Convergence of a Neumann-Dirichlet algorithm for two-body contact problems with non local Coulomb's friction law

DOI:10.1051/m2an:2008003 www.esaim-m2an.org

CONVERGENCE OF A NEUMANN-DIRICHLET ALGORITHM FOR TWO-BODY CONTACT PROBLEMS

WITH NON LOCAL COULOMB’S FRICTION LAW

Guy Bayada

, Jalila Sabil

and Taoufik Sassi

Abstract. In this paper, the convergence of a Neumann-Dirichlet algorithm to approximate Coulomb’s contact problem between two elastic bodies is proved in a continuous setting. In this algorithm, the natural interface between the two bodies is retained as a decomposition zone.

Mathematics Subject Classification. 65N30, 65N55, 65K05.

Received December 23, 2006. Revised September 12, 2007.

1. Introduction

Domain decomposition methods are usually characterized by an artificial splitting of a given domain where a partial differential equation has to be solved. Then, suitable iterations are performed in which a sequence of local problems are considered. Each of these local problems being defined in a subdomain with convenient boundary conditions. One of the potential interest of this approach is that the computation of solutions for the local problems required much less time than the solution for the initial problem and can be made simultaneously by using parallel computers. Moreover various discretizations or various kinds of solvers can be used to deal with each of the local problems. Several methods have been reviewed in [9,17,19].

At first glance, multibody contact problems in elasticity appear to be a natural domain of application for domain decomposition methods. Many numerical procedures have been proposed in the mechanical literature.

They are based on standard discretization techniques for partial differential equations in combination with a special implementation of the non-linear contact conditions (seee.g. [5,6,10–13,20]).

However, so far this kind of approach seems to have been neglected in mathematical literature. One of the first mathematical proof of the convergence for contact decomposition algorithm is proposed in [1]. However, the decomposition boundary is artificial and located inside one of bodies.

More recently, a so-called Neumann-Dirichlet algorithm for the solution of frictionless Signorini contact problem has been proposed and studied in the discretized setting in [14–16]. This new algorithm consists

Keywords and phrases. Domain decomposition methods, contact problems, convergence.

1INSA-LYON, CNRS UMR 5208, UMR 5514 Bˆatiment L´eonard de Vinci, 21 Av. J. Capelle, 69621 Villeurbanne Cedex, France.

[email protected]

2Universit´e Henri Poincar´e, UFR STMP, Facult´e des Sciences et Techniques, Laboratoire LEMTA B.P. 239, 54506 Vandœuvre- les-Nancy Cedex, France. [email protected]

3 Laboratoire de Math´ematiques Nicolas Oresme, LMNO CNRS UMR 6139, Universit´e de Caen, Bd. Mar´echal Juin, 14032 Caen Cedex, France. [email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2008

to solve in each iteration a linear Neumann problem for one body and a unilateral contact problem for the other by using essentially the contact interface as the boundary data transfer. The convergence of this algorithm in the continuous setting has been proved in [4,8]. This last approach has been extended to the contact problem with Coulomb friction. The corresponding discretized Neumann-Dirichlet algorithm and several numerical results are given in [14,16].

The aim of the present paper is to prove the convergence of the Neumann-Dirichlet algorithm for contact problem with Coulomb friction in the continuous setting. The main difficulty in this work is linked with the boundary conditions at the contact interface. They are highly non-linear both in the normal direction (unilateral contact conditions) and in the tangential one (non-differential Coulomb law). A fixed point relaxed procedure is defined for the stresses on the contact surface. For sufficiently small friction coefficient, the convergence of the Neumann-Dirichlet algorithm is proved for the continuous problem. It is also shown, by some numerical calculations, that an optimal relaxation parameter exists and its value is nearly independent of the friction coefficient, the mesh size and of the Young modulus.

The paper is organized as follows:

In Sections 2 and 3, we give a statement of the problem and we propose the natural Neumann-Dirichlet algorithm in the strong formulation. The precise variational formulation of the algorithm is given in Section4.

We then prove in Section5, under some assumptions concerning the data, the convergence of the stresses on the contact surface, which in turn proves the convergence of the whole algorithm. In the last, we present some numerical results asserting the efficiency of our algorithm.

2. Stating the problem

Let us consider two elastic bodies, occupying two bounded domains Ω^α, α = 1,2, of the space R². The boundary Γ^α = ∂Ω^α is assumed piecewise continuous, and composed of three complementary non empty parts Γ^α_u, Γ^α_l and Γ^α_c such that ¯Γ^α_u ∩Γ¯^α_c =∅. Each body Ω^α is fixed on the part Γ^α_u and subjected to surface traction forcesφ^α in (L²(Γ^α_l))². The body forces are denoted byf^αin (L²(Ω^α))². In the initial configuration, both bodies have a common contact portion Γ_c= Γ¹_c = Γ²_c. In other words, we consider the case when contact zone cannot grow during the deformation process. Unilateral contact with non local Coulomb’s friction can take place along the boundary Γ_c. The problem consists in finding the displacement fieldu= (u¹,u²) (where the notationu^αstands foru|Ω^α) and the stress tensor fieldσ= (σ(u¹), σ(u²)) such that: forα= 1,2

⎧⎨

⎩

divσ(u^α) +f^α = 0 in Ω^α, σ(u^α)n^α = φ^α on Γ^α_l,

u^α = 0 on Γ^α_u, (2.1)

where the symbol div denotes the divergence operator of a tensor function and is defined as divσ =

∂σ_ij

∂x_j

i

·

The summation convention of repeated index is adopted. The elastic constitutive law is given by Hooke’s law for homogeneous and isotropic solid:

σ_ij(u^α) =A^α_ijkhe_kh(u^α), e(u^α) =1 2

∇u^α+ (∇u^α)^T

, (2.2)

where A^α(x) = (a^α_ijkh(x))1≤i,j,k,h≤2 ∈(L^∞(Ω^α))¹⁶ is a fourth-order tensor satisfying the usual symmetry and ellipticity conditions in elasticity, ande(u^α) is the strain tensor.

We will use the usual notations for the normal and tangential components of the displacement and stress vector on Γ_c

u^α_N =u^α_in^α_i, u^α_Ti =u^α_i −u^α_Nn^α_i

σ_N^α =σ_ij(u^α)n^α_in^α_j, σ_T^α_i =σ_ij(u^α)n^α_j −σ_N^αn^α, in which we have denoted byn^αthe outward normal unit vector to the boundary.

On the interface Γ_c, the unilateral contact law is described by

σ¹_N =σ²_N =σ_N, σ_T¹ =σ_T² =σ_T, (2.3)

[uN]≤0, σ_N ≤0, σ_N[uN] = 0, (2.4)

where [vN] =v¹.n¹+v².n², is the jump across the interface of any functionv defined on Ω^α. The Coulomb’s law with non-local friction is given by

⎧⎨

⎩

|σ_T| ≤k|S(σ_N)|

|σ_T|< k|S(σ_N)|=⇒[uT] = 0

|σ_T|=k|S(σ_N)|=⇒ ∃ν≥0 [uT] =−νσ_T (2.5) where k(x) is the coefficient of friction on the interface Γ_c: k(x) ∈ L^∞(Γ_c), k(x) ≥ 0 a.e. on Γ_c; S is a regularization operator from the dual ofH¹²(Γ_c) intoL²(Γ_c).

It’s easy to prove (see for example [7]) that (2.5) is equivalent to:

|σ_T| ≤k|S(σ_N)|

k|S(σ_N)||[u_T]|+σ_T[u_T] = 0. (2.6)

3. Neumann-Dirichlet algorithm

Let 0< θ <1 be a parameter that will be determined in order to ensure the convergence of the algorithm.

Letg²₀be a given element on Γ_c. Fori≥1, the sequence of functions (u¹_i)_i≥0 and (u²_i)_i≥0is defined by solving the following problems:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

(I) divσ(u²_i) =f² in Ω²

σ(u²_i)n²=φ² on Γ²_l

u²_i =0 on Γ²_u

σ(u²_i)n²=g²_i−1 on Γ_c

(II) divσ(u¹_i) =f¹ in Ω¹

σ(u¹_i)n¹=φ¹ on Γ¹_l

u¹_i =0 on Γ¹_u

u¹_i.n¹≤ −u²_i.n², σ_{i N}¹ ≤0, σ_i¹

N[u_iN] = 0 on Γ_c

|σ_i¹_T| ≤k|S(σ_i²_N)|

σ_i¹_T[uiT] +k|S(σ_i²_N)||[uiT]|= 0 on Γ_c (III) g²_i :=θσ¹_in¹+ (1−θ)g²_i−1 on Γ_c.

(3.1)

The variational form of this algorithm will be given in (4.9). The proof of its convergence towards the solution of the problem (2.1)–(2.6) will be given in Theorem5.9.

4. The variational formulation

In order to study the previous algorithm, let us first define its variational formulation by introducing the following functional spaces:

Forα= 1,2:

V^α=

v^α∈(H¹(Ω^α))², v^α= 0 on Γ^α_u

, endowed with the norm · _α= · _(H1(Ω^α))². V =V¹×V² provided with the norm · _V =

₂

α=1

· ²_α ¹₂

. K={v∈V, [vN]≤0 a.e.on Γ_c}.

H¹²(Γ_c) =

ϕ∈(L²(Γ_c))²; ∃v∈V^α; α= 1,2; γv_|Γc=ϕ

,provided with the norm ϕ¹_2,Γc= inf{v_α;v∈V^α, γv_|Γc=ϕ},

whereγ is the usual trace operator.

As Γ^α_c ∩Γ^α_u=∅, the spaceH¹²(Γ_c) does not depend onα.

Let us denote by V^α the dual space ofV^α and by·,·_α the duality pairing of the two spaces. The dual norm onV^α is given by:

ψ_Vα = sup

v∈V^α

|ψ,v_α| v_α · Let us introduce for allϕinH¹²(Γ_c) the set:

V₋¹(ϕ) ={v¹∈V¹, v¹.n¹≤ −ϕ a.eon Γ_c}. We will denote by:

a^α(u,v) =

Ω^αA^α_ijkle_ij(u)e_kl(v) dx, a(u,v) = 2 α=1

a^α(u,v),

L^α(v) =

Ω^αf^α.vdx+

Γ^α_l φ^α.vdx, L(v) =²

α=1

L^α(v).

The bilinear forma^α(., .) is continuous and coercive since mes(Γ^α_u)>0, soa^α(., .) satisfies:

∃M^α>0, |a^α(u,v)| ≤M^αu_αv_α ∀u,v∈V^α,

∃m^α>0, a^α(v,v)≥m^αv²_α ∀v∈V^α. (4.1) Let

M = 2 α=1

M^α, m= min

α=1,2m^α, (4.2)

so

|a(u,v)| ≤Mu_Vv_V, ∀u,v∈V,

a(v,v)≥mv²_V ∀v∈V. (4.3)

Let us first recall that the variational formulation of the initial problem (2.1)–(2.6) is given by:

(P₁)

⎧⎨

⎩

FinduinK such that:

a(u,v−u) +

Γc

k|S(σ_N)|(|[v_T]| − |[u_T]|)≥L(v−u) ∀v∈K. (4.4) which has at least one solution and this solution is unique for a sufficiently smallk_L∞(Γc) (seee.g. [7]).

In the following, we will use some lift operators which allow us to build specific functions in Ω^αfrom their values on Γ_c.

Forα= 1,2, let

R^α_D :H¹²(Γ_c) −→ V^α

ϕ −→ R^α_Dϕ. (4.5)

a^α(R^α_Dϕ,v) = 0 ∀v∈V^α such thatγv= 0 on Γ_c γ(R^α_Dϕ) =ϕ on Γ_c,

whose strong formulation is given by:

⎧⎪

⎪⎨

⎪⎪

⎩

divσ^α= 0 in Ω^α, withA^αe(R^α_Dϕ) =σ^α γ(R^α_Dϕ) =ϕ on Γ_c

σ^αn^α= 0 on Γ^α_l γ(R^α_Dϕ) = 0 on Γ^α_u.

(4.6)

Proposition 4.1.

(i)The three normsϕ¹_2,Γc andR^α_Dϕ_α, α= 1,2 are equivalent onH¹²(Γ_c).

(ii)There exists a constant C >0 such that

∀v∈V^α R^β_Dγv_|Γcβ≤Cv_α α, β= 1,2 andα=β.

Proof. (i) Using the definition ofR^α_D, we have:

γ(R^α_D(ϕ)) =ϕ, so from the definition of · ¹_2,Γc, we gain

ϕ¹_2,Γc ≤ R^α_Dϕ₁. (4.7)

Conversely, let us consider the range ofH¹²(Γ_c) byR^α_D W^α=R^α_D

H¹²(Γ_c) .

As for anyϕinH¹²(Γ_c), problem (4.5) has a unique solution, and as the trace restriction on Γ_c of any function inW^αis unique, so the applicationR^α_D is a bijection fromH¹²(Γ_c) inW^α. Let us consider its inverseR^α−1D . It is easy to prove that this last bijection is linear, and we can obtain its continuity from (4.7).

Moreover, asW^α is a closed sub-space ofV^α which is a Banach space, soW^αis also a Banach one.

Using the homeomorphism Banach theorem (see e.g.[18]), we obtain that R^α_D is continuous from H¹²(Γ_c) in W^α,i.e.

R^α_Dϕ₁≤Cϕ¹

2,Γc ∀ϕ∈ H¹²(Γ_c). (4.8)

Consequently, the three normsϕ¹

2,Γc andR^α_Dϕ_α, α= 1,2 are equivalent.

(ii) Let v be inV^α, using inequality (4.8) and the trace theorem, the result is established.

Remark 4.2. For sake of simplicity, we will write in the following R^α_D(γv) instead ofR^α_D(γv_|Γc).

The weak formulation of the algorithm (3.1) is given by:

Letg₀²be a given element inV². Fori≥1, let us define the sequence of functions (u¹_i)_i≥0inV¹ and (u²_i)_i≥0 inV² by solving the following problems:

(Q_i)

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

(I) Findu²_i in V² such that:

a²(u²_i,v) =L²(v) +g_i−1² ,v₂ ∀v∈V² (II) Findu¹_i in V¹(u²_iN) such that:

a¹(u¹_i,w+R¹_D(γu²_i)−u¹_i) +

Γc

k|S(σ_iN(u²_i))|(|w_T| − |[u_iT]|)

≥L¹(w+R¹_D(γu²_i)−u¹_i) ∀w∈V₋¹(0) (III)g²_i,v₂:=

θ

−a¹(u¹_i,R¹_Dγv) +L¹(R¹_Dγv)

+ (1−θ)g²_i−1,v₂∀v∈V².

(4.9)

Remark 4.3. Using inequality (ii) of Proposition 4.1, the expression g_i²,v₂ in (4.9)(III) has sense, i.e. g²_i belongs toV².

5. Convergence

The convergence of the algorithm towards the solution of (P₁) and then of (2.1)–(2.6) is given by a two-step procedure. We prove first, in Proposition5.1, that the convergence of (g_i²)_iinV²induces the strong convergence of (u^α_i)_i. Then, in Proposition5.6, the contraction of an auxiliary operatorT_θ defined by (5.7) is proved. This yields the convergence of the sequence (g_i²)_i by way of Theorem5.9.

Proposition 5.1. Let us suppose that there exists 0 < θ_min < 1 such that for any θ, θ ≥ θ_min, the se- quence(g_i²)_i≥0 converges in V². Then the sequence(u¹_i,u²_i) (strongly) converges inV to the solution(u¹,u²) of (P₁).

Proof. Let (u¹_i,u²_i) and (u¹_j,u²_j) be solutions of the problems (Q_i) and (Q_j) respectively, so from (4.9)(III) we have:

a¹(u¹_i −u¹_j,R¹_Dγv) =−g²_i−1−g_j−1² ,v₂−

g²_i −g_i−1²

θ −g_j²−g_j−1² θ ,v

2

∀v∈V².

Choosingv=R²_Dγ(u¹_i−u¹_j) in the previous inequality and using the fact thatR¹_Dγ(R²_Dγ(u¹_i−u¹_j)) =u¹_i−u¹_j, we get:

u¹_i −u¹_j²₁≤ 1 θ_minm

g²_i−1−g_j−1² _V2+g_i²−g²_i−1V2 +g_j²−g²_j−1V2

R²_Dγ(u¹_i −u¹_j)₂.

Using Proposition4.1(i), and the strong convergence of (g²_i)_i≥0 inV², one obtains the convergence of (u¹_i)_i. The strong convergence of (u²_i)_i can be obtained from the one of (g_i²)_i≥0, by using the coerciveness ofa²(·,·) in (4.9)(I).

Finally, we pass to the limit in equations (II) and (III) of problem (Q_i). The limit of (u^α_i)_i, α= 1,2 appears to be a solution of the two following problems:

(P₂) Findu² inV² such that:

a²(u²,v²) =L²(v²)−a¹(u¹,R¹_D(γv²)) +L¹(R¹_D(γv²)) ∀v²∈V². (5.1)

(P₃)

⎧⎪

⎪⎨

⎪⎪

⎩

Find u¹ in V₋¹(u²_N) such that:

a¹(u¹,w+R¹_D(γu²)−u¹) +

Γc

k|S(σ_N)|(|wT| − |[uT]|)

≥L¹

w+R¹_D(γu²)−u¹

∀w ∈V₋¹(0).

(5.2)

The following Lemma 5.2proves that problems (P₂) and (P₃) imply (P₁) so allowing us to finish the proof of

Proposition5.1.

Lemma 5.2. Assuming that a solution exists for problems(P₂)and(P₃), then there exists also a solution for problem (P₁) (4.4).

Proof. Let us make the following shift for the test function w in (P₃): v¹ = w+R¹_D(γu²). We obtain the following problem (P₃) which is equivalent to problem (P₃):

(P₃)

⎧⎪

⎪⎨

⎪⎪

⎩

Findu¹ inV₋¹(u²_N) such that:

a¹(u¹,v¹−u¹) +

Γc

k|S(σ_N)|

|v¹_T −u²_T| − |[u_T]|

≥L¹(v¹−u¹)

∀v¹∈V₋¹(u²_N).

(5.3)

Letv= (v¹,v²) inK, asv¹.n¹+(v²−u²).n²= [vN]−u²_N ≤ −u²_N on Γ_c, sow=v¹−R¹_Dγ(v²−u²)∈V₋¹(u²_N).

Choosing noww (resp. v²−u²) as test function in (P₃) (resp. in (P₂)), (P₁) is obtained by addition.

In order to prove the convergence of (g²_n)_n, let us introduce the operatorT defined by:

T: V² −→V²

ψ −→T ψ (5.4)

such that:

T ψ,v₂=−a¹(w¹,R¹_Dγv) +L¹(R¹_Dγv)∀v∈V², wherew¹ is the solution of the following obstacle problem:

⎧⎪

⎪⎨

⎪⎪

⎩

Findw¹ inV₋¹(w²_N) such that:

a¹(w¹,v−w¹) +

Γc

k|S(σ_N(w²))|

|vT −w²_T| − |w¹_T−w²_T|

≥ L¹(v−w¹) ∀v∈V₋¹(w²_N)

(5.5)

andw² is the solution of the Neumann problem:

Find w²in V²such that:

a²(w²,v) =L²(v) +ψ,v₂ ∀v∈V². (5.6) We also define the operatorT_θ by:

T_θ: V² −→V²

ψ −→T_θ(ψ) =θT(ψ) + (1−θ)ψ. (5.7)

We shall denote by:

C_R: the maximum of the constants of continuity forR¹_D andR²_D.

C_s: the maximum with respect ofα= 1,2 of the constant of continuity of the applicationS(σ^α_N(·)) which to eachw^αinV^α associatesS(σ_N^α(w^α)), the regularization of σ_N^α(w^α) inL²(Γ_c).

C_tr: the maximum with respect ofα= 1,2 of the constants of continuity of both following trace applications:

T r^α: V^α −→ H¹²(Γ_c), v −→ vT, (5.8)

wherevT is the tangential component ofv.

C_K=k_L∞(Γc)C_sC_tr, wherekis the friction coefficient introduced in (2.6).

Lemma 5.3. Forw² andw˜² inV², let w¹ andw˜¹ be solutions of the problems:

⎧⎪

⎪⎨

⎪⎪

⎩

Findw¹ in V₋¹(w²_N) such that:

a¹(w¹,v−w¹) +

Γc

k|S(σ_N(w²))|

|vT −w²_T| − |[wT]|

≥ L¹(v−w¹) ∀v∈V₋¹(w²_N).

(5.9)

⎧⎪

⎪⎨

⎪⎪

⎩

Findw˜¹ in V₋¹( ˜w²_N) such that:

a¹( ˜w¹,v−w˜¹) +

Γc

k|S(σ_N( ˜w²))|

|v_T −w˜²_T| − |[ ˜w_T]|

≥ L¹(v−w˜¹) ∀v∈V₋¹( ˜w²_N),

(5.10)

then the following estimate holds:

w¹−w˜¹₁≤

(MC_R+C_K)

m +

C_K m

w²−w˜²₂. (5.11)

Proof. We choosev= ˜w¹−R¹_D( ˜w²) +R¹_D(w²) in (5.9) andv=w¹−R¹_D(w²) +R¹_D( ˜w²) in (5.10). By adding both inequalities, we obtain

a¹

w¹−w˜¹,w¹−w˜¹

≤a¹

w¹−w˜¹,R¹_D(w²)−R¹_D( ˜w²)

−

Γc

k

|S(σ_N(w²))| − |S(σ_N( ˜w²))|

(|[w_T]| − |[ ˜w_T]|). (5.12)

The continuity of the applicationsS(σ_N(·)) andT r^αdefined in (5.8) induces

−

Γc

k

|S(σ_N(w²))| − |S(σ_N( ˜w²))|

(|[w_T]| − |[ ˜w_T]|)≤C_K

w¹−w˜¹₁w²−w˜²₂+w²−w˜²²₂ . (5.13) In (5.12), we use for the left-hand side, the coerciveness ofa¹(., .), and for the right-hand side, the continuity of the extensionR^α_D and inequality (5.13) to obtain:

mw¹−w˜¹²₁≤(MC_R+C_K)w¹−w˜¹₁w²−w˜²₂+C_Kw²−w˜²²₂, (5.14) hence

m

w¹−w˜¹₁−MC_R+C_K

2m w²−w˜²_{2 2}

≤

(MC_R+C_K)²

4m +C_K

w²−w˜²²₂. (5.15)

Then the required result is obtained.

Proposition 5.4. The operator T is Lipschitz continuous:

∀ψ,ψ˜ inV² T ψ−Tψ˜_V2 ≤C_Tψ−ψ˜_V2, with

C_T = MC_R m²

MC_R+C_K+ C_Km

. (5.16)

Proof. Letψand ˜ψin V²,w¹ (resp. ˜w¹) andw² (resp. ˜w²) be the solutions of (5.5) and (5.6), so

T ψ−Tψ,˜ v2=−a¹(w¹−w˜¹,R¹_D(γv)) ∀v∈V². (5.17) From the continuity ofa¹(·,·) and Proposition4.1(i), we get:

T ψ−Tψ˜_V2 ≤ MC_Rw¹−w˜¹₁. (5.18) According to (5.6), one has

a²(w²−w˜²,v) =ψ−ψ,˜ v₂ ∀v∈V². The coerciveness of a²(·,·) implies

w²−w˜²₂ ≤ 1

mψ−ψ˜_V2. (5.19)

From (5.18), (5.11) and (5.19), we get the existence of a constantC_T such that:

T(ψ)−T( ˜ψ)_V2 ≤C_Tψ−ψ˜_V2. Let us define the following operatorR²:

R²: V² −→V² ψ −→R²ψ

whereR²ψis the unique solution in V², of the problem:

a²(R²(ψ),v) =ψ,v₂ ∀v∈V². (5.20)

Let us consider the following norm in V²:

|||ψ|||=

a²(R²ψ,R²ψ)¹₂

. (5.21)

Proposition 5.5. |||.||| and._V2 are equivalent norms on the spaceV².

Proof. By usingv =R²(ψ) as test function in (5.20), the proof is the direct consequence of (5.21) and of the continuity and the coerciveness ofa²(·,·), so that we get:

|||ψ||| ≤ √1mψ_V2, (5.22)

√Mm|||ψ||| ≥ ψ_V2. (5.23)

Proposition 5.6. There exist constants C_i,i= 0, ...,5, such thatT_θ is a contraction for any θ in ]C₀, C₁[ if k_L∞(Γc)∈[0, C₂[.

]0, C₀[ if k_L∞(Γc)∈[0, C₃[.

]0, C₀[ if k_L∞(Γc)∈]C₃, C₄[.

]C₆, C₀[ if k_L∞(Γc)∈]C₄, C₅[.

Proof. Letψ and ˜ψ in V², w¹ (resp. ˜w¹) and w² (resp. ˜w²) be the solutions of (5.5) and (5.6). From the definition ofT_θ we have:

|||T_θ(ψ)−T_θ( ˜ψ)|||²=θ²|||T(ψ)−T( ˜ψ)|||²+ (1−θ)²|||ψ−ψ˜|||²+ 2θ(1−θ)a²

R²(T(ψ)−T( ˜ψ)),R²(ψ−ψ)˜

. (5.24) To prove that the operator T_θ is a contraction, we give, in the first step, an estimate for the term a²

R²(T(ψ)−T( ˜ψ)),R²(ψ−ψ)˜

in terms of |||T(ψ)−T( ˜ψ)||| and |||ψ−ψ˜|||. We use the definitions of op- eratorsT andR² and the equivalence between||| · |||and · _V2.

First step. From the uniqueness of the solution of (5.20), we getR²(ψ−ψ) =˜ w²−w˜² so a²

R²(T(ψ)−T( ˜ψ)),R²(ψ−ψ)˜

=a²

R²(T(ψ)−T( ˜ψ)),w²−w˜²

, (5.25)

using the definition (5.20) ofR², we have:

a²

R²(T(ψ)−T( ˜ψ)),w²−w˜²

=

T ψ−Tψ,˜ w²−w˜²

2. (5.26)

The definition (5.4) ofT gives

T ψ−Tψ,˜ w²−w˜²

2=−a¹

w¹−w˜¹,R¹_Dγ(w²−w˜²)

. (5.27)

From inequality (5.12), we obtain:

−a¹

w¹−w˜¹,R¹_Dγ(w²−w˜²)

≤ −a¹

w¹−w˜¹,w¹−w˜¹

−

Γc

k

|S(σ_N(w²))| − |S(σ_N( ˜w²))|

(|[w_T]| − |[ ˜w_T]|). (5.28) In the right-hand side, the first term will be estimated by using the coerciveness ofa¹(., .), inequalities (5.18) and (5.22):

−a¹

w¹−w˜¹,w¹−w˜¹

≤ − m²

M²C_R²|||T(ψ)−T( ˜ψ)|||², (5.29) for the second term, using inequalities (5.11) (see Lem.5.3) and (5.19) in (5.13) and using (5.23), we obtain:

−

Γc

k

|S(σ_N(w²))| − |S(σ_N( ˜w²))|

(|[wT]| − |[ ˜wT]|)≤ C_K m²

MC_R+C_K+

C_Km+ 1 M²

m |||ψ−ψ˜|||². (5.30) Taking inequalities (5.26)–(5.30) into account in (5.25), we obtain:

a²

R²(T(ψ)−T( ˜ψ)),R²(ψ−ψ)˜

≤ − m²

M²C_R²|||T(ψ)−T( ˜ψ)|||² +C_K

m²

MC_R+C_K+

C_Km+ 1 M²

m |||ψ−ψ˜|||². (5.31)

Using (5.31) and (5.24), we obtain:

|||T_θ(ψ)−T_θ( ˜ψ)|||²≤

θ²−m²2θ(1−θ) M²C_R²

|||T(ψ)−T( ˜ψ)|||² +

(1−θ)²+ 2θ(1−θ)C_K m²

MC_R+C_K+

C_Km+ 1 M²

m

|||ψ−ψ˜|||². (5.32) Using (5.16), we get:

|||T_θ(ψ)−T_θ( ˜ψ)|||²≤

θ²−m²2θ(1−θ) M²C_R²

|||T(ψ)−T( ˜ψ)|||² +

(1−θ)²+ 2θ(1−θ)M² m

C_KC_T MC_R +C_K

m²

|||ψ−ψ˜|||². (5.33) Second step. Later on, it is convenient to introduce:

m˜ = min{m,1}, M˜ =M. (5.34)

We immediately have from (5.16)

C_T ≤C˜_T = MC˜ _R

m˜² ( ˜MC_R+C_K+

C_Km).˜ (5.35)

Inequality (5.33) becomes:

|||T_θ(ψ)−T_θ( ˜ψ)|||²≤

θ²−m˜²2θ(1−θ) M˜²C_R²

|||T(ψ)−T( ˜ψ)|||²

+

(1−θ)²+ 2θ(1−θ)M˜² m˜

C_KC˜_T MC˜ _R +C_K

m˜²

|||ψ−ψ˜|||². (5.36)

According to the sign of the coefficient of|||T(ψ)−T( ˜ψ)|||², we distinguish two cases: in the first one, the Lipschitz continuity of the operatorT is used, and in the second one, we only have to cancel this term:

Case 5.7.

For anyθ≥C₀ with C₀= 2 ˜m²

M˜²C_R² + 2 ˜m², (5.37) we haveθ²−m˜²2θ(1−θ)

M˜²C_R² ≥0, so that Proposition5.4implies

θ²−m˜²2θ(1−θ) M˜²C_R²

|||T(ψ)−T( ˜ψ)|||²≤

θ²−m˜²2θ(1−θ) M˜²C_R²

C_T²M˜²

m˜²|||ψ−ψ˜|||². (5.38) Hence, inequality (5.36) becomes

|||T_θ(ψ)−T_θ( ˜ψ)|||²≤h₁(θ)|||ψ−ψ˜|||², (5.39)