁, we get _𝐾𝑣_ℎ𝑥,𝑦₀−ℎ_𝑦−𝑃₀₁𝑣_ℎ𝑥,𝑦₀−ℎ_𝑦𝑑𝑥𝑑𝑦=2ℎ_𝑦𝐹1𝑣_ℎ𝑥,𝑦₀−ℎ_𝑦𝑑𝑥−4ℎ_𝑥ℎ_𝑦𝐹_1𝐹1𝑣_ℎ𝑥,𝑦₀−ℎ_𝑦𝑑𝑥=0. (3.6)
Noticing that (𝑣_ℎ−𝑃₀₁𝑣_ℎ)|_𝐹1 equals (𝑣_ℎ−𝑃₀₃𝑣_ℎ)|_𝐹3 and 𝜕𝑣_ℎ/𝜕𝑥 is only dependent on 𝑥, we can derive that 𝑀₁+𝑀₃=^{𝑥0+ℎ𝑥}_{𝑥0−ℎ𝑥}𝜕𝑢𝜕𝑦𝑥,𝑦₀+ℎ_𝑦−𝜕𝑢𝜕𝑦𝑥,𝑦₀−ℎ_𝑦𝑣_ℎ−𝑃₀₁𝑣_ℎ=𝑑𝑥_{𝑥0+ℎ𝑥𝑥0−ℎ𝑥𝑦0+ℎ𝑦𝑦0−ℎ𝑦}𝜕²𝑢𝜕𝑦²𝑥,𝑦𝑑𝑦𝑣_ℎ−𝑃₀₁𝑣_ℎ=𝑑𝑥_{𝑥0+ℎ𝑥𝑥0−ℎ𝑥𝑦0+ℎ𝑦𝑦0−ℎ𝑦}𝜕²𝑢𝜕𝑦²−𝑃₀𝜕²𝑢𝜕𝑦²𝑣_ℎ−𝑃₀₁𝑣_ℎ=𝜕𝑑𝑦𝑑𝑥₂𝑢𝜕𝑦²−𝑃₀𝜕²𝑢𝜕𝑦²_0,𝐾𝑣_ℎ−𝑃₀₁𝑣_ℎ0,𝐾≤𝑐ℎ^3/2|𝑢|_5/2,𝐾𝑣_ℎ1,𝐾. (3.7)
Similarly, 𝑀₂+𝑀₄≤𝑐ℎ^3/2|𝑢|_5/2,𝐾|𝑣_ℎ|_1,𝐾. By using the same technique as [14, 15], 𝑀 can be estimated as |𝑀|≤𝑐ℎ³|𝑢|_5/2𝑣_ℎℎ. (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∈𝐾²_ℎ the nonconforming FE solution of (3.1). Then one has Π²_ℎ𝑢−𝑢_ℎℎ≤𝐶ℎ^3/2|𝑢|_5/2. (3.9)

Proof. Subtracting (2.1) from (3.1) gives 𝑎_ℎ𝑢−𝑢_ℎ,𝑣_ℎ+_Γ𝑑𝜑(𝑢)−𝜑𝑢_ℎ𝑣_ℎ𝑑𝑠=_𝐾𝜕𝐾𝜕𝑢𝑣𝜕𝑛^ℎ𝑑𝑠. (3.10)
For convenience, we still denote 𝜉=Π²_ℎ𝑢−𝑢_ℎ and 𝜂=𝑢−Π²_ℎ𝑢. Taking 𝑣_ℎ=Π²_ℎ𝑢−𝑢_ℎ in (3.10) yields 𝑎_ℎ𝑢−𝑢_ℎ,𝜉+_Γ𝑑𝜑(𝑢)−𝜑𝑢_ℎ𝜉𝑑𝑠=_𝐾𝜕𝐾𝜕𝑢𝜕𝑛𝜉𝑑𝑠. (3.11)
By Lemma 3.1, we can derive that 𝜉²_ℎ≤𝑎_ℎ(𝜉,𝜉)=𝑎_ℎ𝑢−𝑢_ℎ,𝜉−𝑎_ℎ𝜂,𝜉=−_Γ𝑑𝜑(𝑢)−𝜑𝑢_ℎ𝜉𝑑𝑠−𝑎_ℎ𝜂,𝜉+_𝐾𝜕𝐾𝜕𝑢𝜕𝑛𝜉𝑑𝑠=−_Γ𝑑𝜑(𝑢)−𝜑Π²_ℎ𝑢𝜉𝑑𝑠−_Γ𝑑𝜑Π²_ℎ𝑢−𝜑𝑢_ℎ𝜉𝑑𝑠−𝜇𝜂,𝜉+_𝐾𝜕𝐾𝜕𝑢𝜕𝑛𝜉𝑑𝑠. (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∈𝐻²(Ω) instead of 𝑢∈𝐻^5/2(Ω), the consistency error can be estimated as _𝐾𝜕𝐾𝜕𝑢𝑣𝜕𝑛^ℎ𝑑𝑠≤𝑐ℎ|𝑢|₂𝑣_ℎℎ, (3.13) which can be found in [26]. Then we can derive the following optimal error estimate: 𝑢−𝑢_ℎℎ≤𝐶ℎ|𝑢|₂. (3.14)
Now we start to give the 𝐿²-norm estimate through a duality argument.

Theorem 3.5. Let 𝑢∈𝐾²(Ω) and 𝑢_ℎ∈𝑉²_ℎ be the solutions of (1.1) and (3.1), respectively, there holds 𝑢−𝑢_ℎ0≤𝐶ℎ²|𝑢|₂. (3.15)

Proof. Let 𝜔∈𝐻²(Ω) be the solution of the following auxiliary elliptic problem: −▵𝑤+𝜇𝑤=𝑢−𝑢_ℎ,inΩ,𝑤=0,onΓ−Γ_𝑑,𝜕𝑤(𝑥)𝜕𝑛=−𝛽𝑤,onΓ_𝑑, (3.16) in which 𝛽(𝑥)=(𝜑(𝑢)−𝜑(𝑢_ℎ))/(𝑢−𝑢_ℎ), then ‖𝑤‖₂≤𝑐𝑢−𝑢_ℎ0. (3.17)
By (3.16) and Lemma 3.1, we can derive that 𝑢−𝑢_ℎ²₀=𝑢−𝑢_ℎ,𝑢−𝑢_ℎ=𝑎_ℎ𝑢−𝑢_ℎ+,𝑤_Γ𝑑𝛽𝑤𝑢−𝑢_ℎ𝑑𝑠+_𝐾𝜕𝐾𝜕𝑤𝜕𝑛𝑢−𝑢_ℎ𝑑𝑠=𝑎_ℎ𝑢−𝑢_ℎ,𝑤−Π²_ℎ𝑤+𝑎_ℎ𝑢−𝑢_ℎ,Π²_ℎ𝑤+_Γ𝑑𝛽𝑤𝑢−𝑢_ℎ𝑑𝑠+_𝐾𝜕𝐾𝜕𝑤𝜕𝑛𝑢−𝑢_ℎ𝑑𝑠=𝑎_ℎ𝑢−𝑢_ℎ,𝑤−Π²_ℎ𝑤−_Γ𝑑𝜑(𝑢)−𝜑𝑢_ℎΠ²_ℎ𝑤𝑑𝑠+_Γ𝑑𝛽𝑤𝑢−𝑢_ℎ𝑑𝑠+_𝐾𝜕𝐾𝜕𝑤𝜕𝑛𝑢−𝑢_ℎ𝑑𝑠+_𝐾𝜕𝐾𝜕𝑢Π𝜕𝑛^2ℎ𝑤𝑑𝑠=𝑎_ℎ𝑢−𝑢_ℎ,𝑤−Π²_ℎ𝑤+1𝑘_Γ𝑑𝑢−𝑢_ℎ𝑤−Π²_ℎ𝑤𝑑𝑠+_𝐾𝜕𝐾𝜕𝑤𝜕𝑛𝑢−𝑢_ℎ𝑑𝑠+_𝐾𝜕𝐾𝜕𝑢𝜕𝑛𝑤−Π²_ℎ𝑤𝑑𝑠=𝐽₁+𝐽₂+𝐽₃, (3.18) where 𝐽₁=𝑎_ℎ(𝑢−𝑢_ℎ,𝑤−Π²_ℎ𝑤),𝐽₂=1/𝑘∫_Γ𝑑(𝑢−𝑢_ℎ)(𝑤−Π²_ℎ𝑤)𝑑𝑠,and𝐽₃=∑_𝐾∫_𝜕𝐾𝜕𝑤/𝜕𝑛(𝑢−𝑢_ℎ∑)𝑑𝑠+_𝐾∫_𝜕𝐾𝜕𝑢/𝜕𝑛(𝑤−Π²_ℎ𝑤)𝑑𝑠. These three terms can be estimated one by one as follows. By (3.14), (3.17), and the interpolation theory, 𝐽₁ can be estimated as 𝐽₁=∇𝑢−𝑢_ℎ,∇𝑤−Π²_ℎ𝑤+𝜇𝑢−𝑢_ℎ,𝑤−Π²_ℎ𝑤≤𝑐ℎ²|𝑢|₂|𝑤|₂+𝑐ℎ²𝑢−𝑢_ℎ0|𝑤|₂≤𝑐ℎ²|𝑢|₂𝑢−𝑢_ℎ0+𝑐ℎ²𝑢−𝑢_ℎ²₀. (3.19)
By the trace theorem, (3.17), and Lemma 2.1, one gets 𝐽₂≤1𝑘𝑢−𝑢_ℎ0,Γ𝑑𝑢−Π²_ℎ𝑢_0,Γ𝑑≤𝑐ℎ^5/2|𝑢|₂|𝑤|₂≤𝑐ℎ^5/2|𝑢|₂𝑢−𝑢_ℎ0. (3.20) By (3.13), (3.14), and (3.17), we have 𝐽₃≤𝑐ℎ|𝑢|₂𝑤−Π²_ℎ𝑤_ℎ+𝑐ℎ|𝑤|₂𝑢−𝑢_ℎℎ≤𝑐ℎ²|𝑢|₂|𝑤|₂≤𝑐ℎ²|𝑢|₂𝑢−𝑢_ℎ0. (3.21)
The desired result follows the combination of the above estimates of 𝐽₁, 𝐽₂, and 𝐽₃.

Remark 3.6. As to the 𝐿²-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 𝐾₁,𝐾₂,𝐾₃,𝐾₄∈𝑇_ℎ into one new rectangular element 𝐾₀, whose four edges are 𝐿₁,𝐿₂,𝐿₃, and 𝐿₄. 𝑇_2ℎ represents the corresponding new partition. For the conforming FE scheme, we construct the postprocessing operator Π¹_2ℎ𝑢|_𝐾0∶𝐶(𝐾₀)→𝑃₂(𝐾₀) as follows: Π¹_2ℎ𝑢𝑍_𝑗=𝑢𝑍_𝑗,𝑗=1,2,…,8, (4.1) in which 𝑍_𝑗 is the four vertices and four mid point of edges of 𝐾₀. For the nonconforming FE scheme, we construct the postprocessing Π²_2ℎ operator as Π²_2ℎ𝑢|_𝐾0∈𝑃₂𝐾₀,∀𝐾₀∈𝑇_2ℎ,_𝐿𝑗Π²_2ℎ𝑢−𝑢𝑑𝑠=0,𝑗=1,2,3,4,_{𝐾1∪𝐾3}Π²_2ℎ𝑢−𝑢𝑑𝑥=0,_{𝐾2∪𝐾4}Π²_2ℎ𝑢−𝑢𝑑𝑥=0,∀𝐾₀∈𝑇_2ℎ. (4.2)

It is easy to validate that the interpolation operator is well posed and has the following properties [23]: Π^𝑖_2ℎΠ^𝑖_ℎ𝑢=Π^𝑖_2ℎ𝑢,∀𝑢∈𝐻²(Ω),Π^𝑖_2ℎ𝑢−𝑢_ℎ≤𝑐ℎ^𝑟|𝑢|_𝑟+1,∀𝑢∈𝐻^𝑟+1(Ω)Π,0≤𝑟≤2,_𝑖2ℎ𝑣_ℎℎ≤𝑐𝑣_ℎℎ,∀𝑣_ℎ∈𝐾^𝑖_ℎ. (4.3)

Theorem 4.1. If 𝑢∈𝐻^5/2(Ω) is the exact solution of (1.1), 𝑢_ℎ is the conforming or nonconforming FE solution. The following superconvergence result 𝑢−Π^𝑖_2ℎ𝑢_ℎℎ≤𝑐ℎ^3/2|𝑢|_5/2 (4.4) holds.

Proof. By (4.3), one gets Π^𝑖_2ℎΠ^𝑖_ℎ𝑢−Π^𝑖_2ℎ𝑢_ℎℎ=Π^𝑖_2ℎ(Π^𝑖_ℎ𝑢−𝑢_ℎ)_ℎ≤𝑐Π^𝑖_ℎ𝑢−𝑢_ℎℎ≤𝑐ℎ^3/2|𝑢|_5/2,Π^𝑖_2ℎΠ^𝑖_ℎ𝑢−𝑢_ℎ=Π^𝑖_2ℎ𝑢−𝑢_ℎ≤𝑐ℎ^3/2|𝑢|_5/2. (4.5)
Noticing Π^𝑖_2ℎ𝑢_ℎ−𝑢=Π^𝑖_2ℎ𝑢_ℎ−Π^𝑖_2ℎΠ^𝑖_ℎ𝑢+Π^𝑖_2ℎΠ^𝑖_ℎ𝑢−𝑢, 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 Ω=[0,1]×[0,1] with boundary 𝜕Ω=Γ,𝜇=1,𝜑(𝑢)=𝑢,Γ_𝑑={0}×[0,1],𝑢|_Γ𝑑=(𝑥₁−1/2)²−1/4,𝑢|_Γ−Γ𝑑=0. 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.

Table 1 

The error estimates for conforming FE scheme.

Table 2 

The error estimates for nonconforming FE scheme.

(a)

(b)

Figure 1 

The conforming FE solution (a) and the nonconforming FE solution (b) on the 64×64 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.