A Nitsche-based method for unilateral contact problems: numerical analysis

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A Nitsche-based method for unilateral contact problems:

numerical analysis

Franz Chouly, Patrick Hild

To cite this version:

Franz Chouly, Patrick Hild. A Nitsche-based method for unilateral contact problems: numerical analysis. SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2013, 51 (2), p. 1295-1307. �10.1137/12088344X�. �hal-00717711�

A Nitsche-based method for unilateral contact problems:

numerical analysis

Franz Chouly ^∗ Patrick Hild^†

Abstract

We introduce a Nitsche-based formulation for the finite element discretization of the uni- lateral contact problem in linear elasticity. It features a weak treatment of the non-linear contact conditions through a consistent penalty term. Without any additional assumption on the contact set, we can prove theoretically its fully optimal convergence rate in theH¹(Ω)- norm for linear finite elements in two dimensions, which isO(h^12+ν) when the solution lies in H^32+ν(Ω), 0 < ν ≤ 1/2. An interest of the formulation is that, conversely to Lagrange multiplier-based methods, no other unknown is introduced and no discrete inf–sup condition needs to be satisfied.

Key words: unilateral contact, finite elements, Nitsche’s method.

AMS Subject Classification: 65N12, 65N30, 74M15.

1 Introduction

The numerical implementation of contact and impact problems in solid mechanics generally uses the Finite Element Method (FEM) (see [17, 18, 20, 27, 29, 36, 38]). These problems involve non-linear contact conditions on a part of the boundary that lead naturally to a variational inequality (see, e.g., [14]). In this paper, we limit ourselves to the case of frictionless unilateral contact in plane linear elasticity. Indeed, this case contains the key difficulties related to the non-linear contact conditions. We also limit our study to the approximation with continuous piecewise linear finite elements, which is the most standard approach.

We introduce a special FEM inspired from Nitsche method [31, 2, 35, 4, 21]. Conversely to standard penalization techniques [27], the resulting method is consistent. Moreover, unlike mixed methods (see, e.g., [20]), no additional unknown (Lagrange multiplier) is needed. However, the most interesting in this method is that we are able to prove its optimal convergence in the H¹(Ω)-norm of order O(h^12+ν) provided the solution has a regularity H^32+ν(Ω), 0 < ν ≤1/2.

∗Laboratoire de Math´ematiques - UMR CNRS 6623, Universit´e de Franche Comt´e, 16 route de Gray, 25030 Besan¸con Cedex, France. email: [email protected]

†Institut de Math´ematiques de Toulouse - UMR CNRS 5219, Universit´e Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France. email: [email protected]

Furthermore, we do not need any additional assumption on the contact zone, such as an increased regularity of the contact stress or a finite number of transitions between contact and non-contact.

Besides, the standard FEM for contact consists in a direct conforming approximation of the variational inequality, with the elastic displacement as the only unknown. For this standard FEM and also for all the other approaches such as mixed/hybrid methods (e.g., [7]), stabilized mixed methods (e.g., [23]), penalty methods (e.g., [10]), no such proof of optimal convergence has been established to the best of our knowledge in the case the solution u is in H^32+ν(Ω) (0< ν ≤1/2). The first analyzes in [34, 19, 20] were sub-optimal with a convergence inO(h^12+ν2) and these results were recently improved in [24] to obtain a convergence rate of O(h^12+ν2+ν2) when (0 < ν <1/2), ofO(hp

|ln(h)|) when ν = 1/2 and in [33] where a convergence ofO(h) is obtained when ν = 1/2 +ε (ε >0 arbitrarily small). In fact, additional assumptions on the finiteness of transition points between contact and non-contact are needed to recover optimality (see [6] for the case 0< ν <1/2 and [25] for the caseν = 1/2 in the case of the standard FEM).

Our paper is outlined as follows. In Section 2, we recall the continuous (strong and weak) formulations for unilateral contact problems and introduce our Nitsche-based FEM. In Section 3, we carry out the numerical analysis of this method: we prove its consistency, the existence and uniqueness of solutions, and at least its optimal convergence. Conclusion and perspectives are drawn in Section 4.

Let us introduce some useful notations. In what follows, bold letters likeu,v, indicate vector or tensor valued quantities, while the capital ones (e.g.,V,K. . .) represent functional sets involving vector fields. As usual, we denote by (H^s(.))^d,s∈R, d= 1,2 the Sobolev spaces in one or two space dimensions (see [1]). The usual norm of (H^s(D))^dis denoted by k · k_s,D and we keep the same notation whend= 1 or d= 2. The letter C stands for a generic constant, independent of the discretization parameters.

2 Setting

2.1 The unilateral contact problem

We consider an elastic body Ω in R² where plane small strain assumptions are made. The boundary ∂Ω of Ω is polygonal and we suppose that∂Ω consists in three nonoverlapping parts ΓD, ΓN and the contact boundary ΓC, with meas(ΓD) > 0 and meas(ΓC) > 0. The contact boundary is supposed to be a straight line segment to simplify ; the normal unit outward vector on∂Ω is denotedn. In its initial stage, the body is in contact on Γ_C with a rigid foundation (the extension to two elastic bodies in contact can be easily made, at least for small strain models) and we suppose that the unknown final contact zone after deformation will be included into Γ_C. The body is clamped on Γ_D for the sake of simplicity. It is subjected to volume forces f ∈(L²(Ω))² and to surface loads g∈(L²(ΓN))².

The unilateral contact problem in linear elasticity consists in finding the displacement field

u: Ω→R² verifying the equations and conditions (1)–(2):

divσ(u) +f =0 in Ω, σ(u) =Aε(u) in Ω, u=0 on ΓD, σ(u)n=g on Γ_N,

(1)

where σ = (σij), 1≤i, j≤2,stands for the stress tensor field and divdenotes the divergence operator of tensor valued functions. The notationε(v) = (∇v+∇v^T)/2 represents the linearized strain tensor field andAis the fourth order symmetric elasticity tensor having the usual uniform ellipticity and boundedness property. For any displacement fieldvand for any density of surface forcesσ(v)n defined on∂Ω we adopt the following notation

v=v_nn+v_t and σ(v)n=σ_n(v)n+σ_t(v),

where vt (resp. σt(v)) are the tangential components of v (resp. σ(v)n). The conditions describing unilateral contact without friction on Γ_C are:

un≤ 0 (i) σ_n(u)≤ 0 (ii) σ_n(u)u_n= 0 (iii)

σ_t(u) = 0 (iv)

(2)

We introduce the Hilbert space V and the convex cone K of admissible displacements which satisfy the noninterpenetration on the contact zone ΓC:

V:=

n

v∈ H¹(Ω)2

:v=0 on ΓD

o

, K:={v∈V : vn=v·n≤0 on ΓC}. Define

a(u,v) :=

Z

Ω

σ(u) :ε(v)dΩ, L(v) :=

Z

Ω

f ·vdΩ + Z

ΓN

g·vdΓ,

for anyuandvinV. From the previous assumptions, we know thata(·,·) is bilinear, symmetric, V-elliptic and continuous onV×V. We know also thatL(·) is a continuous linear form on V.

The weak formulation of Problem (1)-(2), as a variational inequality (see [14, 20, 27]), is:

Find u∈Ksuch that:

a(u,v−u)≥L(v−u), ∀v∈K. (3)

Stampacchia’s Theorem ensures that Problem (3) admits a unique solution.

2.2 The Nitsche-based finite element method

Let V^h ⊂ V be a family of finite dimensional vector spaces (see [11]) indexed by h coming from a family T^h of triangulations of the domain Ω (h= max_T∈Thh_T whereh_T is the diameter of T). The family of triangulations is supposed regular, i.e., there exists σ > 0 such that

∀T ∈ T^h, hT/ρT ≤ σ where ρT denotes the radius of the inscribed circle in T. We choose standard continuous and piecewise affine functions, i.e.:

V^h = n

v^h ∈(C(Ω))² :v^h|T ∈(P1(T))²,∀T ∈ T^h,v^h=0on ΓD

o .

Let us introduce the notation [·]₊ for the positive part of a scalar quantitya∈R:

[a]₊ =

a ifa >0, 0 otherwise.

In the rest of this paper, we will make an extensive use of the following properties:

a≤[a]₊, a[a]₊= [a]²₊, ∀a∈R. (4) Note that conditions (4) can be straightforwardly extended to real valued functions. The deriva- tion of a Nitsche-based method comes from the observation that the contact conditions (2) can be reformulated as follows:

Proposition 2.1. Let γ >0. The contact conditions (2) (i)-(iii) on ΓC are equivalent to:

σ_n(u) =−1

γ[u_n−γσ_n(u)]₊. (5)

Proof: Letu be a regular vector field on Ω such that (2) (i)-(iii) holds. The condition (2)-(ii) yields either σ_n(u)<0 or σ_n(u) = 0. Suppose first that σ_n(u)<0. Then (2)-(iii) implies that un= 0. In this case, it holds:

−1

γ[u_n−γσ_n(u)]₊=−1

γ[−γσ_n(u)]₊=σ_n(u).

Then, suppose that σ_n(u) = 0. The condition (2)-(i) can also be expressed as [u_n]₊ = 0, so:

−1

γ[un−γσn(u)]+ =−1

γ[un]+ = 0 =σn(u).

Reciprocally, let u such that (5) holds. Note first that it implies σ_n(u) ≤ 0, so that (2)-(ii) holds. Take first the case σn(u) = 0, then (5) can be rewritten as:

0 =−1 γ[u_n]₊,

which is equivalent to condition (2)-(i). Since σ_n(u) = 0, then condition (2)-(iii) (σ_n(u)u_n= 0) also holds.

We now consider the case σn(u)<0. From (5), [un−γσn(u)]+>0, so that in this case

−γσ_n(u) = [u_n−γσ_n(u)]₊=u_n−γσ_n(u),

from which comes u_n= 0, so that both (2) (i) and (iii) hold.

Remark 2.2. Note that condition (5) is still equivalent to (2)(i)-(iii) onΓ_C whenγ is a positive function defined on ΓC instead of a positive constant.

Let nowube the solution of the unilateral contact problem in its strong form (1)–(2), sufficiently regular so that all the following calculations make sense. From the Green formula and equations (1), (2)-(iv), it holds for every v∈V:

a(u,v)− Z

ΓC

σ_n(u)v_ndΓ =L(v).

Note thatv_n= (v_n−γσ_n(v)) +γσ_n(v), so that:

a(u,v)− Z

ΓC

γ σ_n(u)σ_n(v)dΓ− Z

ΓC

σ_n(u) (v_n−γσ_n(v))dΓ =L(v).

Finally, using condition (5), we obtain:

a(u,v)− Z

ΓC

γ σn(u)σn(v)dΓ + Z

ΓC

1

γ[un−γσn(u)]+(vn−γσn(v))dΓ =L(v). (6) Formula (6) is the starting point of our Nitsche-based formulation. Remark that it may have no sense at the continuous level if ulacks of regularity (u∈V only is not sufficient for instance to justify the above calculations). Nevertheless, and as in the stabilized Lagrange multiplier method [23], we consider in what follows thatγ is a positive piecewise constant function on the contact interface ΓC : for anyx∈ΓC, letT be an element such thatx∈T and set:

γ(x) =γ₀h_T (7)

where γ₀ is a positive constant. This allows to define a discrete counterpart of (6). Let us introduce for this purpose the discrete linear operator

Pγ: V^h → L²(ΓC) v^h 7→ v_n^h−γ σn(v^h) , and also the bilinear form:

Aγ(u^h,v^h) =a(u^h,v^h)− Z

ΓC

γ σn(u^h)σn(v^h)dΓ.

Remark 2.3. With the previous notations, we see that problem (3) could be formally written as follows :







Find a sufficiently regular u∈V such that:

A_γ(u,v) + Z

ΓC

1

γ [P_γ(u)]₊P_γ(v)dΓ =L(v), for all sufficiently regular v∈V.

Our Nitsche-based method then reads:







Findu^h ∈V^h such that:

Aγ(u^h,v^h) + Z

ΓC

1

γ [Pγ(u^h)]+Pγ(v^h)dΓ =L(v^h), ∀v^h∈V^h. (8) 2.3 A reinterpretation of the method

The aim of this subsection is to show some links between the proposed method and the stabilized mixed/hybrid finite element methods (which do not require inf-sup conditions) introduced in [26]. We define the auxiliary variable

λ^h :=−1

γ[Pγ(u^h)]+,

which can be interpreted as the L²-projection of the quantity −_γ¹P_γ(u^h) on the convex cone:

L²₋(Γ_C) :={µ∈L²(Γ_C)|µ≤0 a.e. on Γ_C}.

This means in particular thatλ^h∈L²₋(Γ_C) verifies the following inequality, for allµ∈L²₋(Γ_C):

Z

ΓC

(µ−λ^h)(−1

γP_γ(u^h)−λ^h)dΓ≤0, (9) that is the standard characterization of the projection onto a closed convex set. We then use the definition of the operator Pγ to obtain

Z

ΓC

(µ−λ^h)(−1

γP_γ(u^h)−λ^h)dΓ = Z

ΓC

(µ−λ^h)(−1

γu^h_n+σ_n(u^h)−λ^h)dΓ

=− Z

ΓC

1 γ

(µ−λ^h)u^h_n+γ(µ−λ^h)(λ^h−σn(u^h))

dΓ.

Sinceγ is a positive piecewise constant function on ΓC (see (7)), the above equalities prove that the inequality (9) is equivalent to:

Z

ΓC

(µ−λ^h)u^h_ndΓ + Z

ΓC

γ(µ−λ^h)(λ^h−σn(u^h))dΓ≥0. (10)

The next step is to rewrite the left hand-side of the Nitsche-based method (8) as follows:

Aγ(u^h,v^h) + Z

Γ_C

1

γ [Pγ(u^h)]+Pγ(v^h)dΓ

=a(u^h,v^h)− Z

ΓC

γ σn(u^h)σn(v^h)dΓ− Z

ΓC

λ^h(v^h_n−γ σn(v^h))dΓ

=a(u^h,v^h)− Z

ΓC

λ^hv_n^hdΓ + Z

ΓC

γ(λ^h−σn(u^h))σn(v^h)dΓ.

We combine this last result to the inequality (10) to obtain an equivalent formulation of the Nitsche-based method:















Find (u^h, λ^h)∈V^h×L²₋(Γ_C) such that:

a(u^h,v^h)− Z

Γ_C

λ^hv^h_ndΓ + Z

Γ_C

γ(λ^h−σn(u^h))σn(v^h)dΓ =L(v^h), ∀v^h ∈V^h, Z

ΓC

(µ−λ^h)u^h_ndΓ + Z

ΓC

γ(µ−λ^h)(λ^h−σn(u^h))dΓ≥0, ∀µ∈L²₋(ΓC).

We finally observe that the Nitsche-based method can be regarded as a mixed method with a stabilization term (see [3]). This makes it close to the stabilized method proposed and analyzed in [23], the difference being that in our case, the space for the Lagrange multiplierλ^h isL²₋(ΓC).

A similar analogy has already been noticed in [35] in the case of elliptic problems with Dirichlet boundary conditions.

3 Analysis of the Nitsche-based method

In this section, we carry out the mathematical analysis of the method (8). A difference between Nitsche and classical penalty methods [27] is that this former is consistent, which we first show in §3.1. The proof of well-posedness of the (non-linear) discrete problem (8) is carried out in

§3.2. The error analysis is finally detailed in§3.3. We show that the method converges optimally for a fixed value ofγ₀ and when the mesh size h vanishes.

3.1 Consistency

As well as the Nitsche’s method for second order elliptic problems with Dirichlet boundary condition or domain decomposition [4], our Nitsche-based formulation (8) for unilateral contact is consistent:

Lemma 3.1. The Nitsche-based method for contact is consistent: suppose that the solution u of (1)–(2) is in (H^32+ν(Ω))², withν >0, then u is also solution of

Aγ(u,v^h) + Z

ΓC

1

γ [Pγ(u)]+Pγ(v^h)dΓ =L(v^h), ∀v^h∈V^h.

Proof: Let u be the solution of (1)–(2) and take v^h ∈V^h. Since u∈(H^32+ν(Ω))² and ν >0, we have σn(u) ∈H^ν(ΓC) ⊂L²(ΓC). As a result, Pγ(u) ∈ L²(ΓC) and Aγ(u,v^h) makes sense.

On the one hand, we use the definition of P_γ, of A_γ(·,·) and the reformulation (5) of contact conditions to obtain:

Aγ(u,v^h) + Z

ΓC

1

γ [Pγ(u)]+Pγ(v^h)dΓ

=a(u,v^h)− Z

ΓC

γ σ_n(u)σ_n(v^h)dΓ + Z

ΓC

1

γ (−γσ_n(u))(v^h_n−γ σ_n(v^h))dΓ

=a(u,v^h)− Z

ΓC

σ_n(u)v_n^hdΓ.

On the other hand, with equations (1)–(2) and integration by parts, it holds:

a(u,v^h)− Z

ΓC

σn(u)v_n^hdΓ =L(v^h),

which ends the proof.

3.2 Well-posedness

To prove well-posedness of our Nitsche-based formulation, we need first the following classical lemma that we recall here for convenience of the reader.

Lemma 3.2. Letγ0 be sufficiently small. Then the bilinear formAγ(·,·)is elliptic onV^h: there exists C, independent of the parameter γ₀ and of the mesh size h, such that:

Aγ(v^h,v^h)≥Ckv^hk²_1,Ω, for all v^h ∈V^h.

Proof: Note that we can suppose without loss of generality that Γ_C is a straight line segment parallel to the x−axis. Let E be an edge of a triangle on ΓC and let T ∈ T^h be the element containing E. Consequently we deduce, for anyv^h ∈V^h:

kσ_n(v^h)k_0,E = kσ_yy(v^h)k_0,E

= |E|¹²

|T|¹² kσ_yy(v^h)k_0,T

≤ Ch⁻

1 2

T kσ_yy(v^h)k_0,T

= C

γ γ₀

−¹

2

kσ_yy(v^h)k_0,T.

By summation on all the edges E ⊂ΓC we get kγ¹²σn(v^h)k²_0,Γ

C ≤Cγ0kσ_yy(v^h)k²_0,Ω ≤Cγ0kv^hk²_1,Ω. (11) Hence, from Korn inequality and (11), when γ₀ is small enough, there exists C > 0 such that for any v^h ∈V^h:

a(v^h,v^h)− Z

ΓC

γ(σ_n(v^h))²dΓ≥Ckv^hk²_1,Ω.

We then show that Problem (8) is well-posed using an argument proposed by Brezis for M type and pseudo-monotone operators [9] (see also [30] and [28]):

Theorem 3.3. Suppose that γ0 is sufficiently small. Then the problem (8) admits one unique solution u^h in V^h.

Proof: Using the Riesz representation theorem, we define a (nonlinear) operatorB^h:V^h →V^h, with help of the formula:

(B^hv^h,w^h)_1,Ω=A_γ(v^h,w^h) + Z

ΓC

1

γ [P_γ(v^h)]₊P_γ(w^h)dΓ,

for all v^h,w^h ∈ V^h, and where (·,·)_1,Ω stands for the scalar product in (H¹(Ω))². Note that Problem (8) is well-posed if and only if B is a one-to-one operator.

Due to the properties (4), we observe that, for all a, b∈R:

([a]₊−[b]₊)(a−b) =a[a]₊+b[b]₊−b[a]₊−a[b]₊

≥[a]²₊+ [b]²₊−2[a]+[b]+

= ([a]₊−[b]₊)² ≥0.

Let v^h,w^h ∈V^h, using the above property and then Lemma 3.2, it holds:

(B^hv^h−B^hw^h,v^h−w^h)_1,Ω

=A_γ(v^h−w^h,v^h−w^h) + Z

ΓC

1

γ ([P_γ(v^h)]₊−[P_γ(w^h)]₊)P_γ(v^h−w^h)dΓ

≥A_γ(v^h−w^h,v^h−w^h)

≥Ckv^h−w^hk²_1,Ω.

(12)

Next, let us show that B^h is also hemicontinuous. SinceV^h is a vector space, it is sufficient to show that

[0,1]3t7→ϕ(t) = (B^h(v^h−tw^h),w^h)_1,Ω ∈R is a continuous real function, for all v^h,w^h ∈V^h. Lets, t∈[0,1], we bound:

|ϕ(t)−ϕ(s)|

= |(B^h(v^h−tw^h)−B^h(v^h−sw^h),w^h)_1,Ω|

=

Aγ((s−t)w^h,w^h) + Z

ΓC

1 γ

[Pγ(v^h−tw^h)]+−[Pγ(v^h−sw^h)]+

Pγ(w^h)dΓ

≤ |s−t|Aγ(w^h,w^h) + Z

ΓC

1 γ

[Pγ(v^h−tw^h)]+−[Pγ(v^h−sw^h)]+

|P_γ(w^h)|dΓ.

With help of the bound |[a]₊−[b]+| ≤ |a−b|, for alla, b∈R, and using the linearity of Pγ, we then remark that:

Z

ΓC

1 γ

[P_γ(v^h−tw^h)]₊−[P_γ(v^h−sw^h)]₊

|P_γ(w^h)|dΓ

≤ Z

ΓC

1 γ

P_γ(v^h−tw^h)−P_γ(v^h−sw^h)

|P_γ(w^h)|dΓ

= Z

ΓC

1

γ |(s−t)P_γ(w^h)||P_γ(w^h)|dΓ

= |s−t|

Z

Γ_C

1 γ

Pγ(w^h) 2

dΓ.

It results that:

|ϕ(t)−ϕ(s)| ≤ |s−t|

A_γ(w^h,w^h) +kγ⁻¹²P_γ(w^h)k²_0,Γ

C

,

which means that ϕis Lipschitz, so thatB^h is hemicontinuous. Since property (12) also holds, we finally apply the Corollary 15 (p.126) of [9] to conclude that B^h is a one-to-one operator.

This ends the proof.

3.3 Error analysis

Our Nitsche-based method (8) converges optimally when the mesh parameter h vanishes. This is the object of the following theorems. First, we establish an abstract error estimate.

Theorem 3.4. Suppose that the solution u of Problem (3) belongs to (H^32+ν(Ω))² with ν >0.

Suppose also that the parameter γ0 is sufficiently small. The solution u^h of Problem (8)satisfies the following error estimate:

ku−u^hk_1,Ω+kγ¹²(σn(u) + 1

γ[Pγ(u^h)]+)k_0,ΓC

≤ C inf

v^h∈V^h

(ku−v^hk_1,Ω+kγ⁻¹²(un−v^h_n)k_0,ΓC+kγ¹²σn(u−v^h)k_0,ΓC ,

(13)

with C >0 a constant, independent ofh and u.

Proof: Let v^h ∈ V^h. We first use the V-ellipticity and the continuity of a(·,·), as well as Young’s inequality, to obtain:

αku−u^hk²_1,Ω ≤a(u−u^h,u−u^h)

=a(u−u^h,(u−v^h) + (v^h−u^h))

≤Cku−u^hk_1,Ωku−v^hk_1,Ω+a(u−u^h,v^h−u^h)

≤ α

2ku−u^hk²_1,Ω+C²

2αku−v^hk²_1,Ω+a(u,v^h−u^h)−a(u^h,v^h−u^h), with α >0 the ellipticity constant. We can transform the terma(u,v^h−u^h)−a(u^h,v^h−u^h) sinceu solves (3) and u^h solves (8), which yields:

α

2ku−u^hk²_1,Ω≤ C²

2αku−v^hk²_1,Ω+ Z

ΓC

σ_n(u)

v_n^h−u^h_n dΓ

− Z

ΓC

γ σ_n(u^h)σ_n(v^h−u^h)dΓ + Z

ΓC

1

γ[P_γ(u^h)]₊P_γ(v^h−u^h)dΓ

= C²

2αku−v^hk²_1,Ω+ Z

ΓC

σn(u)

v_n^h−u^h_n

dΓ

− Z

ΓC

γ σn(u^h)σn(v^h−u^h)dΓ + Z

ΓC

1

γ[Pγ(u^h)]+

v^h_n−γσn(v^h)

dΓ

− Z

ΓC

1

γ[P_γ(u^h)]²₊dΓ, (14)

where we use (4). We now write Z

Γ_C

σn(u)

v^h_n−u^h_n

dΓ + Z

Γ_C

1

γ[Pγ(u^h)]+v_n^hdΓ

= Z

ΓC

σ_n(u) + 1

γ[P_γ(u^h)]₊

v^h_n−u_n dΓ +

Z

ΓC

σ_n(u)

u_n−u^h_n dΓ +

Z

ΓC

1

γ[P_γ(u^h)]₊u_ndΓ

≤ Z

ΓC

σn(u) + 1

γ[Pγ(u^h)]+

v^h_n−un

dΓ +

Z

ΓC

σn(u)

un−u^h_n

dΓ,

because of contact condition (2)-(i). We use this last inequality into the global estimate (14) to obtain:

α

2ku−u^hk²_1,Ω

≤C²

2αku−v^hk²_1,Ω+ Z

ΓC

σ_n(u) + 1

γ[P_γ(u^h)]₊

v_n^h−u_n dΓ +

Z

ΓC

σ_n(u)

u_n−u^h_n dΓ

− Z

Γ_C

γ σn(u^h)σn(v^h−u^h)dΓ− Z

Γ_C

1

γ[Pγ(u^h)]+γσn(v^h)dΓ− Z

Γ_C

1

γ[Pγ(u^h)]²₊dΓ

=C²

2αku−v^hk²_1,Ω+ Z

ΓC

σ_n(u) + 1

γ[P_γ(u^h)]₊

v_n^h−u_n

dΓ +T. (15)

The integral term in (15) is bounded thanks to Cauchy-Schwarz and Young inequalities:

Z

ΓC

σ_n(u) + 1

γ[P_γ(u^h)]₊

v_n^h−u_n dΓ

≤ kγ¹²(σn(u) + 1

γ[Pγ(u^h)]+)k_0,ΓCkγ⁻¹²(v_n^h−un)k_0,ΓC

≤ 1 2β1

kγ¹²(σ_n(u) +1

γ[P_γ(u^h)]₊)k²_0,Γ

C+ β₁

2 kγ⁻¹²(v_n^h−u_n)k²_0,Γ

C, (16)

with β1 >0. We then estimate the (four) integrals in (15) coming from T. Using (2)-(iii), we get

T =− Z

ΓC

σn(u)u^h_ndΓ− Z

ΓC

γ σn(u^h)σn(v^h−u^h)dΓ− Z

ΓC

1

γ[Pγ(u^h)]+γσn(v^h)dΓ

− Z

ΓC

1

γ[Pγ(u^h)]²₊dΓ

=− Z

ΓC

σ_n(u)(u^h_n−γσ_n(u^h))dΓ− Z

ΓC

γσ_n(u)σ_n(u^h)dΓ

− Z

ΓC

γ σ_n(u^h)σ_n(v^h−u^h)dΓ− Z

ΓC

1

γ[P_γ(u^h)]₊γσ_n(v^h)dΓ− Z

ΓC

1

γ[P_γ(u^h)]²₊dΓ.

Once again we use (4) and contact condition (2)-(ii) to note that:

− Z

ΓC

σ_n(u)(u^h_n−γσ_n(u^h))dΓ =− Z

ΓC

σ_n(u)P_γ(u^h)dΓ≤ − Z

ΓC

σ_n(u)[P_γ(u^h)]₊dΓ.

It results into:

T ≤ − Z

ΓC

σn(u)[Pγ(u^h)]+dΓ− Z

ΓC

γσn(u)σn(u^h)dΓ

− Z

ΓC

γ σ_n(u^h)σ_n(v^h−u^h)dΓ− Z

ΓC

1

γ[P_γ(u^h)]₊γσ_n(v^h)dΓ− Z

ΓC

1

γ[P_γ(u^h)]²₊dΓ

=− Z

ΓC

σn(u) + 1

γ[Pγ(u^h)]+

[Pγ(u^h)]+dΓ− Z

ΓC

γσn(u)σn(u^h)dΓ

− Z

ΓC

γ σ_n(u^h)σ_n(v^h−u^h)dΓ− Z

ΓC

1

γ[P_γ(u^h)]₊+σ_n(u)

γσ_n(v^h)dΓ +

Z

ΓC

γσ_n(u)σ_n(v^h)dΓ

=− Z

ΓC

γ

σn(u) + 1

γ[Pγ(u^h)]+

1

γ[Pγ(u^h)]++σn(u)

dΓ− Z

ΓC

γσn(u)σn(u^h)dΓ

− Z

ΓC

γ σ_n(u^h)σ_n(v^h−u^h)dΓ + Z

ΓC

γ

σ_n(u) + 1

γ[P_γ(u^h)]₊

σ_n(u−v^h)dΓ +

Z

ΓC

γσ_n(u)σ_n(v^h)dΓ.

As a consequence

T +kγ¹²(σ_n(u) + 1

γ[P_γ(u^h)]₊)k²_0,Γ

C ≤ Z

ΓC

γ

σ_n(u) + 1

γ[P_γ(u^h)]₊

σ_n(u−v^h)dΓ +

Z

ΓC

γσn(u−u^h)σn(v^h−u^h)dΓ. (17) With help of Cauchy-Schwarz and Young inequalities, we bound the first term in (17) :

Z

Γ_C

γ

σn(u) +1

γ[Pγ(u^h)]+

σn(u−v^h)dΓ≤ 1 2β2

kγ¹²(σn(u) + 1

γ[Pγ(u^h)]+)k²_0,Γ

C

+β2

2 kγ¹²σn(u−v^h)k²_0,Γ

C, (18)

with β₂>0.

The second term in (17) is bounded as follows by using (11) and Young inequality:

Z

ΓC

γσ_n(u−u^h)σ_n(v^h−u^h)dΓ

≤ kγ¹²σn(u−u^h)k_0,ΓCkγ¹²σn(v^h−u^h)k_0,ΓC

≤ Cγ

1 2

0kv^h−u^hk_1,Ω

kγ¹²σ_n(u−v^h)k_0,ΓC +kγ¹²σ_n(v^h−u^h)k_0,ΓC

≤ C

γ0kv^h−u^hk²_1,Ω+kγ¹²σn(u−v^h)k²_0,Γ

C

≤ C

γ₀ku−u^hk²_1,Ω+γ₀ku−v^hk²_1,Ω+kγ¹²σ_n(u−v^h)k²_0,Γ

C

. (19)

Putting together (18) and (19) in (17) gives:

T + (1− 1 2β2

)kγ¹²(σ_n(u) + 1

γ[P_γ(u^h)]₊)k²_0,Γ

C ≤C

γ₀ku−u^hk²_1,Ω+γ₀ku−v^hk²_1,Ω + (C+β₂

2 )kγ¹²σ_n(u−v^h)k²_0,Γ

C. (20) We finally combine estimates (16) and (20) into (15) to get:

(α

2 −Cγ₀)ku−u^hk²_1,Ω+ (1− 1 2β₁ − 1

2β₂)kγ¹²(σ_n(u) + 1

γ[P_γ(u^h)]₊)k²_0,Γ

C

≤(C²

2α +Cγ₀)ku−v^hk²_1,Ω+β₁

2 kγ⁻¹²(u_n−v_n^h)k²_0,Γ

C + (C+β₂

2 )kγ¹²σ_n(u−v^h)k²_0,Γ

C. To conclude, take γ0 sufficiently small and for instanceβ1 =β2 = 2.

The optimal convergence of the method is stated below.

Theorem 3.5. Suppose that the solutionu of Problem (3)belongs to (H^32+ν(Ω))² with0< ν ≤ 1/2. Suppose also that the parameter γ₀ is sufficiently small. The solution u^h of Problem (8) satisfies the following error estimate:

ku−u^hk_1,Ω+kγ¹²(σn(u) + 1

γ[Pγ(u^h)]+)k_0,ΓC ≤Ch^12+νkuk3

2+ν,Ω, (21)

with C >0 a constant, independent ofh and u.

Proof: We need to bound the right terms in estimate (13) and we choose v^h = I^hu where I^h stands for the Lagrange interpolation operator mapping onto V^h. The estimation of the Lagrange interpolation error in H¹-norm on a domain Ω is classical (see, e.g., [8, 13]):

ku− I^huk_1,Ω≤Ch^12+νkuk3

2+ν,Ω, (22)

for−1/2< ν ≤1/2.

The estimation of the term kγ⁻¹²(u_n−(I^hu)_n)k_0,ΓC can be done in a very similar manner to [23]. Indeed, let E be an edge of a triangle T ∈ T^h on ΓC:

kγ⁻¹²(un−(I^hu)n)k_0,E ≤Ch⁻

1 2

T h^1+ν_T ku_nk_1+ν,E =Ch^12+νku_nk_1+ν,E, (see [13] for instance). By summation on all the edges and the trace theorem, it results:

kγ⁻¹²(un−(I^hu)n)k_0,ΓC ≤Ch^12+νku_nk_1+ν,ΓC ≤Ch^12+νkuk3

2+ν,Ω. (23)

From [16] (see also the details in [15]), the following estimate also holds:

kγ¹²σ_n(u− I^hu)k_0,ΓC ≤Ch^12+νkuk3

2+ν,Ω. (24)

We conclude by inserting the three estimates (22)–(24) into (13).

Actually we are not able to obtain estimates for the displacements in the L²-norm ku−u^hk_0,Ω (and also ku −u^hk_0,ΓC) by using the Aubin-Nitsche argument as it is achieved in the linear case (see [16]). Note also that the L²(Ω)-norm estimates for contact problems are not easy to prove and there are to our knowledge only few estimates (see [12]). Nevertheless, we can easily obtain the following error estimate on the weighted L²(ΓC)-norm on the normal constraint kγ¹²σ_n(u−u_h)k_0,ΓC (note that σ_n(u_h)6=−¹_γ[P_γ(u^h)]₊ on Γ_C contrary to the continuous case).

Corollary 3.6. Suppose that the solution u of Problem (3) belongs to (H^32+ν(Ω))² with 0 <

ν ≤1/2. Suppose also that the parameter γ0 is sufficiently small. The solution u^h of Problem (8) satisfies the following error estimate:

kγ¹²σ_n(u−u_h)k_0,ΓC ≤Ch^12+νkuk3

2+ν,Ω, (25)

with C >0 a constant, independent ofh and u.

Proof: We use (24), (11), (22) and (21) to establish the bound:

kγ¹²σn(u−u_h)k_0,ΓC ≤ kγ¹²σn(u− I^hu)k_0,ΓC +kγ¹²σn(I^hu−u_h)k_0,ΓC

≤Ch^12+νkuk3

2+ν,Ω+Cγ

1 2

0(kI^hu−uk_1,Ω+ku−u_hk_1,Ω)

≤Ch^12+νkuk3 2+ν,Ω.

Remark 3.7. Although contact problems are known to be limited in regularity (the regularity H^5/2(Ω) can not generally be passed beyond for such inequality problems), the use of quadratic finite element methods can be of interest in particular for the most regular solutions lying in H^s(Ω), 2 < s < 5/2 (as it is considered in e.g., [5, 22, 32, 37]). It is easy to check that the

trace inequality (11) is satisfied in the case of quadratic elements so that Theorem 3.4 still holds.

Moreover, since estimates (22)-(24) remain true in the case of quadratic finite elements (see [15] and [16] for estimate (24)), we come to the conclusion that the optimal convergence rates (21) and (25) hold for 0< ν ≤3/2 when quadratic finite elements are used.

4 Conclusion and perspectives

In this paper we propose a new finite element approximation for the unilateral contact problem in two-dimensional linear elasticity and we achieve the corresponding numerical analysis. This analysis allows us to obtain the first optimal a priori error estimate underH^s regularity (3/2<

s≤2) on the solution without any additional assumption. The numerical experiments as well as the extension to the three-dimensional case should be considered in a forthcoming study.

Acknowledgements.

We thank Yves Renard for helpful discussions and comments.

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