On $P_1$ nonconforming finite element aproximation for the Signorini problem

1.   Introduction

The unilateral contact models are of great practical interest in solid mechanics and many works have been contributed to their numerical analysis. In fact, as a mostly powerful numerical method, the finite element methods for the unilateral contact models have been highlighted in the numerical simulation of variational inequalities for more than fifty years, interested readers please refer to [14,17,22] and the references therein.

It is well known that the unilateral contact problems are typically represented by Signorini's model, which may cause some special difficulties in both mathematical theory and numerical approximation. Most often, linear finite elements are used by the practitioners for the approximation of contact problems with unilateral Signorini boundary conditions. However, the numerical analysis of their convergence has been explored a long way. The first error estimate of conforming linear finite element approximations of this problem with frictionless boundary is probably given by Scarpini and Vivaldi (cf. [21]). They proved $\mathcal{O}(h^{3/4})$ convergence rate with the regularity $u\in H^2(\Omega)$ (for simplicity, we call it "assumption $A_1$"). In the same year, Brezzi, Hager and Raviart (cf. [7]) obtained optimal convergence rate $\mathcal{O}(h)$ with two additional conditions: $u|_{\partial\Omega}\in W^{1, \infty}(\partial\Omega)$ ("assumption $A_2$") and the number of points in the free boundary where the constraint changes from binding to nonbinding is finite ("assumption $A_3$"). In the subsequent more than twenty years, the finite element methods for the these problems developed very quickly, nevertheless, the convergence rates were still based on the results of Scarpini and Brezzi. Until 2000, Belgacem (cf. [1]) presented the quasi-optimal convergence rate $\mathcal{O}(h|\log h|^{1/2})$ by a detailed analysis and novel approach under "assumption $A_1$" and "assumption $A_3$". And later, he and his colleague Renard improved it to $\mathcal{O}(h|\log h|^{1/4})$ (cf. [2]), H$\ddot{\mbox{u}}$eber and Wohlmuth [14] obtained $O(h)$ convergence by nonconforming domain decomposition methods based on dual Lagrange multipliers, also with the additional "assumption $A_3$".

In this work, we consider the nonconforming Crouzeix-Raviart (cf. [9]) finite element approximations to the Signorini problem. Our motivation comes from the numerical investigation, which shows that the convergence rate of $P_1$ nonconforming finite element method is optimal, please refer to section 5. Therefore, a natural interest arises for a better understanding of the convergence properties of the $P_1$ nonconforming element method. Our ultimate aim is to propose some locking-free finite element methods for the Signorini problem in incompressible elasticity. It is well-known that the linear nonconforming Crouzeix-Raviart element can avoid locking phenomena in some incompressible flows (see [6,8,11,15]). What's more, the numerical results presented in [22] have showed that the nonconforming Crouzeix-Raviart element behaves better than its conforming counterpart one when used to solve some variational inequalities with small parameter.

In fact, the nonconforming Crouzeix-Raviart method was firstly considered by Wang in [23] and $O(h^{1/2})$ convergence is obtained therein. Later, Hua and Wang improved it by $\mathcal{O}(h|\log h|^{1/4})$ convergence (cf. [13]) with the additional "assumption $A_3$", exactly the same as the rate of conforming linear finite element method. However, [23] and [13] only consider the case $u\in H^2(\Omega)$. We extend the method to the case $u\in H^{1+\nu}(\Omega)$ with $\frac{1}{2}<\nu\leq1$, which is more reasonable in practice. Some new techniques in the estimate of the consistency error for nonsmooth solution are developed in this paper.

Though nonconforming Crouzeix-Raviart element contains more degrees of freedom and thus involves more expensive computational cost, the results of linear nonconforming finite element method may have some attractive features. For both linear and nonlinear contact condition models, the optimal convergence rate $\mathcal{O}(h)$ of the meanvalue-oriented approximation can be obtained for $u\in H^2(\Omega)$ without "assumption $A_3$". This optimal result can also be obtained for the linear contact condition model by the midpoint-oriented approximation. As far as we know, this is the first time to obtain the optimal convergence rate without any supplementary hypotheses. Meanwhile, the numerical investigation presented in section 5 also shows that the $P_1$ nonconforming finite element method is even slightly better than the conforming one. On the other hand, nonconforming finite element methods have the striking practical advantage that each degrees of freedom belongs to at most two elements, which results in a cheap local communication and the method can be parallelized in a highly efficient manner on MIMD-machines, see e.g. [10] and the references therein.

An outline of this paper is as follows. Section 2 deals with some functional tools and the continuous setting of the Signorini problem. In section 3, we define a meanvalue-oriented type discretized method and obtain the optimal convergence for the case $u\in H^{1+\nu}(\Omega)$ with $\frac{1}{2}<\nu<1$ with "assumption $A_3$". When $u\in H^{2}(\Omega)$, for the meanvalue-oriented approximation, the optimal convergence rate $\mathcal{O}(h)$ can be obtained without "assumption $A_3$". Section 4 is concerned with the convergence properties with a midpoint-oriented type discretized method for the Signorini problem. Finally, in section 5, a numerical experiment is presented, where conforming and nonconforming linear finite elements are compared.

2.   Notations and the Signorini problem

For the sake of the hereafter analysis, we firstly begin with some necessary notations and functional tools, then we give a brief introduce of the Signorini problem.

Let $\Omega\subset R^2$ be a Lipschitz domain whose generic point of $\Omega$ is denoted by $\textbf{x} = (x_1, x_2)$. The Lebesgue space $L^2(\Omega)$ is endowed with the inner product,

and with the norm

Then the standard Sobolev spaces $H^m(\Omega), m\geq1,$ are equipped with the norm

where $\alpha = (\alpha_1, \alpha_2)$ is a multi-index in $N^2$ and the symbol $D^\alpha = \partial^{|\alpha|}/\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}$ denotes a partial derivative. The convention $H^0(\Omega) = L^2(\Omega)$ is adopted. The fractional order Sobolev space $H^\nu(\Omega), \nu\in R_+\setminus N$ is defined by the norm:

where $\nu = m+\theta, m$ is the integer part of $\nu$ and $\theta\in (0, 1)$ is the decimal part. The closure in $H^\nu(\Omega)$ of $\mathcal{D}(\Omega)$ is denoted $H_0^\nu(\Omega)$, where $\mathcal{D}(\Omega)$ is the space of infinitely differentiable functions whose support is contained in $\Omega$.

For any portion $\gamma$ of the boundary $\partial\Omega$ and any $\nu>0$, the Hilbert space $H^\nu(\gamma)$ is defined as the range of $H^{\nu+\frac{1}{2}}(\Omega)$ by the trace operator; it is then endowed with the image norm

Let the space $H^{-\nu}(\gamma)$ stand for the topological dual space of $H^{\nu}(\gamma)$ and the duality paring be denoted $<\cdot, \cdot>_{\nu, \gamma}$.

To be complete with the Sobolev functional tools for subsequent use, recall that for $\nu> \frac{3}{2}$, the trace operator

is continuous from $H^{\nu}(\Omega)$ onto $H^{\nu-\frac{1}{2}}(\partial\Omega)\times H^{\nu-\frac{3}{2}}(\partial\Omega)$.

Suppose the boundary $\partial\Omega$ is a union of three nonoverlapping portions $\Gamma_u, \Gamma_g$ and $\Gamma_C$. The vertices of $\Gamma_C$ are $\{\textbf{c}_1, \textbf{c}_2\}$ and those of $\Gamma_u$ are $\{\textbf{c}'_1, \textbf{c}'_2\}$. The part $\Gamma_u$ of nonzero measure is subjected to Dirichlet conditions while on $\Gamma_g$ a Neumann condition is prescribed, and $\Gamma_C$ is the candidate to be in contact with a rigid obstacle. To avoid technicalities arising from the special Sobolev space $H_{00}^{\frac{1}{2}}(\Gamma_C)$, we assume that $\Gamma_u$ and $\Gamma_C$ do not touch.

For the given data $f\in L^2(\Omega), g\in H^{-\frac{1}{2}}(\Gamma_g)$ and $\phi\in H^{\frac{1}{2}}(\Gamma_C)$, the Signorini problem consists of finding $u$ that verifies in a distributional sense:

where $\textbf{n}$ is the outward unit normal to $\partial\Omega$ and $\Gamma_C^0 = \{x\in \Gamma_C: u(x) = \varphi\}, \Gamma_C^1 = \{x\in \Gamma_C: u(x)>\phi\}$.

Remark 2.1. If the function $\phi\in P_1(\Gamma_C)$, we can call the above model linear contact condition model. Otherwise we call it nonlinear contact condition model. Most papers (see [1,2] only need to consider the case $\phi = 0$, since their analysis can be extended straightforwardly to the case $\phi\neq0$. However, in this paper, we will show that the same approximation method may have different convergence properties for the different contact condition model. Therefore, we must treat them by different techniques.

The functional framework well suited to solve problem (2.2)-(2.5) consists in working with a subset of the following Sobolev space:

equipedd with the seminorm

By the Friedrichs inequality, the semi-norm is actually a norm in $H_{\Gamma_u}^1(\Omega)$, which is equivalent to the natural one $\|\cdot\|_{1, \Omega}$. In the weak formulation, the unilateral contact condition on $\Gamma_C$ is taken into account by incorporating it in the closed cone

The primal variational principle for the Signorini problem produces the following variational inequality:

where

Obviously, the bilinear and linear forms fulfill the Stampacchia theorem's hypothesis, the continuity for both of them and the ellipticity for $a(\cdot, \cdot)$. Thus, the weak formulation (2.6) is well posed and has only one solution in $K(\Omega)$ that depends continuously on the data $(f, g, \phi)$.

3.   A new Crouzeix-Raviart approximation

For simplicity and to avoid more technicalities, suppose the domain $\Omega$ is polygonal in $R^2$. Let $\mathcal{J}_h$ be a regular triangulation of $\Omega$ with a maximum size $h$, and $K\in \mathcal{J}_h$ is the triangular element,

Moreover, the family $\mathcal{J}_h$ is built in such a way that the end points $\{\textbf{c}_1, \textbf{c}_2, \textbf{c}'_1, \textbf{c}'_2\}$ are always chosen as the vertices of some triangular elements.

For any $K\in \mathcal{J}_h, P_k(K)$ stands for the set of polynomials of total degree $\leq k$, and $\forall F\subset \partial K, K\in\mathcal{J}_h$, for any $v\in H^{\frac{1}{2}}(K),$ we define

Then we introduce the Crouzeix-Raviart finite element space corresponding to the partition $\mathcal{J}_h$, which is defined as

It is easy to see that $V_h$ is not a subspace of $H_{\Gamma_u}^1(\Omega)$ and is so-called nonconforming linear finite element.

Suppose the local interpolation $\Pi_K$ on an element $K$ is defined as

and the global interpolation $\Pi_h$,

Set the broken norm as

Obviously, $\|\cdot\|_{h}$ is a norm on $V_h$.

Then, we work with the following finite dimensional closed convex cone,

Note that we use a slight different discrete convex cone space from [13].

Now, we are in a position to define and study the nonconforming finite element approximation to problem (2.6), that is to say,

where

Since the $\|\cdot\|_h$ defined as (3.2) is a norm on $V_h$, by Stampacchia's Theorem, it can be proved that the discrete problem (3.4) has and only has one solution $u_h\in K_h(\Omega).$ Furthermore, the following abstract error estimate holds.

Theorem 3.1. Let $u\in K(\Omega)$ be the solution of the variational Signorini inequality (2.6), and $u_h\in K_h(\Omega)$ be the solution of the discrete problem (3.4). Further assume that $f\in L^2(\Omega), u\in H^{1+\nu}(\Omega), \frac{1}{2}<\nu\leq1$, then we have

Proof. Following the same lines of the proof of the second Strang lemma (cf. [5]), for any $v_h\in K_h(\Omega)$, we have

where

By Green's formula and the Signorini model (2.2)-(2.5),

Let us introduce the interpolation of zero order Raviart-Thomas element $RT$ (cf. [20]), which is defined by

on every element $K$ and $l_i, i = 1, 2, 3$ are three edges of $K$. $RT(\nabla u)$ does make sense because $\nabla u\in H(\mbox{div}, \Omega)$. Moreover, from the definition of $RT(\nabla u)$, we know that $RT(\nabla u)\cdot\textbf{n}$ is constant and continuous on the edges of element, so

By $\int_{F}(\nabla u-RT(\nabla u))\cdot \textbf{n}ds = 0$ and Green's formula, we have

Since $\mbox{div}(RT(\nabla u)) = M_K(\triangle u)$, so

A combination of (3.10) and the error of the interpolation $RT$ gives

As for $I_2$, noticing that $RT(\nabla u)\cdot\textbf{n}|_F = M_F(\frac{\partial u}{\partial\textbf{n}})$,

Now we turn to $I_3$, following the same argument of $I_2$, we can obtain

A combination of (3.6)-(3.13) yields

Then the Young's inequality asserts the desired result.

For the sake of the subsequent analysis, set

then we can divide $\Gamma_{Ch}$ into the following three non-overlapping sets:

Now, we will present the main result of this section.

Theorem 3.2. Let $u\in K(\Omega), u_h\in K_h(\Omega)$ be the solution of (3.4) and (3.6) respectively.

$i.$ Assume that $f\in L^2(\Omega), u, \phi\in H^{1+\nu}(\Omega)$ with $\frac{1}{2}<\nu<1$ and that the number of points in $\Gamma_C$, where the constraint changes from binding to nonbinding, is finite. Then, we have the following optimal error estimate,

$ii.$ Assume that $f\in L^2(\Omega), u, \phi\in H^{2}(\Omega)$. Then, we have the following optimal error estimate,

Proof. In view of Theorem 3.1, we only need to bound the approximation error and $J$, where $J = \sum\limits_{{F\subset \Gamma_C}}\int_{F}\frac{\partial u}{\partial\textbf{n}} (v_h-u_h)ds$. Since $\Pi_hu\in K_h(\Omega)$, we can take $v_h = \Pi_hu$ in (3.5). Then by the classical interpolation result (cf. [9]),

Now, let us concentrate on the bound of $J$, which is also a hard work. Noticing that $(u-\phi)\frac{\partial u}{\partial \textbf{n}}|_{\Gamma_{C}} = 0$, we have

By the definition of the interpolation (3.1), $J_1$ can be written as

Observing that

together with (see Lemma 7.1 of [3])

yield the estimate

We are in a position to bound $J_2$, it can be written as

By the same argument as $J_1$, $J_{21}$ can be estimated as

As for $J_{22},$ since $\frac{\partial u}{\partial \textbf{n}}|_{F} = 0, F\in \Gamma_{Ch}^1$, it can be decomposed as

Considering $(u-\phi)|_F = 0, \forall F\in \Gamma^0_{Ch}$ and $M_F(u_h)\geq M_F(\phi)$, we can derive

Now, the last work is to bound the term $J_{222}$. For a given $F\in\Gamma^-_{Ch}$, if $(\phi-u_h)\leq0$ on $F,$ since $\frac{\partial u}{\partial \textbf{n}}|_{\Gamma_C}\geq0$, then

Therefore, we only consider $F\in\Gamma^-_{Ch}$ on which there is a segment satisfies $(\phi-u_h)>0$. On the other hand, for a such $F$, $M_F(u_h)\geq M_F(\phi)$, namely, $\int_F(\phi-u_h)ds< 0$, so there exists one point $Q_F\in F$, such that $(\phi-u_h)(Q_F) = 0$. Bearing this fact in mind and applying Lemma 8.1 of [3], we have

Let us have a careful analysis of $F\in\Gamma^-_{Ch}$ again. If meas($F\bigcap \Gamma_C^0) = 0$, then $(u-\phi)> 0$ and $\frac{\partial u}{\partial \textbf{n}} = 0$ almost everywhere on $F$, in this case, the above term $-\int_{F}\frac{\partial u}{\partial \textbf{n}} (\phi-u_h)\, ds = 0$. Otherwise, meas($F\bigcap \Gamma_C^0)>0$, noticing that $u-\phi\in H^{1+\nu}(\Omega)\hookrightarrow C^0(\Omega)$, and by the Lebesgue theory, there must be a line segment $F'\subset F\in\Gamma^-_{Ch}$ such that $(u-\phi)|_{F'}\equiv0$. Thus we have $\frac{d(u-\phi)(s)}{ds}|_{F'}\equiv0$. Set $\frac{d(u-\phi)(s)}{ds} = v(s)$, when $\frac{1}{2}<\nu<1$, we can derive

Here the constant $C$ depends on $F'\subset F$, however, since the number of $F\in\Gamma^-_{Ch}$ is finite for the case $\frac{1}{2}<\nu<1$, we can choose the max of them and denote it by a generic constant $C.$

Let us introduce the interpolation of conforming linear finite element $I_h$, noticing that $\int_F\frac{d(u-I_hu)}{ds}ds = 0$ and $\frac{d(I_hu)}{ds}$ is a constant function, then a combination of (3.28) and (3.29) gives

When $\nu = 1$, considering a point $P_F\in F'\subset F$, $v(P_F) = 0$, then we can derive

Combining (3.28) and (3.31), we can estimate $J_{222}$ as

Then a combination of (3.5), (3.19), (3.23), (3.25), (3.27), (3.30) and (3.32) completes the proof.

Remark 3.1. In fact, the results $(ii)$ of Theorem 3.2 can also be extended to the quadrilateral meanvalue-oriented nonconforming rotated $\mathcal{Q}_1$ finite element (cf. [19]) with a minor modification.

4.   Another Crouzeix-Raviart element discretization

In this section, we will discuss an alternative approximation to the numerical model of the contact condition. A distinct idea is to enforce the nonnegativity of the Lagrange degrees of freedom of the discrete solution that are located on the contact region $\Gamma_C$, i.e., $(u_h-\phi)(m_F)\geq0$, where $m_F$ is midpoint of $F$ and $F\in\Gamma_{Ch}$. The approximation of the closed convex cone is defined as

The discrete variational inequality is expressed in the same line as the model presented in previous section and can be described to be:

Using again Stampacchia's Theorem, we know that the approximation problem (4.2) is well posed and the discrete solution is continuous with respect to the data $(f, g, \phi)$. Moreover, the abstract error estimate in Theorem 3.1 is still valid. The convergent properties can be summarized in the following two theorems.

Theorem 4.1. As for the linear contact condition model, that is to say, $\phi\in P_1(\Gamma_C)$, let $u\in K(\Omega)$ be the solution of the variational Signorini inequality (2.6), and $\widetilde{u}_h\in \widetilde{K}_h(\Omega)$ be the solution of the discrete problem (4.2).

$i.$ Assume that $f\in L^2(\Omega), u, \phi\in H^{1+\nu}(\Omega)$ with $\frac{1}{2}<\nu<1$ and that the number of points in $\Gamma_C$, where the constraint changes from binding to nonbinding, is finite. Then, we have the following optimal error estimate,

$ii.$ Assume that $f\in L^2(\Omega), u, \phi\in H^{2}(\Omega)$. Then, we have the following optimal error estimate,

Theorem 4.2. As for the nonlinear contact condition model, let $u\in K(\Omega)$ be the solution of the variational Signorini inequality (2.6), and $\widetilde{u}_h\in \widetilde{K}_h(\Omega)$ be the solution of the discrete problem (4.2).

$i.$ Assume that $f\in L^2(\Omega), u, \phi\in H^{1+\nu}(\Omega)$ with $\frac{1}{2}<\nu<1$ and that the number of points in $\Gamma_C$, where the constraint changes from binding to nonbinding, is finite. Then, we have the following optimal error estimate,

(ii) Assume $f\in L^2(\Omega), u, \phi\in H^2(\Omega)$ and that the number of points in $\Gamma_C$, where the constraints changes from binding to nonbinding, is finite, then we have

(iii) Assume $f\in L^2(\Omega), u, \phi\in H^2(\Omega),$ and $u|_{\Gamma_C}\in H^{2-\frac{1}{p}}(\Gamma_C), p>2$, then we have

Proof of Theorem 4.1. Since $\phi\in P_1(\Gamma_C)$, then by Trapezoidal formula, we have

which implies that $\widetilde{K}_h(\Omega) = K_h(\Omega)$. So Theorem 4.1 is followed by the results of Theorem 3.2.

Regarding the nonlinear contact condition case, $\Pi_hu$ does not belong to $K_h(\Omega)$ any more. Thus we need another interpolation $\widetilde{\Pi}_h$, which is defined to be

and for any $v\in H^2(K)$,

Since $\widetilde{\Pi}_hu\in \widetilde{K}_h(\Omega)$, we can take $\widetilde{\Pi}_hu$ in (3.5). Then by the known interpolation result, we have

Therefore, in order to prove Theorem 4.2, we only need to estimate $\widetilde{J} = \sum\limits_{F\in \Gamma_{Ch}}\int_{F}\frac{\partial u}{\partial \textbf{n}} (\widetilde{\Pi}_hu-\widetilde{u}_h)\, ds$.

Lemma 4.3. Let $u\in K(\Omega)$ be the solution of the variational Signorini inequality (2.6), and $\widetilde{u}_h\in \widetilde{K}_h(\Omega)$ be the solution of the discrete problem (4.2). Assume $f\in L^2(\Omega), u, \phi\in H^{1+\nu}(\Omega)$ with $\frac{1}{2}<\nu<1$ and that the number of points in $\Gamma_C$, where the constraint changes from binding to nonbinding, is finite. Then, we have the following optimal error estimate,

Proof. Follows (3.20), we have

Noticing that $\frac{\partial u}{\partial \textbf{n}}|_{\Gamma^+_{C}} = 0$, $\widetilde{J}_1$ can be presented as

Since $(u-\phi)|_F = 0, \forall F\in \Gamma^0_{Ch}$, then

As for $\widetilde{J}_{12}$, we use the arguments developed in ([1], Lemma 2.4). Setting $p = \frac{1}{\nu}$ and $p' = \frac{1}{1-\nu}$, noticing that $\frac{\partial u}{\partial\textbf{n}}\in H^{\nu-\frac{1}{2}}(\Gamma_C)$, $u\in H^{\nu+\frac{1}{2}}(\Gamma_C)$ and the Sobolev embedding theorem

we have $\frac{\partial u}{\partial\textbf{n}}\in L^{p'}(\Gamma_C)$ and $u\in L^{p}(\Gamma_C)$. Then H$\ddot{\mbox{o}}$lder inequality gives

Resorting to the Gagliardo-Nirenberg inequality yields

Going back to (4.16), and recalling that the number of $F\in \Gamma^-_{Ch}$ is bounded uniformly in $h$, we have

Following the same lines of the estimate of $J_2$ in section 3, one can prove

Then we complete the proof.

Lemma 4.4. Let $u\in K(\Omega)$ be the solution of the variational Signorini inequality (2.6), and $\widetilde{u}_h\in \widetilde{K}_h(\Omega)$ be the solution of the discrete problem (4.2). Assume $f\in L^2(\Omega), u, \phi\in H^2(\Omega)$ and that the number of points in $\Gamma_C$, where the constraints changes from binding to nonbinding, is finite, then we have

Proof. Proceeding as the same lines of Lemma 4.3, the bounds of $\widetilde{J}_{11}$ and $\widetilde{J}_2$ are also valid for the case $\nu = 1$, but we need to reestimate $\widetilde{J}_{12}$. We adopt the techniques developed in [1] (see Lemma 5.1),

where $\frac{1}{p}+\frac{1}{p'} = 1,$ then we set $p' = |\log h|,$ and obtain

The proof of the lemma is completed.

Lemma 4.5. Let $u\in K(\Omega)$ be the solution of the variational Signorini inequality (2.6), and $\widetilde{u}_h\in \widetilde{K}_h(\Omega)$ be the solution of the discrete problem (4.2). Assume $u, \phi\in H^2(\Omega),$ and $u|_{\Gamma_C}\in H^{2-\frac{1}{p}}(\Gamma_C), p>2$, then we have

Proof. We also only need to re-estimate $\widetilde{J}_{12}$. Observing that $\frac{\partial u}{\partial \textbf{n}}\in W^{1-\frac{1}{p}, 2}(\Gamma_C)\hookrightarrow C^0(\Gamma_C)$, then it is easy to know that $\forall F\in \Gamma_{Ch},$ $\frac{\partial u}{\partial \textbf{n}}$ vanishes at least once in $F$. Then by Lemma 8.1 of [3], we have

which implies the desired result.

Proof of Theorem 4.2. We put together Theorem 3.1 and Lemma 4.3, Lemma 4.4 and Lemma 4.5 to obtain point $(i), (ii)$ and $(iii)$ respectively.

Remark 4.1. Point $(iii)$ in Theorem 4.2 is also valid for the linear conforming finite element approximations considered in [1] and [2]. The optimal convergence rate can be recovered without "Assumption $A_3$" here, but with a slightly higher regular condition of the exact solution on $\Gamma_C$. In fact, from [18], we know that the best $u$ is expect to be of $H^\sigma$ with $\sigma<\frac{5}{2}$ in the vicinity of $\Gamma_C$.

Remark 4.2. If $u|_{\Gamma_C}\in W^{1, \infty}(\Gamma_C)$, the optimal convergence rate can also be recovered with the additional "Assumption $A_3$". This result is the same as that of linear conforming finite element approximation, which has been proved by Brezzi, Hager and Raviart in [7].

Remark 4.3. The results for the case $u\in H^2(\Omega)$ presented in this section can also be extended to the quadrilateral midpoint-oriented nonconforming rotated $\mathcal{Q}_1$ finite element (cf. [19]) with a slightly modification.

5.   Numerical test

In order to investigate the numerical behavior of the $P_1$ nonconforming finite element approximation to the Signorini problem, we consider the following equation:

where $\Omega = [0, 1]\times[0, 1]$, $\Gamma_u = [0, 1]\times\{1\}$ is the Dirichlet boundary, $\Gamma_C = [0, 1]\times\{0\}$ is the contact boundary and $\partial\Omega\setminus\{\Gamma_u\cup\Gamma_C\}$ is the Neumman boundary.

Since such a problem does not admit an analytic solution, in order to obtain the convergence order, we must compute a reference solution corresponding to a mesh which is refinement as possible as we can. In this example, we take the discrete solutions of the quadratic triangular finite element with the structured meshes for mesh size $h = \frac{1}{256}$ and $h = \frac{1}{512}$ as the reference solutions (denote by $u_{256}$ and $u_{512}$ respectively). Then we compute $u_{Ch}$ (resp. $u_{NCh}$) by conforming (resp. nonconforming) linear finite element methods using structured meshes for mesh sizes $h = \{\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}\}$, and we compare them with the reference solutions. The detailed numerical results are listed in Table 5.1 and 5.2. Here a.c.e. denotes the average convergence rate.

6.   Conclusion

This paper deals with the convergence properties of the nonconforming Crouzeix-Raviart finite element approximations to the Signorini problem. It is remarkable that the optimal convergence rate $\mathcal{O}(h)$ can be obtained by the meanvalue-oriented discretized method for any $\phi\in H^2(\Omega)$ without the additional assumption that the number of points in $\Gamma_C$, where the constraints changes from binding to nonbinding, is finite. Let us mention that the optimal convergence rate is obtained without any additional assumptions. We note that though the conforming linear finite element method exhibits better numerical results than the nonconforming one for many practical problems (second order elliptic problems etc.), but this may not be true for the Signorini problem in incompressible elasticity. Then an important and attractive direction is to develop a locking-free nonconforming Crouzeix-Raviart finite element method for the Signorini problem in incompressible elasticity, which is a future work of us.