Residual a posteriori error estimators for contact problems in elasticity

Vol. 41, N 5, 2007, pp. 897–923 www.esaim-m2an.org

DOI: 10.1051/m2an:2007045

RESIDUAL A POSTERIORI ERROR ESTIMATORS FOR CONTACT PROBLEMS IN ELASTICITY

Patrick Hild

and Serge Nicaise

Abstract. This paper is concerned with the unilateral contact problem in linear elasticity. We define twoa posteriori error estimators of residual type to evaluate the accuracy of the mixed finite element approximation of the contact problem. Upper and lower bounds of the discretization error are proved for both estimators and several computations are performed to illustrate the theoretical results.

Mathematics Subject Classification. 65N30, 74M15.

Received September 26, 2006. Revised April 17, 2007.

1. Introduction

The finite element method is currently used in the numerical realization of contact problems occurring in several engineering applications (see [18, 19, 24, 25, 30]). An important task consists of evaluating numerically the quality of the finite element computations by using a posteriorierror estimators. There are two important difficulties in developing such tools for contact problems in elasticity: the first one comes from the inequality (unilateral) conditions in the model and the second one is due to the location of these inequality conditions which hold on (a part of) the boundary. For the linear elasticity system with standard boundary conditions (leading to a variational identity), many different approaches leading to various error estimators have been developed and a review of the differenta posteriori error estimators can be found in [29]. Some of these approaches have been chosen and studied for frictionless or frictional unilateral contact problems, in particular in [7, 26, 31] (residual approach using a penalization of the contact condition or the normal compliance law), in [10, 11] (equilibrated residual method) and finally in [14] (residual approach for BEM-discretizations).

Besides, let us mention that many studies dealing with residual estimators for scalar variational inequality problems of the first kind have been achieved in other contexts than elasticity. In particular a great effort was devoted to the obstacle problem (seee.g., [6,8,27] and the references therein). Moreover a residual type estimator for the Signorini problem in its standard formulation can be found in [21] (note that the Signorini problem could be seen as a simplification in the scalar case of the unilateral contact model). For residual estimators dealing with variational inequalities of the second kind we refer the reader to [5] and the references therein. We recall that a variational inequality of the first (resp. second) kind is of the form: u∈C, a(u, v−u)≥L(v−u),∀v∈C (resp. u∈X, a(u, v−u) +j(v)−j(u)≥L(v−u),∀v∈X) whereX is an Hilbert space,C⊂X is a nonempty closed convex set, a(., .) is bilinear,X-elliptic and continuous on X×X, L(.) is linear and continuous onX, j(.) is proper convex and lower semi continuous on X (with values in R∪ {+∞}). More details concerning variational inequalities of the first or second kind can be found ine.g., [2, 17].

Keywords and phrases. Mixed finite element method,a posteriorierror estimates, residuals, unilateral contact.

1 Universit´e de Franche-Comt´e, Laboratoire de Math´ematiques de Besan¸con, CNRS UMR 6623, 16 route de Gray, 25030 Besan¸con, France. hild@math.univ-fcomte.fr

2 Universit´e de Valenciennes et du Hainaut Cambr´esis, MACS, ISTV, 59313 Valenciennes Cedex 9, France.

snicaise@univ-valenciennes.fr

c EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.esaim-m2an.org or http://dx.doi.org/10.1051/m2an:2007045

In the present work we are interested in developing residual estimators for the two-dimensional unilateral contact model in linear elasticity. This problem can be written as a variational inequality of the first kind but also (among others) as a mixed formulation where the unknowns are the displacement field and the contact pressure.

The paper is organized as follows. In Section 2 we introduce the equations modelling the frictionless unilateral contact problem between an elastic body and a rigid foundation. We write the problem using a mixed formulation where the unknowns are the displacement field in the body and the pressure on the contact area. In the third section, we choose a classical discretization involving continuous finite elements of degree one and continuous piecewise affine multipliers on the contact zone. Section 4 is concerned with the study of a first residual estimator which can be seen as the natural one arising from the discrete problem. We obtain both global upper and local lower bounds of the error. In Section 5 we consider a second estimator resulting from another discrete model where the displacement field is the same as in the first model but where the multiplier is modified. The main novelty of the second discrete model is that the multipliers have a constant sign. As in Section 4, we obtain global upper and local lower bounds of the error. Finally in Section 6 we implement both estimators and we compare them on several examples.

Finally we introduce some useful notation and several functional spaces. In what follows, bold letters likeu,v, indicate vector valued quantities, while the capital ones (e.g.,V,K, . . .) represent functional sets involving vector fields.

As usual, we denote by (L²(.))^d and by (H^s(.))^d, s ≥ 0, d = 1,2 the Lebesgue and Sobolev spaces in one and two space dimensions (see [1]). The usual norm of (H^s(D))^d is denoted by · s,D and we keep the same notation whend= 1 ord= 2. For shortness the (L²(D))^d-norm will be denoted by · Dwhend= 1 ord= 2.

In the sequel the symbol| · |will denote either the Euclidean norm inR², or the length of a line segment, or the area of a plane domain. Finally the notationabmeans here and below that there exists a positive constantC independent ofaandb(and of the meshsize of the triangulation) such thata≤C b. The notationa∼bmeans that ab andbahold simultaneously.

2. The unilateral contact problem in elasticity

Let Ω represent an elastic body in R² where plane strain assumptions are assumed. The boundary∂Ω is supposed to be polygonal,i.e., it is the union of a finite number of linear segments. Moreover we suppose that the boundary consists in three nonoverlapping parts Γ_D, Γ_N and Γ_Cwith meas(Γ_D)>0 and meas(Γ_C)>0. The normal unit outward vector on∂Ω is denotedn= (n₁, n₂) and we choose as unit tangential vectort= (−n2, n₁).

In its initial configuration, the body is in contact on Γ_C and we suppose that the unknown final contact zone after deformation will be included in Γ_C. The body is clamped on Γ_D for the sake of simplicity. It is subjected to volume forcesf = (f₁, f₂)∈(L²(Ω))² and to surface forcesg= (g₁, g₂)∈(L²(Γ_N))².

The unilateral contact problem in elasticity consists in finding the displacement field u: Ω→R² verifying the equations and conditions (1)–(6):

divσ(u) +f = 0 in Ω, (1)

where div denotes the divergence operator of tensor valued functions and σ = (σ_ij), 1 ≤ i, j ≤ 2, stands for the stress tensor field. The latter is obtained from the displacement field by the constitutive law of linear elasticity

σ(u) = Aε(u) in Ω, (2)

where A is a fourth order symmetric and elliptic tensor and ε(v) = (∇v+^t∇v)/2 represents the linearized strain tensor field. On Γ_D and Γ_N, the conditions are as follows:

u= 0 on Γ_D, (3)

σ(u)n= g on Γ_N. (4)

Afterwards we choose the following notation for any displacement field vand for any density of surface forces σ(v)ndefined on ∂Ω:

v=v_nn+v_tt and σ(v)n=σ_n(v)n+σ_t(v)t.

The conditions modelling unilateral contact on Γ_C are (seee.g., [13, 15, 16]):

u_n≤0, σ_n(u)≤0, σ_n(u)u_n = 0. (5)

Finally the condition

σ_t(u) = 0 (6)

on Γ_C means that friction is omitted.

In order to derive the variational formulation of (1)–(6), we consider the Hilbert space H_Γ¹_D(Ω) =

v∈H¹(Ω) : v= 0 on Γ_D

, equipped with theH¹(Ω)-norm. We further use the Hilbert space

V= (H_Γ¹_D(Ω))². For our next uses, we introduce the trace spaceH¹²(Γ_C) as follows:

H¹²(Γ_C) =

φ∈L²(Γ_C) :∃u∈H_Γ¹_D(Ω) such that φ=γu on Γ_C , equipped with the norm

φ¹_2,ΓC = inf

u∈H_Γ¹_D(Ω):φ=γuu1,Ω,

whereγis the standard trace operator fromH¹(Ω) toH¹²(∂Ω) (see [1]). The topological dual space ofH¹²(Γ_C) will be denoted byH⁻¹²(Γ_C), whose norm is

ψ₋¹

2,ΓC = sup

φ∈H¹²(ΓC)

ψ, φ

−¹₂,¹₂,ΓC

φ¹_2,ΓC ,

where the notation., .₋1

2,¹₂,ΓC represents the duality pairing betweenH⁻¹²(Γ_C) andH¹²(Γ_C).

The forthcoming mixed variational formulation uses the following convex cone of multipliers on Γ_C M =

µ∈H⁻¹²(Γ_C) : µ, ψ

−¹₂,¹₂,ΓC ≥0 for allψ∈H¹²(Γ_C), ψ≥0 a.e. on Γ_C

. Define

a(u,v) =

Ωσ(u) :ε(v) dΩ, b(µ,v) = µ, v_n

−¹₂,¹₂,ΓC, L(v) =

Ω

f·vdΩ +

ΓN

g·vdΓ, for anyuandvin VandµinH⁻¹²(Γ_C).

The mixed formulation of the unilateral contact problem without friction (1)–(6) consists then in finding u∈Vandλ∈M such that:

a(u,v) +b(λ,v) =L(v), ∀v∈V, b(µ−λ,u)≤0, ∀µ∈M.

(7)

An equivalent formulation of (7) consists in finding (λ,u)∈M ×Vsatisfying L(µ,u)≤ L(λ,u)≤ L(λ,v), ∀v∈V, ∀µ∈M,

whereL(µ,v) = ¹₂a(v,v)−L(v) +b(µ,v).Another classical weak formulation of problem (1)–(6) is a variational inequality: findusuch that

u∈K, a(u,v−u)≥L(v−u), ∀v∈K, (8) whereKdenotes the closed convex cone of admissible displacement fields satisfying the non-penetration condi- tions:

K=

v∈V: v_n≤0 on Γ_C .

The existence and uniqueness of (λ,u) solution to (7) has been stated in [19]. Moreover, the first argumentu solution to (7) is also the unique solution of problem (8) andλ=−σn(u).

3. Mixed finite element approximation

We approximate this problem by a standard finite element method. Namely we fix a family of meshes T_h, h >0, regular in Ciarlet’s sense [9], made of closed triangles. ForK∈T_hwe recall thath_K is the diameter ofK andh= max_K∈Thh_K. The regularity of the mesh implies in particular that for any edgeEofK one has h_E=|E| ∼h_K.

Let us define E_h (resp. Nh) as the set of edges (resp. nodes) of the triangulation and set E_h^int = {E ∈ E_h : E ⊂ Ω} the set of interior edges of T_h (the edges are supposed to be relatively open). We denote by E_h^N ={E ∈E_h :E ⊂Γ_N} the set of exterior edges included into the part of the boundary where we impose Neumann conditions, and similarlyE_h^C={E∈E_h:E⊂Γ_C}is the set of exterior edges included into the part of the boundary where we impose unilateral contact conditions. Set N_h^D =Nh∩Γ_D (note that the extreme nodes of Γ_D belong toN_h^D). LetS denote the set of vertices of Ω and denote byN_h^{N C} the set of nodes which belong to Γ_C∩Γ_N and by N_h^CC the nodes belonging to Γ_C∩ S. Set finally N_h^C = (Nh\ N_h^CC)∩Γ_C (N_h^C contains the nodes in Γ_C which are not vertices of Ω). For an elementK, we will denote byE_K the set of edges ofK and according to the above notation, we setE_K^int=E_K∩E_h^int, E^N_K =E_K∩E_h^N,E_K^C =E_K∩E_h^C.

For an edge E of an elementK, introduce n_K,E = (n₁, n₂) the unit outward normal vector toK alongE and the tangent vectort_K,E =n^⊥_K,E = (−n₂, n₁). Furthermore for each edge E we fix one of the two normal vectors and denote it by n_E and we set t_E =n^⊥_E. The jump of some vector valued functionv across an edge E∈E_h^int at a pointy∈E is defined as

v

E(y) = lim

α→+0v(y+αn_E)−v(y−αn_E), ∀E∈E_h^int. Note that the sign of

v

Edepends on the orientation ofn_E. Finally we will need local subdomains (also called patches). As usual, letω_K be the union of all elements having a nonempty intersection withK. Similarly for a nodexand an edgeE, letω_x=∪K:x∈KK andω_E=∪_x∈Eω_x.

The finite element space used in Ω is then defined by V_h=

v_h∈(C(Ω))²: ∀κ∈T_h, v_h|κ∈(P1(κ))², v_h|ΓD =0

.

We suppose that the contact area consists in several straight line segments, that we denote by Γⁱ_C,1 ≤i≤q such that Γ_C=∪iΓⁱ_C. In order to express the contact constraints by using Lagrange multipliers on the contact zone, we have to introduce the space

W_h=

µ_h:∪iΓⁱ_C →R, µ_h|_Γi

C ∈ C(Γⁱ_C),∃v_h∈V_h s.t. v_h·n=µ_h on ∪iΓⁱ_C

. (9)

The choice of the spaceW_h allows us to define the following closed convex cone:

M_h=

µ_h∈W_h:

ΓC

µ_hψ_hdΓ≥0, ∀ψh ∈W_h, ψ_h≥0

. Remark 3.1. It is easy to check thatM_h⊂M.

The discretized mixed formulation of the unilateral contact problem without friction is to findu_h∈V_h and λ_h∈M_hsatisfying:

a(u_h,v_h) +b(λ_h,v_h) =L(v_h), ∀v_h∈V_h, b(µ_h−λ_h,u_h)≤0, ∀µ_h∈M_h.

(10) Problem (10) consists in finding (λ_h,u_h)∈M_h×V_hsatisfying

L(µ_h,u_h)≤ L(λ_h,u_h)≤ L(λ_h,v_h), ∀v_h∈V_h, ∀µ_h∈M_h,

where L(µh,v_h) = ¹₂a(v_h,v_h)−L(v_h) +b(µ_h,v_h). In order to prove that there is a unique solution to prob- lem (10), and since we are in the finite dimensional case, we only have to check (see [19], Thm. 3.9 and Ex. 3.8) that

vh∈Vsuph,vh=0

b(µ_h,v_h) v_h1,Ω is a norm onW_h. So we have to verify that

{µh∈W_h: b(µ_h,v_h) = 0, ∀v_h∈V_h}={0},

which is satisfied according to the definition ofW_h in (9). As a consequence, we obtain the following statement:

Proposition 3.2. Problem (10)admits a unique solution(λ_h,u_h)∈M_h×V_h.

Proposition 3.3. If (λ_h,u_h) is the solution of (10), then u_h is also the unique solution of the variational inequality: find u_h such that

u_h∈K_h, a(u_h,v_h−u_h)≥L(v_h−u_h), ∀v_h∈K_h, (11) whereK_hdenotes the discrete closed convex cone of admissible displacement fields satisfying the non-penetration conditions, i.e.,

K_h=

v_h∈V_h: v_hn ≤0 onΓ_C . Proof. Takingµ_h= 0 andµ_h= 2λ_h in (10) leads tob(λ_h,u_h) = 0 and to

b(µ_h,u_h) =

ΓC

µ_hu_hn dΓ≤0, ∀µ_h∈M_h.

The latter inequality implies by polarity that u_hn∈ −M_h^∗ (the notationX^∗ stands for the positive polar cone of X for the inner product on W_h induced by b(., .), see [22], p. 119). Let Q_h = {µ_h ∈ W_h : µ_h ≥ 0}.

We haveM_h^∗= (Q^∗_h)^∗ =Q_h sinceQ_h is a closed convex cone. Hence u_hn ∈ −Q_h and u_h∈K_h. Besides (10) andb(λ_h,u_h) = 0 lead to

a(u_h,u_h) =L(u_h) (12)

and for anyv_h∈K_h, we get

a(u_h,v_h)−L(v_h) =−b(λh,v_h) = −

ΓC

λ_hv_hndΓ≥0, (13)

owing toλ_h∈Q^∗_h.

Putting together (12) and (13) implies thatu_h is solution of the variational inequality (11) which admits a

unique solution according to Stampacchia’s theorem.

Remark 3.4. A priori error analyses for this discretization of the unilateral contact problem can be found in [4, 12, 23]. Thea priorierror estimates are of orderh^(1+ν)/2ifu∈(H^3/2+ν(Ω))²,0< ν≤1/2 (see [4, 12]). If an additional assumption dealing with the finiteness of transition points between contact and separation is added then an optimal error estimate of orderh^1/2+ν is obtained (see [23]).

We consider the quasi-interpolation operatorπ_h: for anyv∈L¹(Ω), we defineπ_hv as the unique element in V_h={vh∈ C(Ω) : ∀κ∈T_h, v_h|κ∈P₁(κ), v_h|_ΓD = 0}such that:

π_hv=

x∈Nh\N_h^D

α_x(v)λ_x, (14)

where for anyx∈ Nh\N_h^D,λ_xis the standard basis function inV_hsatisfyingλ_x(x) =δ_x,x, for allx∈ Nh\N_h^D andα_x(v) is defined as follows:

α_x(v) = 1

|ωx|

ωx

v(y) dy, ∀x∈ Nh\ N_h^D.

The following estimates hold (see,e.g., [29]):

Lemma 3.5. For any v∈H_Γ¹_D(Ω)we have

v−π_hvK h_K∇vωK,∀K∈T_h, v−π_hvE h^1/2_E ∇vωE,∀E∈E_h.

Since we deal with vector valued functions we can define a vector valued operator (which we denote again by π_hfor the sake of simplicity) whose components are defined above. Consequently we can directly state the Lemma 3.6. For any v∈V we have

v−π_hvK h_Kv1,ωK,∀K∈T_h, (15) v−π_hvE h^1/2_E v1,ωE,∀E∈E_h. (16)

4. A first error estimator: η 4.1. Definition of the residual error estimators

The element residual of the equilibrium equation (1) is defined by divσ(uh) +f =f onK.

As usual this element residual is replaced by some computable finite dimensional approximation called approx- imate element residual

f_K ∈(Pk(K))². A current choice is to take f_K =

Kf(x)/|K| since for f ∈ (H¹(Ω))², scaling arguments yield f −f_KK h_Kf_1,K and is then negligible with respect to the estimator η defined hereafter. In the same way g is approximated by a computable quantity denotedg_Eon anyE∈E_h^N.

Definition 4.1 (First residual error estimator). The local and global residual error estimators are defined by

η_K = ₆

i=1

η²_{iK 1/2}

, η_1K = h_Kf_KK, η_2K = h^1/2_K

⎛

⎝

E∈E^int_K ∪E_K^N

JE,n(u_h)²_E

⎞

⎠

1/2

,

η_3K = h^1/2_K

⎛

⎝

E∈E^C_K

λh+σ_n(u_h)²_E

⎞

⎠

1/2

,

η_4K = h^1/2_K

⎛

⎝

E∈E^C_K

σt(u_h)²_E

⎞

⎠

1/2

,

η_5K =

⎛

⎝

E∈E^C_K

E

−λ_h+u_hn

⎞

⎠

1/2

,

η_6K =

⎛

⎝

E∈E^C_K

λ_h−²_E

⎞

⎠

1/2

,

η =

K∈Th

η²_{K 1/2}

,

where the notations λ_h+ andλ_h− denote the positive and negative parts of λ_h, respectively;J_E,n(u_h) means the constraint jump ofu_h in normal direction,i.e.,

J_E,n(u_h) =

σ(u_h)n_E

E,∀E∈E_h^int,

σ(uh)n_E−g_E,∀E∈E_h^N. (17)

The local and global approximation terms are defined by

ζ_K =

⎛

⎝h²_K

K⊂ωK

f−f_K²_K+h_E

E⊂E^N_K

g−g_E²_E

⎞

⎠

1/2

, ζ=

K∈Th

ζ_K² _1/2

. (18)

Remark 4.2. In the Definition 4.1, we could also set instead ofη_5K:

ˆ η_5K=

⎛

⎝

E∈E^C_K

E

−λ_h−u_hn

⎞

⎠

1/2

since

ΓCλ_h+u_hn =

ΓCλ_h−u_hn. Note that ˆη_5K =η_5K although

K∈Thηˆ²_5K=

K∈Thη_5K² .

4.2. Upper error bound

Theorem 4.3. Let (λ,u)be the solution of(7) and let(λ_h,u_h)be the solution of(10). Then we have u−u_h1,Ω+λ−λ_h−¹

2,ΓC η+ζ.

Proof. Afterwards we adopt the following notation for the displacement error term:

e_u=u−u_h.

Letv_h∈V_h. From theV-ellipticity ofa(., .) and the equilibrium equations in (7) and (10) we obtain:

e_u²_1,Ω a(u−u_h,u−u_h)

= a(u−u_h,u−v_h) +a(u−u_h,v_h−u_h)

= L(u−v_h)−b(λ,u−v_h)−a(u_h,u−v_h) +b(λ_h−λ,v_h−u_h).

Integrating by parts on each triangle K, using the definition of J_E,n(u_h) in (17) and the complementarity conditionsb(λ,u) =b(λ_h,u_h) = 0 yields:

e_u²_1,Ω

Ω

f·(u−v_h) +b(λ_h,v_h) +b(λ,u_h) +

E∈E_h^N

E

(g−g_E)·(u−v_h)

−

E∈E_h^C

E

(σ(uh)n)·(u−v_h)−

E∈E^int_h ∪E_h^N

E

J_E,n(u_h)·(u−v_h). (19)

Splitting up the integrals on Γ_C into normal and tangential components gives:

e_u²_1,Ω

Ω

f·(u−v_h) +b(λ_h,u) +b(λ,u_h)

+

E∈E^C_h

E

(λ_h+σ_n(u_h))(v_hn−u_n) +

E∈E_h^C

E

σ_t(u_h)(v_ht−u_t)

−

E∈E^int_h ∪E_h^N

E

J_E,n(u_h)·(u−v_h) +

E∈E^N_h

E

(g−g_E)·(u−v_h). (20)

We now need to estimate each term of this right-hand side. For that purpose, we take

v_h=u_h+π_h(u−u_h) (21)

whereπ_h is the quasi-interpolation operator defined in Lemma 3.6.

We start with the integral term. Cauchy-Schwarz’s inequality implies

Ω

f·(u−v_h) ≤

K∈Th

fKu−v_hK,

and it suffices to estimateu−v_hK for any triangleK. From the definition ofv_h and (15) we get:

u−v_hK=e_u−π_he_uKh_Ke_u1,ωK. As a consequence

Ω

f·(u−v_h)

(η+ζ)e_u1,Ω. (22) We now consider the interior and Neumann boundary terms in (20): as previously the application of Cauchy- Schwarz’s inequality leads to

E∈E_h^int∪E_h^N

E

J_E,n(u_h)·(u−v_h)

≤

E∈E_h^int∪E_h^N

JE,n(u_h)Eu−v_hE.

Therefore using the expression (21) and estimate (16), we obtain

u−v_hE = e_u−π_he_uEh^1/2_E e_u1,ωE. Inserting this estimate in the previous one we deduce that

E∈E_h^int∪E^N_h

E

J_E,n(u_h)·(u−v_h)

ηe_u1,Ω. (23)

Moreover

E∈E_h^N

E

(g−g_E)·(u−v_h)

ζe_u1,Ω. (24) The two following terms are handled in a similar way as the previous ones so that

E∈E_h^C

E

(λ_h+σ_n(u_h))(v_hn−u_n)

ηe_u1,Ω (25)

and

E∈E^C_h

E

σ_t(u_h)(v_ht−u_t)

ηe_u1,Ω. (26) Noting thatu_hn≤0 on Γ_C, we have

b(λ,u_h)≤0, (27)

and it remains to estimate one term in (20). Using the discrete complementarity conditionb(λ_h,u_h) = 0 implies

b(λ_h,u) =

ΓC

λ_hu_n =

ΓC

λ_h(u_n−u_hn)

=

ΓC

(λ_h+−λ_h−)(u_n−u_hn)

≤ −

ΓC

λ_h+u_hn−

ΓC

λ_h−(u_n−u_hn)

≤ η²−

ΓC

λ_h−(u_n−u_hn). (28)

The last term in the previous expression is estimated using Cauchy-Schwarz’s and Young’s inequalities:

ΓC

λ_h−(u_n−u_hn) =

E∈E^C_h

E

λ_h−(u_n−u_hn)

≤

E∈E^C_h

λh−Eun−u_hnE

≤

E∈E^C_h

αun−u_hn²_E+ 1

4αλ_h−²_E

,

for anyα >0. A standard trace theorem implies the existence ofC >0 such that

ΓC

λ_h−(u_n−u_hn)

≤ αun−u_hn²_ΓC+ 1 4α

E∈E^C_h

λh−²_E

≤ Cαe_u²_1,Ω+ η²

4α· (29)

Estimates (28) and (29) give

b(λ_h,u)≤Cαe_u²_1,Ω+η²

1 + 1 4α

(30) for anyα >0.

Putting together the estimates (22)–(27) and (30) withαsmall enough in (20), and using Young’s inequality, we come to the conclusion that

u−u_h1,Ωη+ζ. (31) We now search for an upper bound on the discretization errorλ−λ_h corresponding to the multipliers. Let v∈Vandv_h∈V_h. From the equilibrium equations in (7) and (10) we get:

b(λ−λ_h,v) = b(λ,v−v_h)−b(λ_h,v−v_h) +b(λ−λ_h,v_h)

= L(v−v_h)−a(u,v−v_h)−b(λ_h,v−v_h) +a(u_h−u,v_h)

= L(v−v_h)−a(u−u_h,v)−a(u_h,v−v_h)−b(λ_h,v−v_h).

An integration by parts on each elementK gives b(λ−λ_h,v) =

Ω

f·(v−v_h)−a(u−u_h,v)−

E∈E^int_h ∪E^N_h

E

J_E,n(u_h)·(v−v_h)

−

E∈E_h^C

E

(λ_h+σ_n(u_h))(v_n−v_hn)−

E∈E_h^C

E

σ_t(u_h)(v_t−v_ht)

+

E∈E_h^N

E

(g−g_E)·(v−v_h).

Choosingv_h=π_hv whereπ_h is the quasi-interpolation operator defined in Lemma 3.6 and achieving a similar calculation as in (22)–(26) we deduce that

|b(λ−λ_h,v)|(η+ζ+u−u_h1,Ω)v1,Ω

for anyv∈V. As a consequence

λ−λ_h−¹_2,ΓC η+ζ+u−u_h1,Ω. (32) Putting together the two estimates (31) and (32) ends the proof of the theorem.

4.3. Lower error bound

Theorem 4.4. For all elements K, the following local lower error bounds hold:

η_1K u−u_h1,K+ζ_K, (33)

η_2Ku−u_h1,ωK+ζ_K. (34)

Assume that λ ∈L²(Γ_C). For all elements K such that K∩E_h^C =∅, the following local lower error bounds hold:

η_3K

E∈E^C_K

h^1/2λ−λ_hE+u−u_h1,K+ζ_K, (35)

η_4K u−u_h1,K+ζ_K, (36)

η_5K

E∈E^C_K

λ−λ_hE+u−u_hE+λ−λ_h^1/2_E un^1/2_E +u−u_h^1/2_E λ^1/2_E

, (37)

η_6K

E∈E^C_K

λ−λ_hE. (38)

Proof. The estimates ofη_1K andη_2K in (33) and (34) are standard (see,e.g., [28]).

We now estimate η_3K. Writing w_E = w_Enn+w_Ett on E ∈ E_K^C and denoting by b_E the edge bubble function associated with E (i.e., b_E = 4λ_a1λ_a2, when a₁, a₂ are the two extremities of E; we recall thatλ_x is the standard basis function at node xin V_h satisfying λ_x(x) =δ_x,x for any node x, see (14)), we choose w_En= (λ_h+σ_n(u_h))b_E andw_Et= 0 in the elementK containingE (here we made a slight abuse of notation

to simplify) andw_E=0in Ω\K. Therefore λ_h+σ_n(u_h)²_E ∼

E

(λ_h+σ_n(u_h))w_En

= b(λ_h,w_E) +

K

σ(u_h) :ε(w_E)

= b(λ_h,w_E)−

K

σ(u−u_h) :ε(wE) +

K

σ(u) :ε(wE)

= b(λ_h−λ,w_E) +L(w_E)−

K

σ(u−u_h) :ε(wE)

λ−λ_hEw_EE+fKw_EK+u−u_h1,Kw_E1,K. An inverse inequality and estimate (33) imply

h^1/2_K λh+σ_n(u_h)E h^1/2λ−λ_hE+u−u_h1,K+hfK

h^1/2λ−λ_hE+u−u_h1,K+ζ_K.

This estimate gives the estimate ofη_3K in (35). The bound ofη_4K in (36) is obtained as previously by choosing w_En= 0 andw_Et=σ_t(u_h)b_E.

We now considerη_5K. IfE∈E_K^C, letF ⊂E be the part of the edge whereλ_h=λ_h+. So

E

−λh+u_hn =

F

−λhu_hn

=

F

(λ_h−λ)(u_n−u_hn)−

F

λ_hu_n−

F

λu_hn

=

F

(λ_h−λ)(u_n−u_hn)−

F

(λ_h−λ)u_n−

F

λ(u_hn−u_n) λ−λ_hEu−u_hE+λ−λ_hEu_nE+u−u_hEλE. The last estimate implies (37) by taking the square root.

The estimate onη_6K is obvious. Sinceλ≥0 then we have 0≤λ_h−≤ |λ−λ_h|on Γ_C. So λ_h−E≤ λ−λ_hE

and (38) is proved.

Remark 4.5. Assume that u∈ (H²(Ω))² (so λ ∈H¹²(Γ_C)), and that optimal a priorierror estimates hold (note that the question of optimality remains open in thea priori error analysis since the knowna priorierror estimates are not optimal) and define:

η_i =

K∈Th

η²_{iK 1/2}

, 1≤i≤6.

Then one would haveη₁h, η₂h, η₃h, η₄h, η₅h^1/4, η₆h^1/2. Soηh^1/4.

5. A second error estimator: η ˜

The analysis of this error estimator requires a nonstandard definition of the error (in comparison with the already known results in the literature dealing with contact problems). We begin with some preliminaries.