Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows

Mod´elisation Math´ematique et Analyse Num´erique Vol. 35, N 5, 2001, pp. 879–897

EXISTENCE, A PRIORI AND A POSTERIORI ERROR ESTIMATES FOR A NONLINEAR THREE-FIELD PROBLEM

ARISING FROM OLDROYD-B VISCOELASTIC FLOWS

Marco Picasso

and Jacques Rappaz

Abstract. In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B model without convective terms is considered. Assuming the domain to be a convex polygon, existence of a solution is proved for small relaxation times. Continuous piecewise linear finite elements together with a Galerkin Least Square (GLS) method are studied for solving this problem. Existence anda priorierror estimates are established using a Newton-chord fixed point theorem,a posteriorierror estimates are also derived.

An Elastic Viscous Split Stress (EVSS) scheme related to the GLS method is introduced. Numerical results confirm the theoretical predictions.

Mathematics Subject Classification. 65N30, 65N12, 76A10.

Received: January 17, 2001.

1. Introduction

Numerical simulation of viscoelastic flows is of great importance for industrial processes involving plastics, paints or food. The modelling of viscoelastic flows generally consists in supplementing the mass and momentum equations with a rheological constitutive equation relating the velocity and the non Newtonian part of the stress.

When solving viscoelastic flows with finite element methods, the following points should be addressed, see for instance [1] for a review. Firstly, the finite element spaces used to approximate the velocity, pressure and extra-stress fields cannot be chosen arbitrarily, an inf-sup condition has to be satisfied [12, 13, 15, 24, 25].

Secondly, due to the presence of convective terms in both momentum and constitutive equations, adequate discretizations procedure must be used such as discontinuous finite elements, GLS stabilization procedures [5], or the characteristics method. Thirdly, the presence of nonlinear terms prevents numerical methods to converge at high Deborah numbers, this being consistent with theoretical [4, 19, 21, 23, 26] and experimental [22] studies.

In this paper, we focus on the first and last of these three points. In [6], a GLS method with continuous, piecewise linear finite elements was proposed for solving a three fields Stokes’ problem. The method was stable even when the solvent viscosity was small compared to the polymer viscosity. The link with the EVSS method of [12] was proposed. The aim of this paper is to extend the work of [6] to a nonlinear model problem. Existence, a priorianda posteriorierror estimates are derived. Numerical results confirm the theoretical predictions. Even though the model problem studied in this paper is simpler than those considered in [4, 19, 21, 23, 26], we believe that our results are interesting for the following reasons. Firstly, assuming the calculation domain to be a convex

Keywords and phrases. Viscoelastic fluids, Galerkin Least Square finite elements.

∗This work is supported by the Swiss National Science Foundation.

1 D´epartement de Math´ematiques, ´Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland.

c EDP Sciences, SMAI 2001

polygon, we propose an original, but natural, variational setting in order to prove existence of the continuous problem. More precisely, the velocity is inW^2,r, for somer >2, whereas the pressure and the extra-stress are inW^1,r. Secondly, we consider an original stabilized finite element method and we prove existence,a prioriand a posteriori error estimates. Finally, we present some numerical computations that i) confirm the optimality of our theoretical predictions ii) show that the method fails when the computational domain is not convex, thus suggesting that the problem is ill-posed.

The reader should note that the theoretical results presented in this paper can be seen as a first step towards the justification of some numerical methods used to perform mesoscopic calculations [7, 8].

2. The model problem

Let Ω be a bounded domain ofR² with Lipschitz boundary∂Ω. We consider the following problem. Given a force term f, constant solvent and polymer viscositiesη_s>0 and η_p >0, a constant relaxation time λ, find the velocityu, pressurepand extra-stressσ such that

−2ηs div(u) +∇p−divσ=f, divu= 0, 1

2η_pσ− λ 2η_p

(∇u)σ+σ(∇u^T)

−(u) = 0,

(1)

in Ω, where the velocity uvanishes on ∂Ω. Here (u) = ¹₂(∇u+∇u^T) is the rate of deformation tensor and (∇u)σis the the matrix product between∇uandσ.

This model problem is a simplification of viscoelastic models for polymeric liquids. The first equation corresponds to momentum conservation. The total stress is split into three contributions: the pressure−pI, the stress due to the (Newtonian) solvent 2ηs(u), and the extra-stressσ due to the non Newtonian part of the fluid (for instance polymer chains). The third equation is a simplification of the Oldroyd-B constitutive relationship between the extra-stress and the velocity field. For the sake of simplicity, the convective terms in the first and last equations are removed. A future work should take these terms into account.

A theoretical result for Problem (1) withη_s= 0 and with convective terms in the first and last equation has been obtained in [23]. Using an iterative procedure, the solution (u, p,σ) was proved to be inH³×H²×H² providedf ∈H¹was small enough and∂Ω was sufficiently smooth. In this section, we shall prove that Problem (1) has a solution in W^2,r×W^1,r ×W^1,r for some r > 2, provided λ is small enough, when Ω is a convex polygonal domain.

LetL²₀(Ω) be the space of L²(Ω) functions having zero mean, letL²_s(Ω)⁴ be the space of symmetric tensors havingL²(Ω) components. Givenf ∈H⁻¹(Ω)²andg∈L²_s(Ω)⁴, we first consider the following problem (we set λ= 0 and add a source term in the third equation of (1)): find (u, p,σ)∈H₀¹(Ω)²×L²₀(Ω)×L²_s(Ω)⁴ such that

−2η_sdiv (u) +∇p−divσ=f, divu= 0, 1

2ηpσ−(u) =g.

(2)

Eliminatingσ we obtain

−2(η_s+η_p)div (u) +∇p=f+ 2η_pdiv g,

div u= 0. (3)

Since g ∈ L²_s(Ω)⁴, divg ∈ H⁻¹(Ω), and Problem (3) is a classical Stokes problem with solution (u, p) ∈ H₀¹(Ω)²×L²₀(Ω). Setting σ = 2ηp((u) +g), then (u, p,σ) is solution of Problem (2). Thus we define the

operator

T :H⁻¹(Ω)²×L²_s(Ω)⁴−→H₀¹(Ω)²×L²_s(Ω)⁴ (f,g) −→T(f,g) =

def.(u,σ),

where (u,σ) is the solution of (2). Problem (1) can then be formally written as follows: find (u,σ) such that (u,σ) =T

f, λ

2ηp

(∇u)σ+σ(∇u)^T

. (4)

The question is now to find the correct spaces for the above problem. Indeed, since (u,σ) area priori only in H₀¹(Ω)²×L²_s(Ω)⁴, then (∇u)σ+σ(∇u)^T is only inL¹(Ω), thus div ((∇u)σ+σ(∇u)^T)∈/ H⁻¹(Ω) !

Let us recall some regularity properties of Stokes’ problem. Let µ > 0 and consider the solution (w, q) ∈ H₀¹(Ω)²×L²₀(Ω) of

−2µdiv(w) +∇q=f, div w= 0.

According to Proposition 2.3 of [27], if the boundary ∂Ω is C² and if f ∈L^r(Ω)², thenw ∈W^2,r(Ω)², for all 1 < r <∞. According to Theorem 7.3.3.1 of [18] and Proposition 5.3 of [17], if Ω is a convex polygon, then there exists r >2 (rdepends on the largest angle of the polygon) such that, iff ∈L^r(Ω)², thenw∈W^2,r(Ω)² and kwk_W^2,r ≤CkfkL^r. In the sequel,this being crucial for our analysis, we will assume that Ω is a convex polygonand we will considerr as above.

We set

X= (W^2,r(Ω)∩H₀¹(Ω))²×W_s^1,r(Ω)⁴,

where W_s^1,r(Ω)⁴ is the space of symmetric tensors having components in W^1,r(Ω). We know that W^1,r(Ω), r >2, is an algebra, namely ifϕ, ψ∈W^1,r(Ω) then the productϕψ∈W^1,r(Ω) and there existsC such that

kϕψk_W^1,r_(Ω)≤Ckϕk_W^1,r_(Ω)kψk_W^1,r_(Ω) ∀ϕ, ψ∈W^1,r(Ω).

Therefore, if (u,σ)∈X then

g= λ 2η_p

(∇u)σ+σ(∇u)^T

∈W_s^1,r(Ω)⁴,

and div g∈L^r(Ω)². Thus, iff ∈L^r(Ω)², then T

f, λ

2ηp

(∇u)σ+σ(∇u)^T

∈X.

From now on, we will restrictT toL^r(Ω)²×W_s^1,r(Ω)⁴ that is T :L^r(Ω)²×W_s^1,r(Ω)⁴−→X

(f,g) −→T(f,g) =

def.(u,σ), (5)

where (u,σ) is the solution of (2). The operatorT is bounded since

kT(f,g)k_X≤C(kfk_L^r+kgk_W^1,r) ∀(f,g)∈L^r(Ω)²×W_s^1,r(Ω)⁴. Finally, going back to (4), we are looking forU= (u,σ)∈X such that

F(λ,U) = 0, (6)

whereF :R×X→X is defined by

F(λ,U) =U−T

f, λS(U) , andS:X→W_s^1,r(Ω)⁴ by

S(U) = 1 2η_p

(∇u)σ+σ(∇u)^T

. (7)

We have the following result.

Lemma 2.1. For all λ≥0, the operatorF(λ,·) :X→X isC¹, with Frechet derivative given by DUF(λ,U)V= (v,τ)−T

0, λ

2η_p

(∇u)τ+τ(∇u)^T + (∇v)σ+σ(∇v)^T ,

for allU= (u,σ)∈X,V= (v,τ)inX.

Proof. It suffices to note that the operatorS:X →W_s^1,r(Ω)⁴ isC¹ with Frechet derivative DS(U)V= (∇u)τ+τ(∇u)^T+ (∇v)σ+σ(∇v)^T.

We now observe that, when λ= 0, (6) reduces to a classical three fields Stokes’ problem. Thus, there is a uniqueU₀= (u₀,σ0)∈X such thatF(0,U₀) = 0. Using the implicit function theorem, we can then prove the following result.

Theorem 2.2. There exists λ0 > 0 and δ > 0 such that, for all λ ≤ λ0, there exists a unique U(λ) = (u(λ),σ(λ)) ∈ X such that F(λ,U(λ)) = 0 and kU(λ)−U₀kX ≤ δ. Moreover, the mapping λ ∈ [0, λ0] → U(λ)∈X is continuous.

Proof. Whenλ= 0 there is a uniqueU₀= (u₀,σ0)∈X such that F(0,U₀) = 0. Moreover, using Lemma 2.1, we have DUF(0,U₀) =I, which is an isomorphism onto X. The result is then an immediate consequence of the implicit function theorem.

3. A Galerkin Least Square method

In [6], a GLS method with continuous, piecewise linear finite elements was studied for solving (1) in the linear case, i.e. when λ = 0. The method was proved to be stable and convergent even when the solvent viscosity η_s was small compared to the polymer viscosity η_p. A priori error estimates where derived in the space W =H₀¹(Ω)²×L²₀(Ω)×L²_s(Ω)⁴. Finally, this GLS method was shown to be equivalent to some EVSS method. The aim of the paper is to extend these results to the nonlinear case, that is whenλ6= 0.

For any h >0, letTh be a mesh of Ω into triangles K with diametershK less thanh. We assume that the mesh satisfies the regularity and inverse assumptions in the sense of [10]. We considerWhthe finite dimensional subspace ofW consisting in continuous, piecewise linear velocities, pressures, and stresses on the meshT_h. More preciselyW_h⊂W is defined by W_h=V_h×Q_h×M_h where

Vh= n

v∈ C⁰(Ω)²;v_|K∈(P1)², ∀K∈ Th

o∩H₀¹(Ω)²,

Qh=

nq∈ C⁰(Ω);q_|K ∈P1, ∀K∈ Th

o∩L²₀(Ω),

Mh=

nτ ∈ C⁰(Ω)⁴;τ_|K∈(P1)⁴, ∀K∈ Th

o∩L²_s(Ω)⁴.

(8)

In order to approach the solution of (1) we consider the following problem: find (u_h, ph,σh)∈Wh such that 2ηs((uh),(v))−(ph,div v) + (σh,(v))−(f,v)

−(divu_h, q)− X

K∈Th

αh²_K 2η_p

−2η_sdiv (u_h) +∇p_h−divσ_h−f,∇q

K

− 1

2ηpσh− λ

2ηp(∇u_hσh+σh∇u^T_h)−(uh),τ + 2ηpβ(v)

= 0,

(9)

for all (v, q,τ)∈Wh. Here (·,·) denotes theL²(Ω) scalar product for scalars, vectors and tensors, for instance (σ,τ) = X²

i,j=1

Z

Ω

σ_ij τ_ij dx ∀σ,τ ∈L²(Ω)⁴.

Also,α >0 and 0< β <2 are dimensionless stabilization parameters. Note that, since the velocity is piecewise linear, then div(u_h) is zero in each of the mesh triangles.

In the next section, existence of a solution to (9) is proved, as well asa priorierror estimates. In Section 5, a posteriorierror estimates based on the equation residual are derived. In Section 6, an EVSS method related to the GLS method is presented. Numerical results on both GLS and EVSS schemes are reported in Section 7.

4. Existence and

error estimates

Let us consider the discrete counterpart of the operatorT, namely the operator Th:L²(Ω)²×L²_s(Ω)⁴−→H₀¹(Ω)²×L²_s(Ω)⁴

(f,g) −→Th(f,g) =

def.(u_h,σh)∈Vh×Mh, (10) where (u_h, ph,σh)∈Wh satisfies

2ηs((uh),(v))−(ph,div v) + (σh,(v))−(f,v)

−(divu_h, q)− X

K∈Th

αh²_K 2η_p

−2η_sdiv (u_h) +∇p_h−divσ_h−f,∇q

K

− 1

2ηpσh−g−(uh),τ+ 2ηpβ(v)

= 0,

(11)

for all (v, q,τ)∈Wh. We then have the following result.

Lemma 4.1. The operatorT_his well defined and is uniformly bounded with respect toh. Moreover, there exists h0>0 andC such that, for all (f,g)∈L^r(Ω)²×W_s^1,r(Ω)⁴ we have

kT(f,g)−T_h(f,g)k_H¹_×L² ≤Ch

kfk_L^r+kgk_W^1,r

∀h≤h0.

Proof. We introduce, as in [6], the bilinear form Bh(u_h, ph,σh;v, q,τ) corresponding to the left hand side of the weak formulation (11) whenf andgare zero. The operatorThis then defined byTh(f,g) = (u_h,σh) where (u_h, ph,σh)∈Whsatisfies

Bh(u_h, ph,σh;v, q,τ) = (f,v)−

g,τ+ 2ηpβ(v)

− X

K∈Th

αh²_K

2ηp (f,∇q)_K

for all (v, q,τ)∈Wh. It is proved in [6] thatBh:Wh×Wh→Rsatisfies the uniform (with respect toh) inf-sup condition in the normk · k_H¹_×L²_×L². ThusTh is well defined and we have, for allh >0

ku_h, ph,σhk_H¹_×L²_×L² ≤Ckf,gk_L²_×L², and thus

kThk_L(L²_×L²_,H¹_×L²₎≤C,

where C does not depend on h. Moreover, if (f,g) ∈ L^r(Ω)²×W_s^1,r(Ω)⁴, then we can introduce T(f,g) = (u,σ)∈X, where (u,σ) is the solution of (2) and whereT is defined in (5). We have

kT(f,g)−Th(f,g)k_H¹_×L²=ku−u_h,σ−σhk_H¹_×L²

≤ ku−rhu,σ−rhσk_H¹_×L²+krhu−u_h, rhσ−σhk_H¹_×L², whererh denotes Lagrange interpolant on scalars, vectors or tensors. Note that, consideringrhu,rhpandrhσ has a meaning since (u, p,σ) ∈ W^2,r(Ω)²×W^1,r(Ω)×W_s^1,r(Ω)⁴, and thus u, pand σ are continuous on Ω.

Using classical interpolation results [10] together with the well posedness of the operatorT, we have ku−rhu,σ−rhσk_H¹_×L²≤Ch

|u|_H²+|σ|_H¹_∩C⁰

≤Ch

kuk_W^2,r +kσk_W^1,r

≤Ch

kfk_L^r+kgk_W^1,r .

(12)

On the other side, using the fact thatBh satisfies the uniform inf-sup condition onWh, there is a constantC independent ofhsuch that

krhu−u_h, rhσ−σhk_H¹_×L² ≤C sup

06=(v,q,τ)∈Wh

Bh(rhu−u_h, rhp−ph, rhσ−σh;v, q,τ)

kv, q,τk_H¹_×L²_×L² · (13) The scheme (11) is consistent in the sense of [14], that is

B_h(u−u_h, p−p_h,σ−σ_h;v, q,τ) = 0 ∀(v, q,τ)∈W_h, and thus

kr_hu−u_h, r_hσ−σ_hk_H¹_×L²≤C sup

06=(v,q,τ)∈Wh

Bh(u−rhu, p−rhp,σ−rhσ;v, q,τ) kv, q,τk_H¹_×L²_×L² · From the definition ofBh we have, for all (v, q,τ)∈Wh,

B_h(u−r_hu, p−r_hp,σ−r_hσ;v, q,τ)

= 2(ηs+ηpβ)((u−rhu),(v))

−(p−rhp,div v) + (1−β)(σ−rhσ,(v))

−( div(u−r_hu), q)− X

K∈Th

αh²_K

2ηp (f− ∇r_hp+ divr_hσ,∇q)_K

− 1

2η_p(σ−rhσ,τ) + ((u−rhu),τ).

Using Cauchy-Schwarz and Young inequalities, there is a constantCindependent ofhsuch that

|Bh(u−rhu, p−rhp,σ−rhσ;v, q,τ)|

≤C

ku−rhu, p−rhp,σ−rhσk_H¹_×L²_×L² kv, q,τk_H¹_×L²_×L²

+ X

K∈Th

h²_K

kfk_L²_(K)+kr_hpk_H¹_(K)+kr_hσk_H¹_(K)

k∇qk_L²_(K)

.

We now use the inverse inequality h_Kk∇qk_L²_(K)≤Ckqk_L²_(K), the fact that kr_hpk_H¹ ≤Ckr_hpk_W^1,r ≤Ckpk˜ _W^1,r krhσk_H¹ ≤Ckrhσk_W^1,r ≤Ckσk˜ _W^1,r, forhsufficiently small, and standard interpolation estimates [10] to obtain

|Bh(u−rhu, p−rhp,σ−rhσ;v, q,τ)|

≤Ch

ku, p,σk_W^2,r_×W^1,r_×W^1,r kv, q,τk_H¹_×L²_×L² +

kfk_L^r+kpk_W^1,r +kσk_W^1,r kqk_L²

. Using the well posedness of the operatorT we obtain

|B_h(u−r_hu, p−r_hp,σ−r_hσ;v, q,τ)| ≤Ch

kfk_L^r+kgk_W^1,r

kv, q,τk_H¹_×L²_×L², for all (v, q,τ)∈Wh. The above inequality in (13), together with (12) finally yields

ku−u_h,σ−σhk_H¹_×L² ≤Ch

kfkL^r+kgk_W^1,r , which is the desired result.

We now would like to rewrite the nonlinear GLS scheme (9) in an abstract framework, as we did in (6) for the continuous Problem (1). We now introduceX_h=V_h×M_hequipped with thek · k_H¹_×L² norm and we want to rewrite the nonlinear GLS scheme (9) as

F_h(λ,U_h) = 0 with U_h= (u_h,σ_h). (14)

HereFh:R×Xh→Xh is defined by

Fh(λ,U_h) =U_h−Th

f, λS(U_h) ,

whereS is still formally defined by

S(Uh) =S(u_h,σh) = 1 2η_p

(∇u_h)σh+σh(∇u_h)^T .

However, sinceX_h6⊂X, we have to extend S onW₀^1,r(Ω)²×W_s^1,r(Ω)⁴, and since W_s^1,r(Ω)⁴ ⊂ C⁰(Ω), we can considerS :W₀^1,r(Ω)²×W_s^1,r(Ω)⁴→L^r_s(Ω)⁴. Clearly, this operatorS isC¹.

We have the following result.

Lemma 4.2. Let U(λ),0≤λ≤λ0, be given by Theorem 2.2. There existsC such that, for allh >0, for all λ≤λ0 we have

kFh(λ, rhU(λ))k_H¹_×L²≤Ch, (15)

kDUFh(λ, rhU(λ))−DUFh(λ,V)k_L(Xh)≤Cλ

hkrhU(λ)−Vk_H¹_×L², (16) for allV∈X_h.

Proof. From the definition ofF_h and sinceF(λ,U(λ)) = 0, we have F_h(λ, r_hU(λ)) =r_hU(λ)−U(λ)−T_h

f, λS(r_hU(λ)) +T

f, λS(U(λ)) , so that

kFh(λ, rhU(λ))k_H¹_×L²

≤ kU(λ)−r_hU(λ)k_H1×L²+(T−T_h)

f, λS(U(λ))

H¹×L²

+Th

0, λ[S(U(λ))−S(rhU(λ))]

H¹×L².

Using Lemma 4.1 for the second and third terms in the right hand side of the above inequality together with interpolation results, we obtain

kFh(λ, rhU(λ))k_H¹_×L²

≤ChkU(λ)k_X+Ch

kfk_L^r+λkS(U(λ))k_W^1,r +CλkS(U(λ))−S(r_hU(λ))k_L²,

(17)

C being independent ofhand λ∈[0, λ0]. A simple calculation shows that

kS(U(λ))k_W^1,r ≤CkU(λ)k²_X, (18) C being independent ofhand λ∈[0, λ0]. On the other hand, we also have

2ηp

S(U)−S(rhU)

=∇uσ+σ∇u^T −(∇r_hu)r_hσ−r_hσ(∇r_hu)^T

=∇(u−rhu)σ+ (∇rhu)(σ−rhσ) +σ∇(u−rhu)^T + (σ−rhσ)(∇rhu)^T, so that

kS(U(λ))−S(rhU(λ))k_L² ≤CkU(λ)−rhU(λ)k_H¹_×L²kU(λ)kX

≤ChkU(λ)k²_X, (19)

C being independent ofhand λ. Finally (19) and (18) in (17) yields (15) forλ∈[0, λ0].

Let us now prove (16). For allV= (v,τ)∈Xh, for allW= (w,γ)∈Xh, we have DUFh(λ, rhU(λ))−DUFh(λ,V)

W=−Th

0, λh

DS(rhU(λ))W−DS(V)Wi .

Using Lemma 4.1 we obtain

DUFh(λ, rhU(λ))−DUFh(λ,V)

W

H¹×L²≤Cλ

DS(rhU(λ))−DS(V) W

L². We have

DS(r_hU)−DS(V) W

L²

= 1

2ηpk∇(r_hu−v)γ+γ∇(r_hu−v)^T+∇w(r_hσ−τ) + (r_hσ−τ)∇w^Tk_L²

≤ 1 ηp

k∇(rhu−v)kL^rkγkL^q+k∇wkL^rkrhσ−τkL^q

,

withq=_r−2^2r . Forp >2, the following inverse inequalities hold kvkL^p≤ C

h^p−2p kvk_L², k∇vkL^p≤ C

h^p−2p k∇vk_L², for all continuous, piecewise linear function v, see [10]. This yields

DS(r_hU)−DS(V) W

L² ≤C1

hkr_hU−Vk_H¹_×L²kWk_H¹_×L², C being independent ofλandh. This last inequality yields (16).

Before proving existence of a solution to (14) we still need to check thatD_UF_his invertible is the neighbour- hood of U(λ).

Lemma 4.3. Let U(λ), 0 ≤ λ≤ λ0, be given by Theorem 2.2. There exists 0 < λ1 ≤λ0 such that, for all λ≤λ1 and for allhwe have

kD_UF_h(λ, r_hU(λ))⁻¹k_L(Xh)≤2.

Proof. By definition ofFh, we have

D_UF_h(λ, r_hU(λ)) =I−T_h

0, λDS(r_hU(λ)) , whereDS(r_hU(λ)) is such that

DS(rhU)V= 1 2ηp

∇(rhu)τ +τ∇(rhu)^T+∇v(rhσ) + (rhσ)∇v^T ,

for allV= (v,τ)∈X_h. Thus we can write

D_UF_h(λ, r_hU(λ)) =I−G_h with G_h=T_h(0, λDS(r_hU(λ))).

If we prove that kGhk_L(Xh) ≤1/2 for sufficiently small values of λ, then DUFh(λ, rhU(λ)) is invertible and kDUFh(λ, rhU(λ))⁻¹k_L(Xh) ≤2. Using Lemma 4.1, it suffices to prove that DS(rhU(λ)) : Xh → L²(Ω)⁴ is uniformly bounded with respect toh. Proceeding as in the proof of the previous Lemma we have

kDS(r_hU)Vk_L² = 1 2ηp

∇(r_hu)τ+τ∇(r_hu)^T +∇v(r_hσ) + (r_hσ)∇v^T

L²

≤C

k∇ukL^∞kτk_L²+k∇vk_L²kσkL^∞

≤C˜

kuk_W^2,rkτk_L²+k∇vk_L²kσk_W^1,r ,

C, ˜C being independent ofλandh. In other words we have

kDS(rhU(λ))Vk_L² ≤CkU(λ)kXkVk_H¹_×L², and, going back toG_h we obtain

kGh(V)k_H¹_×L² ≤Cλk˜ U(λ)kXkVk_H¹_×L². Finally, forλsufficiently small we obtainkG_hk_L(Xh)≤1/2 and the result follows.

We are now in position to state the main result of the section

Theorem 4.4. Let U(λ), 0≤λ≤λ0, be given by Theorem 2.2. There exists 0 < λ≤λ0, hand δ >0 such that, for all0≤λ≤λ, for all0< h≤h, there exists a uniqueU_h(λ)in the ball of X_h centered atr_hU(λ)with radiusδhin the normH¹×L², satisfying

Fh(λ,U_h(λ)) = 0.

Moreover, the mapping λ∈[0, λ]→U_h(λ)∈Xh is continuous and there existsC >0 such that the following a priori error estimate holds

kU(λ)−U_h(λ)k_H¹_×L² ≤Ch ∀λ≤λ ∀h≤h.

Remark 4.5. The statement of the above existence result is similar (although not the same) to those of [4, 21], in which the convective term in the extra-stress constitutive equation are considered (see also [26] for analogous results on a second-grade fluid). However, a different technique is used in this paper, allowing a priori and a posteriorierror estimates to be obtained more easily, with other assumptions.

In order to prove this theorem, we use the following abstract result.

Theorem 4.6 (Th. 2.1 of [9]). Let Y andZ be two real Banach spaces with normsk·kY andk·kZ respectively.

Let G:Y →Z be aC¹ mapping and v∈Y be such that DG(v)∈ L(Y;Z)is an isomorphism. We introduce the notations

=kG(v)kZ,

γ=kDG(v)⁻¹k_L(Z;Y), L(α) = sup

x∈B(v,α)

kDG(v)−DG(x)k_L(Y;Z), withB(v, α) ={y∈Y;kv−ykY ≤α}, and we are interested in findingu∈Y such that

G(u) = 0. (20)

We assume that 2γL(2γ)≤1. Then Problem (20) has a unique solution uin the ball B(v,2γ) and, for all x∈B(v,2γ), we have

kx−ukY ≤2γkG(x)kZ. (21) Proof of Theorem 4.4. We apply Theorem 4.6 with Y = Xh, Z = Xh, G = Fh, v = rhU(λ) and the norm k · k_H¹_×L² inXh. The mappingG:Y →Z isC¹and, according to Lemma 4.2, forλsufficiently small there is a constantC1independent ofλandhsuch that≤C1h. According to Lemma 4.3, forλsufficiently smallγ≤2.

According to Lemma 4.2, there is a constantC2 independent ofλandhsuch thatL(α)≤C2αλ/h. Thus, we have

2γL(2γ)≤2.2C2(2.2.C1h)λ

h= 16C1C2λ.

Thus, for λsufficiently small 2γL(2γ)≤1 and Theorem 4.6 applies. There exists a unique U_h(λ) in the ball B(v,2γ) such thatFh(λ,U_h(λ)) = 0 and we have

kr_hU(λ)−U_h(λ)k_H¹_×L² ≤4C1h.

It suffices to use the triangle inequality

kU(λ)−U_h(λ)k_H¹_×L² ≤ kU(λ)−rhU(λ)k_H¹_×L²+krhU(λ)−U_h(λ)k_H¹_×L²,

and standard interpolation [10] results to obtain thea prioriestimates. The fact that the mappingλ→U_h(λ) is continuous is a direct consequence of the implicit function theorem.

5.

error estimates

Let us consider again the operator

T_h:L²(Ω)²×L²_s(Ω)⁴−→H₀¹(Ω)²×L²_s(Ω)⁴ (f,g) −→Th(f,g) =

def.(u_h,σh)∈Vh×Mh,

where (u_h, ph,σh)∈Whsatisfies (11). We now introduce a residual based error estimator forTh. The notations are those of [2]. For any triangleKof the triangulationTh, letEK be the set of its three edges. For each interior edge ` of Th, let us choose an arbitrary normal direction n, let [·]` denote the jump across edge `. For each edge`ofTh lying on the boundary∂Ω, we set [·]`= 0. The local error estimator corresponding to (11) is then defined by

µ²_K(f,g) = 1 η_s+η_p

h²_Kk−2η_sdiv (u_h) +∇p_h−divσ_h−fk²_L2(K)

+1 2

X

`∈EK

|`| k[2η<sub>s</sub>(u<sub>h</sub>)n]<sub>`</sub>k²_L2(`)

+ (η_s+η_p)kdivu_hk²_L2(K)

+η_p 1

2η_pσ_h−g−(u_h) ²

L²(K)

.

We have the followinga posteriorierror estimate for operatorT−T_h.

Lemma 5.1. There existsC andh0>0such that, for all (f,g)∈L^r(Ω)²×W_s^1,r(Ω)⁴ we have kT(f,g)−Th(f,g)kH¹×L²≤C X

K∈Th

µ²_K(f,g)

!_1/2

∀h≤h0.

Proof. We set T(f,g) = (u,σ) where (u, p,σ) is the solution of (2), we also set Th(f,g) = (u_h,σh) where (u_h, p_h,σ_h) is the solution of (11). We recall thatW =H₀¹(Ω)²×L²₀(Ω)×L²_s(Ω)⁴. LetB:W×W →Rbe the bilinear form corresponding to the weak formulation of (2), namely

B(u, p,σ;v, q,τ) = 2η_s((u),(v))−(p,div v) + (σ,(v))

−(divu, q)− 1

2η_p(σ,τ) + ((u),τ),

for all (u, p,σ) and (v, q,τ) inW. In Lemma 2 of [6], it is proved thatBsatisfies an inf-sup condition, therefore we have

ku−u_h, p−p_h,σ−σ_hk_H¹_×L²_×L² ≤C sup

06=(v,q,τ)∈W

B(u−u_h, p−p_h,σ−σ_h;v, q,τ) kv, q,τk_H¹_×L²_×L² , C being independent ofh. For all (v, q,τ)∈W we have

B(u−u_h, p−ph,σ−σh;v, q,τ) = (f,v)−(g,τ)−B(u_h, ph,σh;v, q,τ). IntroducingBh:Wh×Wh→Ras in the proof of Lemma 4.1, we have

B(u−u_h, p−ph,σ−σh;v, q,τ) = (f,v−v_h)−(g,τ−τh)−B(u_h, ph,σh;v−v_h, q−qh,τ−τh) + (Bh−B)(u_h, ph,σh;v_h, qh,τh),

for all (v_h, qh,τh)∈Wh, that is:

B(u−u_h, p−p_h,σ−σ_h;v, q,τ)

=−2η_s((u_h),(v−v_h)) + (p_h,div (v−v_h))−(σ_h,(v−v_h)) + (f,v−v_h) + (divu_h, q−qh)− X

K∈Th

αh²_K 2ηp

−2ηsdiv(uh) +∇ph−div σh−f,∇qh

K

+ 1

2ηpσ_h−g−(u_h),τ −τ_h−2η_pβ(v_h)

.

We then proceed as in [3, 28], integrate by parts on each triangleK∈ Th the first three terms in the right hand side of the above equation, and choosev_h=R_hv (whereRh is Cl´ement’s interpolant [11]),qh= 0, τh = 0 to conclude.

We are now in position to statea posteriorierror estimates for the solution to (9). Let us briefly recall the notations. Problem (1) is written asF(λ,U) = 0 with U= (u,σ)∈X and F(λ,U) =U−T(f, λS(U)), the operatorsT andS being defined in (5) and (7). Problem (9) is written asF_h(λ,U_h) = 0 withU_h= (u_h,σ_h)∈ X_h6⊂X andF_h(λ,U_h) =U_h−T_h(f, λS(U_h)), the operatorT_h being defined in (10). According to Theorem 2.2, forλ sufficiently small, there is a uniqueU(λ) such thatF(λ,U(λ)) = 0. According to Theorem 4.4, for hand λsufficiently small, there exists a unique U_h(λ) in a neighbourhood ofrhU(λ) depending onh (in the normk · k_H¹_×L²) such thatFh(λ,U_h(λ)) = 0. Moreover, whenhgoes to zero,U_h(λ) converges toU(λ) in the normk · k_H¹_×L².

Theorem 5.2. There existsλ0,h0 andC >0 such that, for allλ≤λ0, for allh≤h0,

kU(λ)−U_h(λ)k_H¹_×L²≤C X

K∈Th

µ²_K

f, λS(U_h(λ))!^1/2

. (22)

Proof. Using the definition ofF andF_h we have U−U_h=T

f, λS(U)

−Th

f, λS(Uh)

=T

0, λ

S(U)−S(U_h)

+ (T−T_h)

f, λS(U_h)

. (23)