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

Authors: Dongyang Shi; Hongbo Guan; Xiaofei Guan
Journal: Journal of Applied Mathematics
 (2012)

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## 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 nonconformingEQrot 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 EQrot FE. At last, some numerical results are provided to verify the theoretical analysis.


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## Body

## 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(h3/2). The authors of [3] first obtained the error bound O(h3/2-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(h3/4|logh|1/4) if the displacement field is of H2 regularity and also showed that if stronger but reasonable regularity is available (u∈W2,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(h3/2) when the exact solution u∈H5/2(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]:
(1.1)findu∈K*,suchthata(u,v-u)+j(v)-j(u)≥(f,v-u),∀v∈K*,
where Ω⊂R2 is a bounded convex polygonal domain; K* is defined as follows:
(1.2)K*={v∈H1(Ω)∣v=0,onΓ-Γd;v≥0,∂v∂n≥0,v∂v∂n=0,onΓd=Γd0∪Γd+},
in which Γ=∂Ω,Γd⊂Γ and Γd0={x∈Γd∣v(x)=0}, Γd+={x∈Γd∣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
(1.3)ψ(t)=∫0t‍φ(τ)dτ,φ(τ)={g,τ≥kg,τk,|τ|≤kg,-gτ≤-kg,
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(h1/2-ε) error estimate of energy norm for linear FE; the authors of [21] got the O(h1/2) error estimate in energy norm by improving the result of [20] for u∈H3/2(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∈H5/2(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(h3/2) is derived under the broken energy norm. In Section 3, the nonconforming EQrot 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
(2.1)findu∈K*,suchthata(u,v)+∫Γd‍φ(u)vds=(f,v),∀v∈K*,
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,
(2.2)|φ(a)-φ(b)|≤1k|a-b|,(2.3)(φ(a)-φ(b))(a-b)≥0.
Let Th be a rectangular partition with a maximum size h in (x,y) plane, K∈Th a general element;Vh1 and Vh2 are the conforming bilinear FE space and the nonconforming EQrot FE space. We denote by Πh1 and Πh2 the associated interpolation operators on Vh1 and Vh2, respectively. In the meantime, we denote Khi by a convex set associated with K* in Vhi (i=1,2) as follows:
(2.4)Kh1={vh∈Vh1∣vh=0onΓ-Γd},Kh2={vh∈Vh2∣∫F‍vhds=0,F⊂Γ-Γd,∫F‍vhds≥0,F⊂Γd},
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 allu∈H2(Ω),F⊂∂K, there holds ∥u-Πhiu∥0,F≤Ch3/2|u|2,K.Lemma 2.2.
Letu∈H5/2(Ω), then for vh∈Kh1, there holds
(2.5)(∇(u-Πh1u),vh)=O(h3/2)|u|5/2|vh|1,
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
(2.6)findu∈Kh1,suchthata(uh,vh)+∫Γd‍φ(uh)vhds=(f,vh),∀vh∈Kh1.Theorem 2.3.
Letu∈H5/2(Ω) be the exact solution of (1.1) and uh∈Kh1 the bilinear FE solution of (2.6), then there holds
(2.7)|Πh1u-uh|1≤ch3/2|u|5/2,
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=vh in it, one can get
(2.8)a(u-uh,vh)+∫Γd‍(φ(u)-φ(uh))vhds=0.
Let ξ=Πh1u-uh and η=u-Πh1u. Taking vh=ξ in the above equation, there yields
(2.9)a(u-uh,ξ)+∫Γd‍(φ(u)-φ(uh))ξds=0.
By the definition of a(v,v), we have
(2.10)|ξ|12≤a(ξ,ξ)=a(u-uh,ξ)-a(η,ξ)=-∫Γd‍(φ(u)-φ(uh))ξds-a(η,ξ)=-∫Γd‍(φ(u)-φ(Πh1u))ξds-∫Γd‍(φ(Πh1u)-φ(uh))ξds-(∇η,∇ξ)-μ(η,ξ).
Noticing (2.3), we have -∫Γd‍(φ(Πh1u)-φ(uh))ξds≤0; thus
(2.11)|ξ|12≤I1+I2,
in which I1=-∫Γd‍(φ(u)-φ(Πh1u))ξds,I2=-(∇η,∇ξ)-μ(η,ξ).
From (2.2) and Lemma 2.1, I1 can be estimated as
(2.12)|I1|≤ck∫Γd‍|η||ξ|ds≤∥η∥0,Γd∥ξ∥0,Γd≤ch3/2|u|2|ξ|1.
Applying the interpolation theory and Lemma2.2, we get
(2.13)|I2|≤ch3/2|u|5/2|ξ|1.
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
(3.1)findu∈Kh2,suchthatah(uh,vh)+∫Γd‍φ(uh)vhds=(f,vh),∀vh∈Kh2,
where ah(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]).
Ifu∈H2(Ω),vh∈Kh2, one has
(3.2)(∇(u-Πh2u),∇vh)=0.By using the similar technique in [26], one now states and proves the following important conclusion.Lemma 3.2.
For allu∈H5/2(Ω),vh∈Kh2, there holds
(3.3)∑K‍∫∂K‍∂u∂nvhds≤ch3/2|u|5/2∥vh∥h,
where ∥vh∥h=(∑K∈Th|vh|1,K2)1/2.Proof.
LetZ1=(x0-hx,y0-hy),Z2=(x0+hx,y0-hy),Z3=(x0+hx,y0+hy), and Z4=(x0-hx,y0+hy) be the four vertices of K, Fi=ZiZi+1¯(i=1,2,3,4,mod4). We define operators P0 and P0i as
(3.4)P0v=1|K|∫K‍vdx,P0iω=1|Fi|∫Fi‍ωds,
respectively, where |K| and |Fi| denote the measures of K and Fi, respectively.
It can be checked that(3.5)∑K‍∫∂K‍∂u∂nvhds=∑K‍[-∫F1‍∂u∂y(vh-P01vh)dx+∫F2‍∂u∂x(vh-P02vh)dy+∫F3‍∂u∂y(vh-P03vh)dx-∫F4‍∂u∂x(vh-P04vh)dy]+∑F⊂Γd‍∫F‍∂u∂nvhds≐∑K‍∑i=14‍Mi+M.
By the definition ofP01, we get
(3.6)∫K‍(vh(x,y0-hy)-P01vh(x,y0-hy))dxdy=2hy∫F1‍vh(x,y0-hy)dx-4hxhy|F1|∫F1‍vh(x,y0-hy)dx=0.
Noticing that(vh-P01vh)|F1 equals (vh-P03vh)|F3 and ∂vh/∂x is only dependent on x, we can derive that
(3.7)M1+M3=∫x0-hxx0+hx‍[∂u∂y(x,y0+hy)-∂u∂y(x,y0-hy)](vh-P01vh)dx=∫x0-hxx0+hx‍[∫y0-hyy0+hy‍∂2u∂y2(x,y)dy](vh-P01vh)dx=∫x0-hxx0+hx‍∫y0-hyy0+hy‍(∂2u∂y2-P0∂2u∂y2)(vh-P01vh)dydx=∥∂2u∂y2-P0∂2u∂y2∥0,K∥vh-P01vh∥0,K≤ch3/2|u|5/2,K|vh|1,K.
Similarly,M2+M4≤ch3/2|u|5/2,K|vh|1,K. By using the same technique as [14, 15], M can be estimated as
(3.8)|M|≤ch3|u|5/2∥vh∥h.
Thus the desired result follows.Theorem 3.3.
Letu∈H5/2(Ω) be the exact solution of (1.1) and uh∈Kh2 the nonconforming FE solution of (3.1). Then one has
(3.9)∥Πh2u-uh∥h≤Ch3/2|u|5/2.Proof.
Subtracting (2.1) from (3.1) gives
(3.10)ah(u-uh,vh)+∫Γd‍(φ(u)-φ(uh))vhds=∑K‍∫∂K‍∂u∂nvhds.
For convenience, we still denoteξ=Πh2u-uh and η=u-Πh2u. Taking vh=Πh2u-uh in (3.10) yields
(3.11)ah(u-uh,ξ)+∫Γd‍(φ(u)-φ(uh))ξds=∑K‍∫∂K‍∂u∂nξds.
By Lemma3.1, we can derive that
(3.12)∥ξ∥h2≤ah(ξ,ξ)=ah(u-uh,ξ)-ah(η,ξ)=-∫Γd‍(φ(u)-φ(uh))ξds-ah(η,ξ)+∑K‍∫∂K‍∂u∂nξds=-∫Γd‍(φ(u)-φ(Πh2u))ξds-∫Γd‍(φ(Πh2u)-φ(uh))ξds-μ(η,ξ)+∑K‍∫∂K‍∂u∂nξds.
Noticing Lemma3.2 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 assumeu∈H2(Ω) instead of u∈H5/2(Ω), the consistency error can be estimated as
(3.13)∑K‍∫∂K‍∂u∂nvhds≤ch|u|2∥vh∥h,
which can be found in [26]. Then we can derive the following optimal error estimate:
(3.14)∥u-uh∥h≤Ch|u|2.
Now we start to give theL2-norm estimate through a duality argument.Theorem 3.5.
Letu∈K2(Ω) and uh∈Vh2 be the solutions of (1.1) and (3.1), respectively, there holds
(3.15)∥u-uh∥0≤Ch2|u|2.Proof.
Letω∈H2(Ω) be the solution of the following auxiliary elliptic problem:
(3.16)-▵w+μw=u-uh,inΩ,w=0,onΓ-Γd,∂w∂n=-β(x)w,onΓd,
in which β(x)=(φ(u)-φ(uh))/(u-uh), then
(3.17)∥w∥2≤c∥u-uh∥0.
By (3.16) and Lemma 3.1, we can derive that
(3.18)∥u-uh∥02=(u-uh,u-uh)=ah(u-uh,w)+∫Γd‍βw(u-uh)ds+∑K‍∫∂K‍∂w∂n(u-uh)ds=ah(u-uh,w-Πh2w)+ah(u-uh,Πh2w)+∫Γd‍βw(u-uh)ds+∑K‍∫∂K‍∂w∂n(u-uh)ds=ah(u-uh,w-Πh2w)-∫Γd‍(φ(u)-φ(uh))Πh2wds+∫Γd‍βw(u-uh)ds+∑K‍∫∂K‍∂w∂n(u-uh)ds+∑K‍∫∂K‍∂u∂nΠh2wds=ah(u-uh,w-Πh2w)+1k∫Γd‍(u-uh)(w-Πh2w)ds+∑K‍∫∂K‍∂w∂n(u-uh)ds+∑K‍∫∂K‍∂u∂n(w-Πh2w)ds=J1+J2+J3,
where J1=ah(u-uh,w-Πh2w),J2=1/k∫Γd‍(u-uh)(w-Πh2w)ds,andJ3=∑K∫∂K‍∂w/∂n(u-uh)ds+∑K∫∂K‍∂u/∂n(w-Πh2w)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
(3.19)J1=(∇(u-uh),∇(w-Πh2w))+μ(u-uh,w-Πh2w)≤ch2|u|2|w|2+ch2∥u-uh∥0|w|2≤ch2|u|2∥u-uh∥0+ch2∥u-uh∥02.
By the trace theorem, (3.17), and Lemma 2.1, one gets
(3.20)J2≤1k∥u-uh∥0,Γd∥u-Πh2u∥0,Γd≤ch5/2|u|2|w|2≤ch5/2|u|2∥u-uh∥0.
By (3.13), (3.14), and (3.17), we have
(3.21)J3≤ch|u|2∥w-Πh2w∥h+ch|w|2∥u-uh∥h≤ch2|u|2|w|2≤ch2|u|2∥u-uh∥0.
The desired result follows the combination of the above estimates ofJ1, J2, and J3.Remark 3.6.
As to theL2-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 elementsK1,K2,K3,K4∈Th into one new rectangular element K0, whose four edges are L1,L2,L3, and L4. T2h represents the corresponding new partition. For the conforming FE scheme, we construct the postprocessing operator Π2h1u|K0:C(K0)→P2(K0) as follows:
(4.1)Π2h1u(Zj)=u(Zj),j=1,2,…,8,
in which Zj is the four vertices and four mid point of edges of K0. For the nonconforming FE scheme, we construct the postprocessing Π2h2 operator as
(4.2)Π2h2u|K0∈P2(K0),∀K0∈T2h,∫Lj‍(Π2h2u-u)ds=0,j=1,2,3,4,∫K1∪K3‍(Π2h2u-u)dx=0,∫K2∪K4‍(Π2h2u-u)dx=0,∀K0∈T2h.It is easy to validate that the interpolation operator is well posed and has the following properties [23]:
(4.3)Π2hiΠhiu=Π2hiu,∀u∈H2(Ω),∥Π2hiu-u∥h≤chr|u|r+1,∀u∈Hr+1(Ω),0≤r≤2,∥Π2hivh∥h≤c∥vh∥h,∀vh∈Khi.Theorem 4.1.
Ifu∈H5/2(Ω) is the exact solution of (1.1), uh is the conforming or nonconforming FE solution. The following superconvergence result
(4.4)∥u-Π2hiuh∥h≤ch3/2|u|5/2
holds.Proof.
By (4.3), one gets
(4.5)∥Π2hiΠhiu-Π2hiuh∥h=∥Π2hi(Πhiu-uh)∥h≤c∥Πhiu-uh∥h≤ch3/2|u|5/2,∥Π2hiΠhiu-u∥h=∥Π2hiu-u∥h≤ch3/2|u|5/2.
NoticingΠ2hiuh-u=Π2hiuh-Π2hiΠhiu+Π2hiΠhiu-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,Γd={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.
h
1/2
1/4
1/8
1/16
1/32
1/64
∥
Π
h
1
u
-
u
h
∥
h
2.1780E−02
6.6596E-03
1.8851E-03
5.1760E-04
1.3861E-04
3.5405E-05
order
/
1.8084
1.8796
1.9084
1.9324
1.9786
∥
u
-
Π
2
h
1
u
h
∥
h
5.3923E-03
1.5926E-03
4.1215E-04
1.0371E-04
2.5691E-05
6.1214E-06
order
/
1.8401
1.9657
1.9935
2.0092
2.0486Table 2
The error estimates for nonconforming FE scheme.
h
1/2
1/4
1/8
1/16
1/32
1/64
∥
Π
h
2
u
-
u
h
∥
h
1.1264E-01
4.6572E-02
1.8679E-02
8.0004E-03
3.1977E-03
1.3906E-03
order
/
1.5552
1.5790
1.5280
1.5817
1.5164
∥
u
-
Π
2
h
2
u
h
∥
h
8.1523E-02
3.1059E-02
1.0260E-02
3.1725E-03
9.4359E-04
2.7340E-04
order
/
1.6201
1.7399
1.7983
1.8336
1.8578
∥
u
-
u
h
∥
0
1.0995E-02
2.7109E-03
6.4263E-04
1.5941E-04
3.9258E-05
9.3430E-06
order
/
2.0139
2.0539
2.0078
2.0151
2.0498The conforming FE solution (a) and the nonconforming FE solution (b) on the64×64 mesh.
(a)
(b)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.



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*Source: 156095-2012-11-26.xml*