Mixed finite element approximation of 3D contact problems with given friction : error analysis and numerical realization

M2AN, Vol. 38, N 3, 2004, pp. 563–578 DOI: 10.1051/m2an:2004026

MIXED FINITE ELEMENT APPROXIMATION OF 3D CONTACT PROBLEMS WITH GIVEN FRICTION: ERROR ANALYSIS

AND NUMERICAL REALIZATION

Jaroslav Haslinger

and Taoufik Sassi

Abstract. This contribution deals with a mixed variational formulation of 3D contact problems with the simplest model involving friction. This formulation is based on a dualization of the set of admissible displacements and the regularization of the non-differentiable term. Displacements are approximated by piecewise linear elements while the respective dual variables by piecewise constant functions on a dual partition of the contact zone. The rate of convergence is established provided that the solution is smooth enough. The numerical realization of such problems will be discussed and results of a model example will be shown.

Mathematics Subject Classification. 65N30, 74M15.

Received: January 9, 2004.

1. Introduction

In the theory of variational inequalities, problems involving contact and friction conditions are of a permanent interest (we refer to [8] and to [18] for a large review of the main unilateral contact models). Finite elements are the most currently used methods for the approximation of contact problems involving friction (see [14, 17] and the references therein). In the primal variational formulation (where the displacement is the only unknown), discretized non-penetration conditions constitute the key point of the approximation model. These conditions can be relaxed and expressed in a weaker sense [4, 14, 16]. In the mixed variational formulation where the displacement and the normal stresses on the contact zone are treated as independent variables, the unilateral conditions have been expressed by using Lagrange multipliers (see [14]). Both variational formulations lead to a non-differentiable problem. To overcome this difficulty one can introduce (see [12]) another set of Lagrange multipliers related to the tangential component of the stress on the contact zone transforming the non-differentiable problem into a smooth one.

The purpose of this contribution is to extend this approach to 3D contact problems with given friction by using piecewise constant discretizations of Lagrange multipliers on the contact zone. Unlike to 2D problems, constraints imposed on the “tangential” Lagrange multipliers are now quadratic. This fact complicates not only

Keywords and phrases. Mixed finite element methods, unilateral contact problems with friction,a priorierror estimates.

1 Department of Numerical Mathematics, Charles University 12116 Praha 2, Czech Republic. e-mail:mfkfk@met.mff.cuni.cz

2Laboratoire de Math´ematiques Nicolas Oresme, CNRS UMR UFR Sciences Campus II, Bd Mar´echal Juin, 14032 Caen Cedex, France. e-mail:sassi@math.unicaen.fr

c EDP Sciences, SMAI 2004

the theoretical analysis but also the practical realization. We establisha priorierror estimates and propose the efficient numerical method.

The paper is organized as follows. Firstly, we introduce the mathematical model of the Signorini problem with given friction. In Section 2 we present the continuous mixed variational formulation of the problem.

In the third section, we propose a well-posed finite element discretization in order to approximate the mixed formulation. In Section 4a priori error estimates for the mixed finite element approximations are established.

Finally, in Section 5 we present a numerical approach which is based on the dual variational formulation (i.e.

the formulation in terms of the Lagrange multipliers) combined with a linearization of the quadratic constraints.

Numerical results of a model example will be shown.

2. Preliminary and notations

The Euclidean norm of a point x∈Rⁿ,n≥2 will be denotedx in what follows.

Let Ω ⊂ Rⁿ be a bounded domain with the Lipschitz boundary ∂Ω. The symbol H^k(Ω), k ≥ 1 integer, stands for the classical Sobolev space equipped with the norm:

vk,Ω:=





0≤|α|≤k

D^αv²_0,Ω





12

using the standard multi-index notation (the following convention is adopted: H⁰(Ω) := L²(Ω), _0,Ω :=

L²(Ω)). Fractional Sobolev spacesH^τ(Ω),τ ∈R⁺\Nare defined by H^τ(Ω) ={v∈H^k(Ω)| v_τ,Ω<+∞}, where

vτ,Ω:=

v²_k,Ω+

|α|=k

Ω

Ω

(D^αv(x)−D^αv(

y

|x−

y

withk being the integer part ofτ andθits decimal part (see [1, 11]).

On any portion Γ⊆∂Ω we introduce the spaceH¹²(Γ) as follows:

H¹²(Γ) = ψ∈L²(Γ)| ψ¹

2,Γ <+∞

,

where

ψ¹_2,Γ :=

ψ²_0,Γ+

Γ

Γ

(ψ(x)−ψ(

y

|x−

y

. (2.1)

It is well-known that there exists a linear continuous mappingE∈L(H¹²(Γ), H¹(Ω)) such that Eψ|Γ =ψ on Γ,

whereEψ|Γ denotes the trace ofEψon Γ.

The topological dual space ofH¹²(Γ) is denotedH⁻¹²(Γ), ,

12,Γ stands for the duality pairing and µ₋¹_2,Γ:= sup

ψ∈H¹²(Γ), ψ=0

µ, ψ

12,Γ

ψ¹_2,Γ (2.2)

is the norm inH⁻¹²(Γ).

The Cartesian product ofmprevious spaces and their elements are denoted by bold characters. The respective norms are introduced as follows:

vk,Ω:=

_m

=1

v²_{k,Ω 1/2}

,v= (v1,· · ·, vm)∈H^k(Ω)

ψ¹_2,Γ:=

_m

=1

ψ²1 2,Γ

_1/2

,ψ= (ψ1,· · ·, ψm)∈H¹²(Γ)

µ₋¹

2,Γ:=

_m

=1

µ²₋1 2,Γ

_1/2

,µ= (µ1,· · ·, µ_m)∈H⁻¹²(Γ).

A special attention will be paid to the case whenk= 1 andn=m= 3,i.e. Ω⊂R³and H¹(Ω) = (H¹(Ω))³. LetV ⊂H¹(Ω) be a subspace of functions vanishing on a non-empty portion Γuopen in∂Ω:

V = v = (v1, v2, v3)∈H¹(Ω)| vi= 0 on Γu, i= 1,2,3

.

In addition to the classical norm _1,Ω, we introduce the energetic norm| |_1,Ωin V corresponding to the scalar product

((u,v))_1,Ω:=

Ω

3 i,j=1

εij(u)εij(v)dx, whereεij(u) is thei, jth component of the linearized strain tensor

ε(u) := 1

2(∇u+ (∇u)^T).

From the Korn’s inequality it follows that _1,Ωand| |_1,Ωare equivalent inV.

Let Γ⊆∂Ω\Γu be a portion of ∂Ω such that dist(Γ,Γu)>0¹ and denote byH_Γ¹² the space of tracesv_|Γ forv∈V. Besides the norm 1

2,Γ defined by (2.1) we introduce the following quotient norm:

|ψ|¹_2,Γ:= inf

v∈V,v=ψonΓ|v|1,Ω. It is readily seen that

|ψ|¹_2,Γ =|w_ψ|1,Ω, wherew_ψ∈V is the unique solution of the elliptic problem:

((w_ψ,v))_1,Ω= 0 ∀v∈V₀={z∈V| z=0on Γ}, w_ψ=ψ on Γ.

From the Banach theorem on the isomorphism it follows that 1

2,Γ and| |¹_2,Γ are equivalent inH_Γ¹². LetH⁻_Γ¹² be the dual space ofH_Γ¹² and denote| |₋¹_2,Γ the dual norm corresponding to| |₂¹_,Γ:

|µ|₋¹

2,Γ:= sup

ψ∈H_Γ¹²,ψ=0

µ,ψ

12,Γ

|ψ|¹_2,Γ ·

1This assumption is not necessary but it simplifies the presentation.

It is readily seen that

|µ|₋¹_2,Γ=|w_µ|1,Ω, (2.3)

wherew_µ∈V is the unique solution of the elliptic problem:

((w_µ,v))_1,Ω= µ,v

12,Γ ∀v∈V. (2.4)

In addition, ₋1

2,Γ and| |₋¹_2,Γ are equivalent in H⁻_Γ¹².

3. Variational formulation of the Signorini problem with given friction

Let us consider an elastic body occupying a domain Ω⊂R³with the Lipschitz boundary∂Ω which is split into three non-empty, non-overlapping parts Γu, Γg and Γ_c,dist(Γ_c,Γu)>0: ∂Ω = Γu∪Γg∪Γ_c. For the sake of simplicity of our presentation we shall suppose that Ω = (0, a)×(0, b)×(0, c), where a, b, c > 0 are given and Γ_c = (0, a)×(0, b)× {0}. The zero displacements are prescribed on Γu while surface tractions of density g∈L²(Γg) act on Γg. The body is unilaterally supported along Γ_c by the rigid half-space

S={x= (x1, x2, x3)∈R³ |x3≤0}.

Besides constraints on the deformation of Ω we shall take into account effects of friction. We restrict ourselves to the simplest model, namely the model withgiven friction. Finally the body is subject to volume forces of densityf ∈L²(Ω). Our aim is to find an equilibrium state of Ω.

The classical formulation of the previous problem consists in finding a displacement vectoru= (u1, u2, u3) satisfying the equations and conditions (3.1)–(3.4):





divσ(u) +f = 0 in Ω, σ(u)n−g = 0 on Γg, u = 0 on Γu.

(3.1)

The symbol σ(u) = (σij(u))³_i,j=1 stands for a symmetric stress tensor which is related to the linearized strain tensorε(u) by means of a linear Hooke’s law:

σ(u) =Aε(u), (3.2)

where A is a fourth order symmetric tensor satisfying the usual ellipticity conditions in Ω. Further nis the outward unit normal vector to ∂Ω and div denotes the divergence operator acting on tensor functions (the summation convention is adopted):

divσ = ∂σij

∂xj

₃

i=1·

Let u_n := unn, u_t := u−unn be the normal, tangential component of the displacement vector u and σ_n(u) := σn(u)n, and σ_t(u_t) := σ(u)n−σn(u)n the normal, tangential component of the stress vector σ(u)n, respectively, where un :=u·n and σn(u) := (σ(u)n)·n. Here· denotes the scalar product of two vectors.

The unilateral and friction conditions prescribed on Γ_c read as follows:

un≤0, σn(u)≤0, unσn(u) = 0 on Γ_c, (3.3)









σ_t(u) ≤s a.e. on Γ_c,

ifσ_t(u)(x)< s(x) thenu_t(x) =0, x∈Γ_c, ifσ_t(u)(x)=s(x)>0 then there existsν(x)≥0 such thatu_t(x) =−ν(x)σ_t(u(x)), x∈Γ_c.

(3.4) Heres∈L²(Γ_c), s≥0 is a given function andσ_t(u) is the Euclidean norm ofσ_t(u).

Remark 3.1. In the classical Coulomb’s law of friction [2, 6] the given slip bound sis replaced by the term F|σn(u)|which is not knowna priori. The symbolFstands for the coefficient of Coulomb friction.

Remark 3.2. Taking into account the fact thatn= (0,0,−1) on Γ_c, we have:

un =−u3, σn(u) =σ33(u), σ_t(u) = (−σ13(u),−σ23(u),0).

Remark 3.3. Lett₁(x), t₂(x), x∈Γ_c be two unit vectors such that the triplet{n(x),t₁(x),t₂(x)} forms a local orthonormal basis inR³with origin atx∈Γ_c. Then any vector functionw: Γ_c−→R³can be represented in the local coordinate system{n(x),t₁(x),t₂(x)} as follows:

w(x) = (wn(x),w_t(x))∈R×R²,

wherewn(x) =w(x)·n,w_t(x) = (wt1(x), wt2(x)),wtj(x) =w(x)·t_j(x), j= 1,2. In particular, we shall use this representation for the trace of displacement vectorsv on Γ_c in what follows,i.e. v(x) = (vn(x),v_t(x))∈ R×R²,x∈Γ_c and

v_t=

|vt1|²+|vt2|²¹₂

a.e. on Γ_c.

To give the variational formulation of our problem we first introduce the Hilbert spaceV: V = v∈H¹(Ω)| v=0on Γu

,

and its closed convex subsetK ofkinematically admissible displacements:

K=

v∈V | vn≤0 on Γ_c

·

Next, we define:

a(u,v) :=

ΩAε(u)ε(v) dx,

L(v) :=

Ω

f·v dx+

Γg

g·vdΓ,

j(v) :=

Γc

sv_tdΓ.

Theprimal variational formulationof the contact problem with given friction reads as follows:

Findu∈Ksuch that

J(u)≤J(v) ∀v∈K, (3.5)

where

J(v) = 1

2a(v,v)−L(v) +j(v) is the total potential energy functional.

On the basis of the previous assumptions we have:

Theorem 3.1. Problem (3.5) has a unique solution.

In order to release the unilateral constraintvn ≤0 on Γ_c and to regularize the non-differentiable termj we use a duality approach. To this end we introduce several notations. Let γn : V −→ L²(Γ_c) be the trace mapping defined by

γnv :=vnon Γ_c, v∈V. Further let

H_Γ¹²_c:=γnV, H_Γ⁻_c¹² := the dual ofH_Γ¹²_c, H_Γ¹²

c:= (H_Γ¹²_c)³, H⁻_Γ¹²

c := (H_Γ⁻_c¹²)³, Mn= µn∈H_Γ⁻_c¹², µn ≥0 on Γ_c

,

M_t = µ_t= (µt1, µt2)∈L²(Γ_c), µ_t ≤s a.e.on Γ_c

= µ_t∈L²(Γ_c)|

Γc

µ_t·ψdΓ−

Γc

sψdΓ≤0 ∀ψ∈L²(Γ_c)

.

Remark 3.4. In accordance with the previous section, the spaces H_Γ¹²

c, H⁻_Γ¹²

c will be endowed with two equivalent norms 1

2,Γc,| |¹_2,Γc and ₋1

2,Γc,| |₋¹_2,Γc, respectively.

It is easy to verify that

minK J(v) = min

V sup

Mn×Mt

L(v, µn,µ_t), whereL : V ×Mn×M_t−→Ris the Lagrangian defined by

L(v, µn,µ_t) =J(v) + µn, vn

12,Γc+

Γc

µ_t·v_tdΓ,

and ., .

12,Γc stands for the duality pairing betweenH_Γ⁻_c¹² andH_Γ¹²_c.

By amixed formulation of (3.5) we call a problem of finding asaddle-pointofLonV ×Mn×M_t: Find (w, λn,λ_t)∈V ×Mn×M_tsuch that

L(w, µn,µ_t)≤L(w, λn,λ_t)≤L(v, λn,λ_t) ∀(v, µn,µ_t)∈V ×Mn×M_t,

or equivalently 









Find (w,λ)∈V ×Msuch that a(w,v) +b(λ,v) =L(v), ∀v∈V, b(µ−λ,w)≤0, ∀µ∈M,

(3.6)

whereλ:= (λn,λ_t), M:=Mn×M_t and b(µ,v) :=

µn, vn

12,Γc+

Γc

µ_t ·v_tdΓ, µ∈M, v∈V. The relation between (3.5) and (3.6) follows from the next theorem.

Theorem 3.2. There exists a unique solution (w,λ)of (3.6). In addition, w=u inΩ, λn=−σn(u), λ_t=−σ_t(u) onΓ_c, whereu∈K is the solution to (3.5).

Proof is obvious.

Convention: the solution to (3.6) will be denoted by (u, λn,λ_t) in what follows.

4. Finite element approximation

The present section is devoted to a finite element approximation of the saddle-point problem (3.6). The key point lies in the finite element discretization of the closed convex setMof Lagrange multipliers which leads to a well-posed discrete problem and gives also a good convergence rate for approximate solutions.

We start with discretizations ofV,Mn andM_t.

Let{T_h}, h→0+ be aregularfamily (see [5]) of partitions of Ω into tetrahedronsκ. With anyT_h∈ {T_h} we associate the following space of continuouspiecewise linearfunctions:

Vh= vh∈C(Ω)| vh|κ∈P1(κ) ∀κ∈T_h, vh= 0 on Γu

, V_h= (Vh)³.

Further, let {R_H}, H →0+ bea strongly regular family of rectangulations of Γ_c with H >0 being the norm ofR_H. On everyR_H ∈ {R_H}we construct the space of piecewise constantfunctions:

LH= µH ∈L²(Γ_c)| µH|R∈P₀(R)∀R ∈ R_H ,

L_H = (LH)³ and the following convex sets:

MHn= µHn∈LH| µHn≥0 a.e. on Γ_c

, (4.1)

M_Ht= µ_Ht∈(LH)²|

Γc

µ_Ht·ψ_H dΓ−

Γc

sψ_HdΓ≤0∀ψ_H∈(LH)²

. (4.2)

Recall that · stands for the Euclidean norm inR². The setM_Htis the externalapproximation ofM_t, while MHnis theinternalapproximation ofMn.

The discretization of (3.6) is defined in a standard way:











Findu_h∈V_handλ_H∈M_Hsuch that 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,

(4.3)

where λ_H := (λHn,λ_Ht) andM_H := MHn×M_Ht. To ensure the existence and uniqueness of a solution to (4.3), the following stability condition is needed:

µ_H∈L_H| b(µ_H,v_h) = 0∀v_h∈V_h

={0}. (4.4)

In the forthcoming convergence analysis, we will need more information on the compatibility between the spaces L_H andV_h. We shall introduce a stronger assumption, namely the Ladyzhenskaya-Babuska-Brezzi condition which in our particular case takes the form

v_hsup∈V_h

b(µ_H,v_h)

|v_h|_1,Ω ≥β|µ_H|₋¹_2,Γc ∀µ_H∈L_H, (4.5) where β > 0 is independent ofh and H. The dual norm | |₋¹

2,Γc in (4.5) is evaluated by means of (2.3).

Next we shall suppose that the auxiliary problem (2.4) isregularin the following sense:











there existsε∈(0,1/2) such that for every µ∈H⁻_Γ^12+ε

c := the dual ofH_Γ^12−ε

c

the solutionw_µ of (2.4) belongs toV ∩H^1+ε(Ω) and w_µ1+ε,Ω≤C(ε)µ₋¹

2+ε,Γc,

holds with a positive constant C(ε) which depends solely on ε > 0. If it is so then using exactly the same approach as in [12] one can prove the next lemma.

Lemma 4.1. Let us suppose that (2.4) is regular, {R_H} is a strongly regular family of rectangulations of Γ_c and the ratio h/H is sufficiently small. Then the Ladyzhenskaya-Babuska-Brezzi condition (4.5) is satisfied.

5. Error analysis

In this section we establish a priori error estimates for the mixed finite element approximation (4.3). We start with the following lemma [3]:

Lemma 5.1. Let (u,λ), (u_h,λ_H) be the solution of (3.6), (4.3), respectively. Then for any v_h ∈ V_h and µ_H∈M_H it holds:

a(u−u_h,u−u_h) ≤ a(u−u_h,u−v_h) +b(λ−µ_H,u_h−u) +b(λ−λ_H,u−v_h)

+b(λ−µ_H,u) +b(λ_H−λ,u). (5.1)

Proof. Letv_h be an element ofV_h. Then

a(u−u_h,u−u_h) =a(u−u_h,u−v_h) +a(u−u_h,v_h−u_h). From the equations in (3.6) and (4.3), we obtain:

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

= b(λ,u_h−v_h) +b(λ_H,v_h−u_h). Further

a(u−u_h,u−u_h) =a(u−u_h,u−v_h) +b(λ,u_h−v_h) +b(λ_H,v_h−u_h), and finally

a(u−u_h,u−u_h) = a(u−u_h,u−v_h) +b(µ_H−λ,u−u_h) +b(λ−λ_H,u−v_h) +b(λ−µ_H,u) +b(λ_H−λ,u) +b(µ_H−λ_H,u_h).

The inequality in (4.3) implies thatb(µ_H−λ_H,u_h)≤0 for anyµ_H∈M_H. From this the error estimate (5.1)

follows.

We now derive an upper bound foru−u_h1,Ω.

Lemma 5.2. Let (u,λ),(u_h,λ_H) be the solution of (3.6), (4.3), respectively. Suppose that u ∈H²(Ω) and the fourth order tensorA defining the Hooke’s law is sufficiently smooth. Then

u−u_h²_1,Ω ≤ C

hλ−λ_H−¹_2,Γc+H^12+ελ−λ_H−¹_2+ε,Γc+H³² +H

, (5.2)

whereε >0 is arbitrarily small and C >0 is a positive constant which does not depend on h,H.

Proof. We shall estimate each term on the right hand side of (5.1). But before proving these estimates, let us recall some approximation results. Let I_h be the Lagrange interpolation operator with values in V_h. There exists a constantC >0 which does not depend onhand such that for allv ∈H²(Ω) (see [5]) it holds:

v−I_hv1,Ω≤Chv2,Ω. (5.3)

Next, we introduce the projection operatorΠ_H : L²(Γ_c) −→L_H by ΠHϕ∈L_H,

Γc

(ΠHϕ−ϕ)· µ_H dΓ = 0, ∀µ_H∈L_H. (5.4) The operator ΠH has the following approximation property (see [19]): there exists a positive constant C independent ofH such that ∀ϕ∈H^ν_Γc,ν =¹₂, 1 and anyµ∈[0,¹₂), it holds that

ϕ−ΠHϕµ,Γc ≤ C H^ν−µϕν,Γc. (5.5)

Moreover, ifϕ∈L²(Γ_c) then

ϕ−ΠHϕ₋¹_2+ε,Γc ≤ C H^12−εϕ−ΠHϕ0,Γc, ε∈[0,¹₂). (5.6) Finally, let us note that the trace theorem implies that

λ¹_2,Γc≤Cu2,Ω, (5.7)

provided that the tensorAis sufficiently smooth (the constantC >0 depends only onA).

Next we estimate each term appearing on the right hand side of (5.1) by choosing: v_h = I_hu and µ_H = Π_Hλ= (πHnλn,π_Htλ_t), whereπHn andπ_Ht are the components ofΠ_H inLH and (LH)², respectively. It is readily seen thatΠ_Hλ∈M_H as follows from the definition ofM,M_H and (5.4).

(i) The first term is evaluated by using continuity ofa(., .) and (5.3):

a(u−u_h,u−v_h)≤Chu−u_h1,Ω. (ii) The second is estimated by using (5.5), (5.6), (5.7) and the trace theorem:

b(λ−µ_H,u_h−u) ≤ Cλ−Π_Hλ₋¹

2,Γcu_h−u_1,Ω≤CHu_h−u_1,Ω. (iii) The third term is estimated as follows:

b(λ−λ_H,u−v_h)≤Cλ−λ_H−¹

2,Γcu−v_h1,Ω≤Chλ−λ_H−¹

2,Γc.

(iv) To evaluate the fourth term, we invoke the definition of theL²-projection operators:

b(λ−µ_H,u) =

Γc

(λn−πHnλn)undΓ +

Γc

(λ_t−π_Htλ_t)·u_tdΓ

=

Γc

(λn−πHnλn)(un−πHnun)dΓ +

Γc

(λ_t−π_Htλ_t)·(u_t−π_Htu_t)dΓ

≤ λn−πHnλn0,Γcun−πHnun0,Γc

+λ_t−π_Htλ_t0,Γcu_t−π_Htu_t0,Γc

≤ CH^3/2, making use of (5.5) and (5.7).

(v) To estimate the last term, let us observe that the complementarity conditionλnun= 0a.e. on Γ_c yields:

b(λ_H−λ,u) =

Γc

λHnundΓ +

Γc

(λ_Ht−λ_t)·u_tdΓ. Sinceun≤0 andλHn≥0a.e. on Γ_c, we have

Γc

λHnundΓ≤0. Therefore

b(λ_H−λ,u) ≤

Γc

(λ_Ht−λ_t)·u_tdΓ. (5.8)

We now shall estimate (5.8). Noticing that

Γc

µ_Ht·ψ_H dΓ−

Γc

sψ_HdΓ≤0∀ψ_H∈(LH)²,

and using that−λ_t·u_t+su_t= 0a.e. on Γ_c (see (3.4) and Th. 3.2), we have:

Γc

(λ_Ht−λ_t)·u_tdΓ =

Γc

(λ_Ht−λ_t)·(u_t−π_Ht(u_t))dΓ +

Γc

(λ_Ht−λ_t)·π_Ht(u_t)dΓ +

Γc

λ_t·u_tdΓ−

Γc

su_tdΓ

≤

Γc

(λ_Ht−λ_t)·(u_t−π_Ht(u_t))dΓ +

Γc

s(π_Ht(u_t) − u_t)dΓ +

Γc

λ_t·(u_t−π_Ht(u_t))dΓ

≤

Γc

(λ_Ht−λ_t)·(u_t−π_Ht(u_t))dΓ +

Γc

s(π_Ht(u_t)−u_t)dΓ +

Γc

λ_t·(u_t−π_Ht(u_t))dΓ

≤ λ_Ht−λ_t−¹_2+ε,Γcu_t−π_Ht(u_t)¹_2−ε,Γc

+s0,Γcu_t−π_Ht(u_t)0,Γc+λ_t0,Γcu_t−π_Ht(u_t)0,Γc

≤ CH^12+ελ_H−λ₋¹

2+ε,Γc+CH (5.9)

making use of (5.5) and

µ_t(H^j

Γc)² ≤ µ_(H^j

Γc)³ for anyµ= (µn,µ_t)∈H^j_Γ

c, j=±1 2·

From (i)–(v) and theV-ellipticity of the bilinear forma(., .), we easily arrive at (5.2).

The next lemma gives the error estimate for the Lagrange multiplier.

Lemma 5.3. Let all the assumptions of Lemma 5.2 together with (4.5) be satisfied. Then

λ−λ_H−¹_2,Γc≤Cu−u_h1,Ω+CH, (5.10) λ−λ_H−¹

2+ε,Γc ≤CH^−εu−u_h1,Ω+CH^1−ε, (5.11) whereε >0 is arbitrarily small and C is a positive constant which does not depend on hand H.

Proof. Let us consider the equations in (3.6) and (4.3). SinceV_h⊂V, we have a(u,v_h) +b(λ,v_h) = L(v_h) ∀v_h∈V_h, a(u_h,v_h) +b(λ_H,v_h) = L(v_h) ∀v_h∈V_h. Therefore

a(u−u_h,v_h) +b(λ−λ_H,v_h) = 0 ∀v_h∈V_h, and

b(λ_H−Π_Hλ,v_h) = a(u−u_h,v_h) +b(λ−Π_Hλ,v_h)

≤ Cu−u_h1,Ωv_h1,Ω+CHλ¹_2,Γcv_h1,Ω ∀v_h∈V_h. The inf-sup condition (4.5) yields:

β˜ λ_H−ΠHλ₋¹_2,Γc≤ sup

v_h∈V_h

b(λ_H−ΠHλ,v_h)

|v_h|_1,Ω ≤ Cu−u_h1,Ω+CHλ₂¹_,Γc. (5.12) Here we used the Korn’s inequality and the fact that the dual norms ₋1

2,Γc and| |₋¹_2,Γc are equivalent in H⁻_Γ¹²

c . The constant ˜β >0 appearing on the left of (5.12) is independent ofhand H.

The triangle inequality

λ−λ_H−¹_2,Γc≤ λ−ΠHλ₋¹_2,Γc+ΠHλ−λ_H−¹_2,Γc together with (5.12) and (5.5) proves (5.10).

Let us prove (5.11). The triangle and inverse inequality yield:

λ−λ_H−¹

2+ε,Γc≤ λ−Π_Hλ_H−¹

2+ε,Γc + Π_Hλ−λ_H−¹

2+ε,Γc

≤C

H^1−ε+H^−εΠHλ−λ_H−¹_2,Γc ,

making use of (5.6). The rest of the proof now follows from (5.12)

We finally obtain the following global result giving an upper bound for our mixed finite element approxima- tion.

Theorem 5.1. Let(u,λ),(u_h,λ_H)be the solution to (3.6), (4.3), respectively. Suppose that u∈H²(Ω), the fourth order tensor A defining the Hooke’s law is sufficiently smooth and the Ladyzhenskaya-Babuska-Brezzi condition (4.5) is satisfied. Then

u−u_h1,Ω+λ−λ_H−¹

2,Γc≤CH¹², (5.13)

where the positive constant C does not depend onhandH.

Proof. Assembling (5.2), (5.10) and (5.11) we arrive at (5.13) making use thath≤CH, whereC >0 does not

depend onhandH.

Remark 5.1. Let us suppose that the slip boundsis a positive constant. Then the error estimate (5.13) can be improved. Indeed, instead of (4.2) we define the setM_Ht as follows:

M_Ht= µ_Ht∈(LH)²| µ_Ht ≤s a.e. on Γ_c · ThenM_Ht is theinternalapproximation ofM_tso that

Γc

µ_Ht·ψdΓ−s

Γc

ψdΓ≤0 ∀ψ∈L²(Γ_c). (5.14)

It is very easy to verify that ifµ_t∈M_tthenπ_Htµ_t∈M_Htso that the estimates (i)–(iv) of Lemma 5.2 remain valid. Since−λ_t·u_t+su_t= 0a.e. on Γ_c, the last term can be written as follows (see (5.8)):

b(λ_H−λ,u) ≤

Γc

(λ_Ht·u_t−su_t) dΓ≤0

making use of (5.14) withµ_Ht:=λ_Htandψ:=u_t. Therefore this term which is responsible for the lower order of convergence can be omitted in (5.1). Under the assumptions of Theorem 5.1 we have

u−u_h1,Ω+λ−λ_H−¹

2,Γc≤CH³⁴.

6. Numerical realization

In this section we describe in brief the numerical method for the realization of 3D contact problems with given friction. This method is based on the the dual formulationof (3.5).

Let (u, λn,λ_t) be the saddle-point ofLonV ×Mn×M_t with the notation introduced in Section 2. Then L(u, λn,λ_t) = min

V sup

Mn×Mt

L(v, µn,µ_t)

= max

Mn×Mt

infV L(v, µn,µ_t). Denote

S(µn,µ_t) :=−inf

V L(v, µn,µ_t)

the dual functional. The dual variational formulation to (3.5) reads as follows:

Find (µ^∗_n,µ^∗_t)∈Mn×M_tsuch that S(µ^∗_n,µ^∗_t)≤S(µn,µ_t) ∀(µn,µ_t)∈Mn×M_t.

(6.1) It is well-known [9] that (6.1) has a unique solution and, in addition (µ^∗_n,µ^∗_t) = (λn,λ_t), where (λn,λ_t) is a part of the solution to (3.6). It is also readily seen [13] that Sis a quadratic functional. Therefore (6.1) is a