GLS and EVSS methods for a three-field Stokes problem ... - Aalto Math

Comput. Methods Appl. Mech. Engrg. 190 (2001) 3893±3914

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GLS and EVSS methods for a three-®eld Stokes problem arising from viscoelastic ¯ows J. Bonvin a,1, M. Picasso a,*, R. Stenberg b b

a D epartement de Math ematiques, Ecole Polytechnique F ed erale de Lausanne, 1015 Lausanne, Switzerland Department of Mathematics, Tampere University of Technology, Korkeakoulunkatu 1, P.O. Box 692, 33101 Tampere, Finland

Received 2 February 2000

Abstract In this paper, order one ®nite elements together with Galerkin least-squares (GLS) methods are used for solving a three-®eld Stokes problem arising from the numerical study of viscoelastic ¯ows. Stability and convergence results are established, even when the solvent viscosity is small compared to the viscosity due to the polymer chains. An iterative algorithm decoupling velocity±pressure and stress calculations is proposed. The link with the modi®ed elastic viscous split stress (EVSS) method studied in (M. Fortin, R. Guenette, R. Pierre, Comput. Methods Appl. Mech. Engrg. 143 (1997) 79±95; R. Guenette, M. Fortin, J. Non-Newtonian Fluid Mech. 60 (1995) 27±52) is presented. Numerical results are in agreement with theoretical predictions, and with those presented in (M. Fortin, R. Guenette, R. Pierre, Comput. Methods Appl. Mech. Engrg. 143 (1997) 79±95). Ó 2001 Elsevier Science B.V. All rights reserved.

1. Introduction Numerical simulation of viscoelastic ¯ows is of interest for many industrial applications such as extrusion or injection processes. Macroscopic modelling of viscoelastic ¯ows usually consists in supplementing the mass and momentum equations with a di€erential constitutive equation involving the stress and the velocity ®eld. When considering this set of partial di€erential equations in the frame of a ®nite element method, three problems may arise. The ®rst one is related to the ``Babuska±Brezzi'' or ``inf±sup'' condition which restricts the choice of the ®nite element spaces used to discretize the velocity, pressure and stress ®elds. The second problem is due to the presence of convective terms in the di€erential constitutive equation. It can be solved by using appropriate upwinding procedures, such as streamline upwind Petrov Galerkin (SUPG) [3], Galerkin least-squares (GLS) [4,5], discontinuous ®nite elements [6±8] or characteristicGalerkin methods [9]. The third problem arises form the strong non-linear nature of the model. Existence and uniqueness results for viscoelastic models are established only at low Deborah numbers (see for instance [10] for a review), this being consistent with the fact that numerical procedures fail to converge at high Deborah numbers. The goal of this paper is to focus on the ®rst of these three problems, namely the inf±sup condition and, for this purpose, a three-®eld Stokes problem is introduced. Various approaches have been considered to

*

Corresponding author. Tel.: +41-216-9342-97; fax: +41-216-9343-03. E-mail address: marco.picasso@ep¯.ch (M. Picasso). 1 Supported by the Swiss National Fund for Scienti®c Research. 0045-7825/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 0 0 ) 0 0 3 0 7 - 8

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obtain stable and convergent methods. The ®rst of these approaches considers the usual Galerkin method and makes use of ®nite element spaces (for the velocity, pressure and stress) satisfying the inf±sup condition. For instance, continuous approximations were proposed in [3] and justi®ed in [11], discontinuous approximations in [6,8]. An alternative is to use equal-order approximations for the velocity, pressure and stress, while adding stabilization terms to the usual weak formulation in a GLS fashion, see for instance [5] for computations and [12] for mathematical studies. The third possible manner to cope with the inf±sup condition (half way between the two methods hereabove) consists in splitting the stress into two parts, an ``elastic'' part and a ``viscous'' part. The velocity and pressure spaces still have to satisfy the inf±sup condition but no additional compatibility condition apply on the two split stresses. This so-called elastic viscous split stress (EVSS) method was introduced in [2,13,14] and a numerical analysis was proposed in [1] for a three-®eld Stokes problem arising from viscoelastic models. Recently, many extensions have been proposed and we refer to [15] for a review. For instance, an adaptive splitting of the stress is proposed in [16], together with SUPG techniques to account for convection, whereas discontinuous stresses were considered in [7]. In this paper, GLS methods are proposed and the connection with EVSS methods is proposed for a three-®eld Stokes problem arising from viscoelastic models. Continuous, piecewise linear velocity, pressure and stress are considered on triangles. Stabilization terms are added and, as for the modi®ed EVSS method of [1], it is proved that the method remains stable even when the solvent viscosity is small. Two GLS schemes are presented. The ®rst GLS method corresponds to the case when all the possible stabilization terms are added in a least-squares fashion, following the ideas of [17]. Stability and convergence results are established using techniques borrowed from [12,17,18]. The stability parameters are chosen in order to ensure convergence, even when the solvent viscosity is much smaller than the polymer viscosity. In the second GLS method, only a selected number of these terms are added. Stability and convergence results are also given. The reasons for introducing this second, ``reduced'' GLS method are the following. Firstly, the number of stabilizing terms is reduced, which simpli®es the implementation. It must be kept in mind that our goal is to apply this GLS method to ``real-life'' constitutive relationships, the simplest being the Oldroyd-B equation. Secondly, this reduced GLS method is closely linked to the modi®ed EVSS method of [1,2] which is now very popular in the ®eld of non-newtonian ¯uid mechanics. Finally, as for the modi®ed EVSS method, the velocity±pressure and stress computations can be decoupled, which reduces the memory requirements. The outline of the paper is the following. The three-®eld Stokes model is presented in Section 2. The ®rst GLS method is presented and studied in Section 3, the second GLS method in Section 4. The link with the modi®ed EVSS method of [1,2] is proposed in Section 5. An iterative procedure to decouple the velocity± pressure and stress computations is presented in Section 6. In Section 7, numerical results are reported, matching those of [1], and con®rming our theoretical predictions.

2. A three-®eld Stokes problem Let X be a bounded polygonal domain of R2 with boundary oX. We consider the following formulation of the Stokes problem. Given force terms f 1 , f2 , f 3 , constant solvent and polymer viscosities gs P 0 and gp > 0, ®nd the velocity u, pressure p and extra-stress r such that 2gs div …u† ‡ rp

div r ˆ f 1 ;

div u ˆ f2 ;

1 r 2gp

…u† ˆ f 3

…1†

in X, where the velocity u vanishes on oX and …u† ˆ …1=2†…ru ‡ ruT † is the rate of deformation tensor. The aim of the paper is to propose a numerical procedure for solving (1) with continuous, piecewise linear ®nite elements, which is stable and convergent for any values of gs P 0 and gp > 0. The reason for choosing the above formulation of Stokes problem is due to the fact that, in number of viscoelastic codes, the velocity±pressure computations are decoupled from the extra-stress computations at each iteration, thus reducing memory requirements and possibly CPU time. This decoupling is based on the usual splitting of

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the total stress for a polymeric liquid into three contributions: the pressure pI, the stress due to the (newtonian) solvent 2gs …u†, and the extra-stress due to the polymer chains r, which we shall simply refer to as the ``stress'' in this paper. For instance, when considering the steady Oldroyd-B model, the convective term q…u
r†u should be added in the ®rst equation of (1), and the last equation of (1) should be replaced by the extra-stress constitutive equation
r ‡ k …u
r†r rur rruT ˆ 2gp …u†; …2† where the relaxation time k characterizes the elastic properties of the ¯uid. Note that when gs ˆ 0, Maxwell's model is obtained, and we want to design methods which are still stable and convergent. The variational formulation for problem (1) consists in seeking …u; p; r† 2 W such that B…u; p; r; v; q; s† ˆ F …v; q; s†

…3†

for all …v; q; s† 2 W . Here B is the symmetric, bilinear form de®ned by B…u; p; r; v; q; s† ˆ 2gs ……u†; …v††

…p; div v† ‡ …r; …v††

…div u; q†

1 …r; s† ‡ ……u†; s† 2gp

…4†

and F is the form de®ned by F …v; q; s† ˆ …f 1 ; v†

…f2 ; q†

…f 3 ; s†;

where …
;
† stands for the usual L2 …X† inner product for scalars, vectors or tensors. The appropriate functional space W is de®ned by W ˆ H01 …X†2  L20 …X†  L2 …X†4s : 2

Here, H01 …X† stands for the space of square integrable velocities de®ned on X, vanishing on the boundary and with ®rst derivatives also square integrable, L20 …X† stands for the space of square integrable functions having zero average, and L2 …X†4s denotes the space of 2  2 symmetric tensors whose components are square integrable. Well posedness of the continuous problem. Let k
kW be the norm on W de®ned by ku; p; rk2W ˆ 2gk…u†k2 ‡

1 1 kpk2 ‡ krk2 ; 2g 2gp

…5† 2

4

where g ˆ gs ‡ gp is the total viscosity of the ¯uid. Assume that …f 1 ; f2 ; f 3 † 2 L2 …X†  L2 …X†  L2 …X†s and let k
k be the norm on this space de®ned by 0 12 1 …f ; v† 1 A ‡ 2gkf2 k2 ‡ 2gp kf 3 k2 : …6† kf 1 ; f2 ; f 3 k2 ˆ @ sup 2g 06ˆv2H 1 …X†2 k…v†k 0

We have the following existence and unicity result for the exact solution …u; p; r† of (3). Theorem 1 (Existence and uniqueness of the exact solution). Let gs P 0 and gp > 0. Problem (3) admits a unique solution …u; p; r† 2 W . Moreover, the solution depends continuously on the data and there exists a positive constant C, independent of gs and gp ; such that ku; p; rkW 6 Ckf 1 ; f2 ; f 3 k :

…7†

This theorem results from the Babuska±Ladyzhenskaya theorem (see for instance [19]) and is true since the inf±sup condition holds for B, this being stated in the following lemma. Lemma 2 (Stability of the continuous problem). Let gs P 0 and gp > 0. There exist two positive constants C1 and C2 ; independent of gs and gp ; such that

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…div v; p† P C1 kpk k…v†k 06ˆv2H 1 …X†2 sup

8p 2 L20 …X†;

…8†

0

sup

06ˆs2L2 …X†4s

…s; …u†† P C2 k…u†k ksk

2

8u 2 H01 …X† :

…9†

Therefore, there exists a positive constant C3 ; independent of gs and gp ; such that sup

06ˆ…v;q;s†2W

B…u; p; r; v; q; s† P C3 ku; p; rk 8…u; p; r† 2 W : kv; q; skW

…10†

Before proving this lemma, let us note that condition (8) is nothing but the standard inf±sup condition for the usual two-®eld Stokes problem (see for instance [19,20]), whereas (9) is trivial (given u, choose s ˆ …u††. From the numerical point of view, conditions (8) and (9) must also be satis®ed in the ®nite element subspaces. Choosing equal order approximations for u, p, and r (for instance, continuous, piecewise linears), does not lead to a stable scheme, as in the case of the classical (two ®elds) Stokes problem. A possible remedy to this situation is to enrich the ®nite element subspaces for the velocity and stress in order to satisfy both stability conditions (8) and (9), see for instance [3,8,9,11]. The GLS method is designed to avoid both compatibility conditions (8) and (9) by adding extra terms to the bilinear form B and to the linear form F. More precisely, the extra terms are least-square-like expressions of the equation residual, weighted on each triangle of the ®nite element mesh. The two methods that are proposed in this paper consist in adding such stabilization terms to (3), while maintaining order one approximations for the velocity, the pressure and the stress. The EVSS method of [1,2] is such that (8) is satis®ed while (9) is avoided by a technique very close to GLS. In other words, the EVSS formulation of [1,2] o€ers an alternative which consists in using compatible velocity±pressure ®nite elements and low order stresses, but with the extra cost of having to add an extra variable to stabilize the discretization of (1), as will be pointed out in Section 5. Proof of Lemma 2. First, note that for any …u; p; r† 2 W we have B…u; p; r; u; p; r† ˆ 2gs k…u†k2 ‡

1 krk2 : 2gp

…11†

2 Condition (8) implies that for any p 2 L20 …X† there exists ~v 2 H01 …X† , such that

…div~v; p† P C1 kpkk…~v†k: Choosing ~v such that k…~v†k ˆ 1=2gkpk yields …div~v; p† P

C1 kpk2 : 2g

Similarly, condition (9) implies that for any u 2 H01 …X†2 there exists ~s 2 L2 …X†4s such that k~sk ˆ 2gp k…u†k and   ~s; …u† P 2gp C2 k…u†k2 : We want to prove that there exists a constant C3 such that, for all …u; p; r† 2 W , we have B…u; p; r; u

d~v; p; r ‡ d~s† P C3 ku; p; rkW ku

d~v; p; r ‡ d~skW

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for some value d. Using the Cauchy±Schwarz inequality we have     r; …~v† B…u; p; r; ~v; 0; ~s† ˆ 2gs …u; …~v† ‡ …p; div~v† P

2gs k…u†kk…~v†k ‡

C1 2 kpk 2g

krkk…~v†k

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  1 …r; ~s† ‡ …u; ~s 2gp 1 2 krkk~sk ‡ C2 2gp k…u†k : 2gp

Consider Young's inequality, which writes 2ab 6 …1=c†a2 ‡ cb2 , for all real numbers a; b and for any c > 0. Using this inequality and then the de®nitions of ~v and ~s, we get 2gs c C1 1 k…u†k2 2gs 1 k…~v†k2 ‡ kpk2 krk2 2c2 2c1 2 2g c2 1 1 1 c3 2 2 2 2 k…~v†k k~sk ‡ C2 2gp k…u†k krk 2gp 2c3 2gp 2 2      c 3  gs 1 c 1 gs c 2 2 C1 ˆ 2gp C2 k…u†k ‡ kpk2 2g 2 c1 2g 4g

B…u; p; r; ~v; 0; ~s† P

1 2gp



 gp 1 ‡ krk2 : 2c3 c2

From (11) and the previous inequality we get the following lower bound:      d c3  2 B…u; p; r; u d~v; p; r; ‡d~s† P 2gs 1 ‡ 2gp d C2 k…u†k 2c1 2      gp d c1 g s c2 1 1 2 2 C1 ‡ 1 d ‡ krk : kpk ‡ 2g 2gp 2c3 c2 2g 4g Let us choose c1 > 0; c2 > 0; c3 > 0 and d > 0 such that 1

d > 0; 2c1

C2

c3 > 0; 2

C1

c 1 gs 2g

c2 > 0; 4g



1

gp 1 d ‡ 2c3 c2

 > 0:

If we choose for instance c1 ; c2 and c3 such that c1 ˆ

C1 ; 2

c2 ˆ C1 g

and

c3 ˆ

C2 2

and using the fact that gs =g 6 1 and gp =g 6 1, we obtain     d 3dC2 1 dC1 2 2 kpk B…u; p; r; u d~v; p; r ‡ d~s† P 2gs 1 ‡ 2gp k…u†k ‡ C1 2g 2 4    1 1 1 2 ‡ 1 d ‡ krk : 2gp C2 C1 In the inequality hereabove, the coecients are all positive if d is chosen suciently small, i.e., such that   1! 1 1 : 0 < d < min C1 ; ‡ C2 C1 Thus, for any …u; p; r† 2 W we have found …v; q; s† ˆ …u

d~v; p; r ‡ d~s† 2 W such that

B…u; p; r; v; q; s† P C4 ku; p; rk2W ; where

 C4 ˆ min 1

d 3dC2 dC1 ; ;1 ; C1 4 2



1 1 d ‡ C2 C1



is clearly independent of gs and gp . The previous inequality together with the fact that ku

d~v; p; r ‡ d~skW 6 …1 ‡ d†ku; p; rkW

yield inequality (10) with C3 ˆ C4 =…1 ‡ d†.

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3. A ®rst GLS method The GLS method was originally developed for the Stokes problem in [18]. It was then extended to di€usion±convection problems in [21]. In [17], the GLS method was presented in a general framework and applied to Navier±Stokes equations. In [4,12,22], the GLS method was applied to the three-®eld Stokes problem (1) with gs ˆ 0. However, it seems that nobody has studied in detail the GLS method for solving formulation (1) of the Stokes problem, for any values gs P 0 and gp > 0. In this paper, only continuous, piecewise linears will be considered. For any h > 0, let Th be a mesh of X into triangles K with diameters hK less than h. As usual, the mesh is assumed to be regular in the sense of [23], which means that the smallest angle is bounded below by some positive constant independent of the mesh size h. We consider Wh the ®nite dimensional subspace of W consisting in the continuous, piecewise linear velocities, pressures, and stresses on the mesh Th . More precisely Wh  W is de®ned by Wh ˆ Vh  Qh  Th ; where

n o 2 2 2 Vh ˆ v 2 C0 …X† j vjK 2 …P1 † ; 8K 2 Th \ H01 …X† ; n o Qh ˆ q 2 C0 …X† j qjK 2 P1 ; 8K 2 Th \ L20 …X†; n o 4 4 4 Th ˆ s 2 C0 …X† j sjK 2 …P1 † ; 8K 2 Th \ L2 …X†s :

…12†

We now use the general framework of [17] in order to present a GLS formulation corresponding to (3). First, we rewrite (1) as L1 …u; p; r† ˆ f 1 ;

L2 …u; p; r† ˆ f2 ;

L3 …u; p; r† ˆ f 3

…13†

with L1 …u; p; r† ˆ

2gs div …u† ‡ rp

div r;

L2 …u; p; r† ˆ div u;

L3 …u; p; r† ˆ

1 r 2gp

…u†:

Following [17], the GLS method corresponding to the weak formulation (3) may be stated as follows: Find …uh ; ph ; rh † 2 Wh such that Bh …uh ; ph ; rh ; v; q; s† ˆ Fh …v; q; s†

…14†

for all …v; q; s† 2 Wh . Here the symmetric, bilinear form Bh and the form Fh are de®ned by Bh …uh ; ph ; rh ; v; q; s† ˆ B…uh ; ph ; rh ; v; q; s† ‡

3 X X iˆ1 K2Th

Fh …v; q; s† ˆ F …v; q; s† ‡

3 X X iˆ1 K2Th

ji …Li …uh ; ph ; rh †; Li …v; q; s††K ;

ji …f i ; Li …v; q; s††K ;

where …
;
†K denotes the L2 …K† inner product. The stabilization coecients ji ; i ˆ 1; 2; 3 are designed to obtain optimal convergence. Following [4,12], we choose: j1 ˆ

ah2K ; 2gp

j2 ˆ 0;

j3 ˆ 2gp b;

where a > 0 and 0 < b < 1 are dimensionless parameters. Using (13), the bilinear form Bh is then de®ned by X ah2 K Bh …uh ; ph ; rh ; v; q; s† ˆ B…uh ; ph ; rh ; v; q; s† … 2gs div …uh † ‡ rph div rh 2g p K2Th ! 1 1 2gs div …v† ‡ rq div s†K ‡ 2gp b rh …uh †; s …v† …15† 2gp 2gp

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and the form Fh by X ah2 K …f 1 ; 2gs div …v† ‡ rq 2g p K2Th

Fh …v; q; s† ˆ F …v; q; s†

1 div s†K ‡ 2gp b f 3 ; s 2gp

! …v† :

Note that, since we have chosen j2 ˆ 0, the GLS term corresponding to the second equation of (1), namely …div u; div v†, is not included. This is due to the fact that this term essentially produces terms which are already present in ……u†; …v††. The GLS scheme (14) with Bh given by (15) is said to be consistent in the following sense. If the solution 2 4 …u; p; r† 2 W of the weak formulation (3) is regular enough, that is, if …u; p; r† 2 H 2 …X†  H 1 …X†  H 1 …X† , then …u; p; r† satis®es (1) in the L2 sense in each triangle, and we have Bh …u; p; r; v; q; s† ˆ Fh …v; q; s† 8…v; q; s† 2 Wh : Consequently, if …uh ; ph ; rh † 2 Wh is the solution of (14), we obtain Bh …u

uh ; p

ph ; r

rh ; v; q; s† ˆ 0

8…v; q; s† 2 Wh :

…16†

We are now in a position to present some theoretical results concerning stability and convergence of the method. In Lemma 3, stability is ®rst established in the discrete norm k
kh de®ned, for all …v; q; s† 2 Wh by 2

2

kv; q; skh ˆ 2gk…v†k ‡

1 X 2 1 2 2 hK krqkK ‡ ksk : 2g K2Th 2gp

…17†

Then stability and convergence are established in the (natural) k
kW norm, following [12,24]. Our goal is to prove convergence of the method, keeping as much information as possible on the in¯uence of physical parameters gs ; gp . Again, we want to prove that the method is reliable for any values gs P 0 and gp > 0, even when gs is much smaller than gp . The following inverse inequality for the stress will be needed. Since the mesh is regular in the sense of [23], there exists a positive constant CI , independent of the mesh size, such that CI

X K2Th

2

h2K kdiv skK 6 ksk

2

8s 2 Th :

…18†

In the sequel, it will always be assumed that CI represents the largest positive constant satisfying inequality (18). We then have the following stability result. Lemma 3 (Stability in norm k
kh ). Let gs P 0 and gp > 0. Let Bh be defined by (15) with a > 0 and 0 < b < 1. There exists a constant C > 0, independent of h, gs and gp , but depending on a; b and on the shape of the mesh triangles, such that for all …v; q; s† 2 Wh 2

Bh …v; q; s; v; q; s; † P Ckv; q; skh :

…19†

Therefore (14) has a unique solution. Proof of Lemma 3. Let …v; q; s† be an element of Wh . Since v is piecewise linear, we have div …v† ˆ 0 in each triangle K of the mesh. Using (15) we then obtain Bh …v; q; s; v; q; s† ˆ …2gs ‡ 2gp b†k…v†k2 ‡

X ah2 1 b K ksk2 ‡ krq 2gp 2gp K2Th

div sk2K :

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By Young's inequality, we obtain 1 b 2 2 ksk Bh …v; q; s; v; q; s† P …2gs ‡ 2gp b†k…v†k ‡ 2gp   a X 2 2 ‡ h …1 c†krqkK ‡ 1 2gp K2Th K

  1 2 kdiv skK c

for any c > 0. To retrieve control on the pressure we have to choose c < 1. Using (18) we have   X   1 1 2 2 hK kdiv skk P 1 CI 1 ksk2 : c c K2Th We then obtain the following inequality: 2

Bh …v; q; s; v; q; s† P …2gs ‡ 2gp b†k…v†k ‡

X ah2 K …1 2g p K2Th

2

c†krq kK ‡

 1 1 2gp

 b ‡ a~ 1

1 c



2

ksk ; …20†

where we have set a~ ˆ a=CI . It then remain to choose c such that   1 1 1 b 0 < c < 1 and 1 b ‡ a~ 1 > 0; i:e:; 1 < < 1 ‡ : c c a~ If we choose for instance c such that   1 1 1 b 1 b ˆ 1‡1‡ ; ˆ1‡ ~ c 2 2~ a a a simple calculation shows that 1

 b ‡ a~ 1

1 c

 ˆ

1

b 2

and

1

cˆ

1‡

!

2~ a 1

1

:

b

…21†

Introducing (21) into (20) yields  2 Bh …v; q; s; v; q; s† P …2gs ‡ 2gp b†k…v†k ‡ 1 ‡

2a CI …1 b†



1

X ah2 1 b 2 2 K krq kK ‡ ksk : 4g 2g p p K2Th

Since 0 < b < 1 and gp 6 g we get 2

Bh …v; q; s; v; q; s† P b2gk…v†k ‡



1 2 ‡ a CI …1 b†



1

1 X 2 1 b 1 2 2 hK krq kK ‡ ksk 2g K2Th 2 2gp

…22†

and thus (19) holds with C ˆ min b;

1

b 2

 ;

1 2 ‡ a CI …1 b†

 1! :

Finally, existence and uniqueness of the solution of (14) is a direct consequence of (19). Indeed, (19) implies that the kernel of Bh is zero. Therefore the corresponding mapping is injective, and thus bijective since Wh is a ®nite dimensional subspace of W.

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Remark 4. Our numerical interpretation of Lemma 3 is the following. When a > 0 and 0 < b < 1, the velocity is controlled for any solvent viscosity gs , even when gs ˆ 0. However, a careful look at the constant C of (19) shows that when gs and b approach zero simultaneously, the method fails. Similarly, when a approaches zero, or when b approaches one, the method also fails. These predictions are compared to numerical experiments in Section 7. By Lemma 3, stability can be proved in the (natural) k
kW norm de®ned in (5), using arguments similar to those introduced in [12]. Lemma 5 (Stability in norm k
kW ). Let gs P 0 and gp > 0. Let Bh be defined by (15) with a > 0 and 0 < b < 1. There exists a positive constant C, independent of h, gs and gp , but depending on X; a; b and the shape of the triangles, such that, for all …u; p; r† 2 Wh , we have Bh …u; p; r; v; q; s† P Cku; p; rkW : kv; q; skW 06ˆ…v;q;s†2Wh sup

…23†

This stability result is a consequence of Lemma 3 together with Lemma 3.3 in [12]. The proof of Lemma 5 is not given here since it is very similar to that of Lemma 2, as well as to that of Lemma 3.2 in [12]. Using Lemma 5, convergence in norm k
kW can be proved. Theorem 6 (Convergence in norm k
kW ). Let gs P 0 and gp > 0. Let …u; p; r† 2 W be the solution of (3), let …uh ; ph ; rh † 2 Wh be the solution of (14), where Bh is defined by (15) with a > 0 and 0 < b < 1. Moreover, assume that …u; p; r† 2 H 2 …X†2  H 1 …X†  H 1 …X†4 . Then, there exists a positive constant C, independent of h, gs and gp , but depending on X; a; b and the shape of the triangles, such that ku

uh ; p

ph ; r

rh k2W

g 6C 1 ‡ s gp

!2 h2

gkuk2H 2 …X†2

! 1 1 2 2 ‡ kpkH 1 …X† ‡ krkH 1 …X†4 : g gp

…24†

In order to prove Theorem 6, we need as in [12] a technical lemma. Lemma 7. Let gs P 0 and gp > 0. Consider the bilinear form Bh defined by (15) with a > 0 and 0 < b < 1. Then, there exists a positive constant C, independent of h, gs and gp , but depending on X; a; b and the shape of the triangles, such that, for all  Y  2 4 …u; p; r† 2 W \ H 2 …K†  H 1 …K†  H 1 …K† K2Th

and for all …v; q; s† 2 Wh the following inequality holds: ! gs jBh …u; p; r; v; q; s†j 6 C 1 ‡ jGh …u; p; r†jkv; q; skW ; gp

…25†

where Gh is defined by 1 1 1 X 2 2 2 2 2 2 jGh …u; p; r†j ˆ gk…u†k ‡ kpk ‡ krk ‡ h kL1 …u; p; r†kK : g gp g K2Th K

…26†

Proof of Lemma 7. Let us de®ne w1 ˆ …u; p; r† and w0 ˆ …v; q; s†. Using (15) and since div …v† ˆ 0 on each triangle, we have

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Bh …w1 ; w0 † ˆ …2gs ‡ 2gp b†……u†; …v††

…p; div v† ‡ …1

a X 2 h …L1 w1 ; rq 2gp K2Th K

 ……u†; s†

b†…r; …v††

…div u; q†

1 b …r; s† ‡ …1 2gp

b†

div s†K :

Using the discrete Cauchy±Schwarz inequality we have 2

Bh …w1 ; w0 † 6 …2gs ‡ 2gp b†k…u†k ‡ 1 b ‡ krk2 ‡ 2gp …1 2gp 

1 1 b 2 2 2 krk ‡ 2gkdiv uk kpk ‡ 2g 2gp a X 2 b†k…u†k ‡ 2 h kL1 w1 k2K 2gp K2Th K

!1=2

2

…2gs ‡ 2gp b†k…v†k2 ‡ 2gkdiv vk2 ‡ 2gp …1

b†k…v†k2 ‡

!  1=2 1 b a X 2 2 2 2 ‡ ksk ‡ h krq kK ‡ kdiv skK : 2gp 2gp K2Th k

1 1 b kqk2 ‡ ksk2 2g 2gp

We need now the following inverse inequality for the pressure. Since the mesh is regular in the sense of [23], there exists a positive constant CIp independent of the mesh size, such that X 2 2 CIp h2K krq kK 6 kqk 8q 2 Qh : …27† K2Th

Using Korn's inequality and the inverse inequalities for the stress (18) and for the pressure (27) we obtain !1=2 1 1 b a X 2 2 2 2 jBh …w1 ; w0 †j 6 2g…1 ‡ C †k…u†k ‡ kpk ‡ krk ‡ h kL1 w1 kK 2g gp gp K2Th K !1=2 1 ‡ ag=CIp gp 1 b ‡ a=CI 2 2 2 K kqk ‡  2g…1 ‡ C †k…v†k ‡ ksk ; 2g 2gp K

2

where C K is the constant of the Korn inequality. The above estimate and the de®nition of norm k
kW yield (25) with C proportional to !   a a gs K 1‡ max 1; C ; a; p ; : CI CI gp Proof of Theorem 6. Let us set w ˆ …u; p; r†; wh ˆ …uh ; ph ; rh † and choose qh w the interpolant de®ned by qh w ˆ …rh u; Rh p; Rh r†; where rh denotes the standard Lagrange interpolant on vectors, Rh and Rh are the interpolant of Clement [25,26] on scalars and tensors, respectively. Using Young's inequality one can write kw

wh k2W 6 2kw

qh wk2W ‡ 2kqh w

wh k2W :

…28†

By Lemma 5, there exists a constant C1 (independent of h; gs and gp , but depending on X; a; b and on the shape of the triangles) and w0 2 Wh satisfying Bh …qh w

wh ; w0 † P C1 kqh w

wh kW kw0 kW :

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Using the consistency, Eq. (16), we have Bh …qh w

w; w0 † P C1 kqh w

wh kW kw0 kW :

On the other hand, by Lemma 7 there exists a constant C2 (independent of h; gs and gp , but depending on X; a; b and the shape of the triangles) such that ! gs Bh …qh w w; w0 † 6 C2 1 ‡ jGh …w qh w†jkw0 kW : gp By putting together the last two estimates we obtain ! C2 gs jGh …w qh w†j: kqh w wh kW 6 1‡ C1 gp

…29†

Estimate (29) into (28) yields kw

wh k2W

6 2kw

qh wk2W



C2 ‡2 C1

2

g 1‡ s gp

!2 jGh …w

qh w†j2 :

From the de®nitions of w; qh w; k
kW and Gh …
†, we have !2   C22 gs 1 2 2 kw wh kW 6 2 2 ‡ 2 gk…u rh u†k ‡ kp 1‡ g C1 gp ! 1 X 2 2 ‡ h kL1 …w qh w†kK : g K2Th K

…30†

2

Rh pk ‡

1 kr gp

Rh rk

2

Using standard interpolation estimates (see for instance [23] for rh , and [25,26] for Rh and Rh †, and considering the forms of the constants C1 and C2 , we obtain (24) with C independent of gs and gp . Finally, convergence can be proved in the L2 norm for the velocity by using the Aubin±Nitsche trick. Let …u; p; r† 2 W be the solution of (3) and assume that it is regular enough, in the sense that there exists a positive constant C, independent of h; gs and gp , but depending on X, such that 1 1 1 gkuk2H 2 …X†2 ‡ kpk2H 1 …X† ‡ krk2H 1 …X†4 6 C kf 1 ; f2 ; f 3 k2 : g gp g

…31†

Let …uh ; ph ; rh † 2 Wh be the solution of (14), where Bh is de®ned by (15) with a > 0 and 0 < b < 1. Then, ~ independent of h; gs and gp , but depending on X; a; b and on the shape of there exists a positive constant C, the triangles, such that !4 ! g 1 1 ku uh k 6 C~ 1 ‡ s h2 kukH 2 …X†2 ‡ kpkH 1 …X† ‡ krkH 1 …X†4 : …32† g gp gp

4. A reduced GLS method The GLS method introduced in the previous section is stable and convergent, but requires several stabilization terms to be added. If an Oldroyd-B model was used instead of the Stokes model, the number of stabilization terms would be much greater since convective and tensor derivatives would have to be added. Our aim is thus to ®nd which of the GLS terms are important for the stability, without loosing on

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consistency. For this purpose, the following reduced GLS method is proposed. Find …uh ; ph ; rh † 2 Wh such that Bh …uh ; ph ; rh ; v; q; s† ˆ Fh …v; q; s†

…33†

for all …v; q; s† 2 Wh , where the (non-symmetric) bilinear form Bh is de®ned by X ah2 K … 2gs div …uh † ‡ rph 2gp K2Th !

Bh …uh ; ph ; rh ; v; q; s† ˆ B…uh ; ph ; rh ; v; q; s† ‡ 2gp b

1 rh 2gp

…uh †;

div rh ; rq†K

…v†

…34†

and the form Fh by Fh …v; q; s† ˆ F …v; q; s†

X ah2 K …f 1 ; rq†K ‡ 2gp b…f 3 ; …v†† 2g p K2Th

with the dimensionless parameters a > 0 and 0 < b < 2. Let us point out that scheme (33) is consistent in the sense that, if the solution …u; p; r† 2 W of the weak formulation (3) is regular enough and if …uh ; ph ; rh † 2 Wh is the solution of (33), then Eq. (16) holds for all …v; q; s† 2 Wh . In the case of this reduced GLS scheme, we shall ®rst prove stability in the discrete norm k
kh . The stability and convergence in the norm k
kW will simply be recalled without proofs. Lemma 8 (Stability in norm k
kh ). Let gs P 0 and gp > 0. Let Bh be defined by (34) with 0 < b < 2 and 0 < a < 2CI , where CI is the largest positive constant that satisfies the inverse inequality (18). Then, there exists a constant C > 0, independent of h; gs and gp , but depending on X; a; b and the shape of the triangles, such that, for all …v; q; s† 2 Wh we have 2

Bh …v; q; s; v; q; s† P Ckv; q; skh :

…35†

Therefore (33) has a unique solution. Proof of Lemma 8. By de®nition of Bh we have 2

Bh …v; q; s; v; q; s† ˆ …2gs ‡ 2gp b†k…v†k ‡ ‡

X ah2  2 K krqkK 2g p K2Th

1 2 ksk 2gp

b…s; …v††

 …div s; rq†K :

Using Young's inequality, we have …s; …v†† P

2gp c1 k…v†k2 2

1 ksk2 ; 2gp 2c1

…div s; rq†K P

c2 krqk2K 2

1 kdiv sk2K ; 2c2

where c1 and c2 are positive constants to be chosen. Using (18) and the above two inequalities we obtain !   c1  1 b a~ 2 2 1 Bh …v; q; s; v; q; s† P 2gs ‡ 2gp b 1 ksk k…v†k ‡ 2gp 2 2c2 2c1 a  c2  X 2 2 ‡ 1 h krqkK ; …36† 2gp 2 K2Th K where we have set a~ ˆ a=CI . Clearly, we have to choose c1 and c2 such that 0 < c1 < 2;

0 < c2 < 2

and

1

a~ 2c2

b >0 2c1

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for the three terms in the right-hand side of (36) to be positive. A possibility is to choose c1 and c2 such that 1 2

b >0 2c1

and

1 2

a~ > 0; 2c2

i.e., 0 < b < c1 < 2

and

0 < a~ < c2 < 2:

Choosing for instance c1 ˆ 1 ‡ …b=2† and c2 ˆ 1 ‡ …~ a=2†, a simple calculation shows that ! c1 2 b c2 2 a~ b 1 2 b 2 a~ a~ > 0; 1 > 0 and 1 ˆ ˆ ˆ 1 ‡ > 0: 4 4 2 2 2c2 2c1 2 2 ‡ b 2 ‡ a~

…37†

Finally (37) in (36) yields (35), with C proportional to    a a ;a 1 : min b…2 b†; 1 2CI 2CI The same argument as for Lemma 3 ensures existence and uniqueness of the solution of (33). Remark 9. As for the previous GLS method, our numerical interpretation of Lemma 8 is the following. The velocity is controlled for any gs , even when gs ˆ 0. Moreover, a careful analysis of the constant of (35) shows that the control on the velocity is maximum when b ˆ 1 and that the control on the pressure is maximum when a ˆ CI . When b approaches 2 or when a approaches 2CI , the method can be expected to fail. These predictions are compared to experiments in Section 7. As in the previous section, stability and convergence can be proved for the reduced GLS scheme (33) in the norm k
kW de®ned in (5). Using de®nition (34) instead of (15) for the bilinear form Bh and assuming that 0 < a < 2CI and 0 < b < 2; the stability result (23) of Lemma 5 and the convergence estimate (24) of Theorem 6 hold unaltered for the reduced GLS scheme of this section. The proofs are not given here, since they are very similar to those of Section 3. 5. Link between reduced GLS and modi®ed EVSS methods In this section, a modi®ed version of the so-called modi®ed EVSS method of [1] is presented. Then the link with the reduced GLS method of the previous section is presented. The modi®ed EVSS method analysed in [1] consists in adding to the problem an extra unknown for stability purposes. More precisely, the method consists in solving, instead of (1), the following set of partial di€erential equations: 2…gs ‡ d†div …u† ‡ rp div r ‡ 2ddiv d ˆ f 1 ; 1 r …u† ˆ f 3 ; d …u† ˆ 0: 2gp

div u ˆ f2 ; …38†

Here d is the rate of deformation introduced for stability purposes, and d is a positive parameter that must be chosen carefully to ensure stability. Again, for viscoelastic ¯ows, the third equation of (38) should be replaced by the extra-stress constitutive equation, for instance (2) if the Oldroyd-B model is considered. In [1], it was proved that the modi®ed EVSS scheme was stable and convergent. The ®nite element spaces for the velocity and pressure had to satisfy the inf±sup condition (8), but there was no additional condition on the ®nite element space for the stresses. This discussion aims at showing that the reduced GLS stabilization method of Section 4 is related to some modi®ed EVSS scheme. More precisely, let Vh ; Qh ; Th be the continuous, piecewise linear ®nite element spaces de®ned in (12), the following method is considered for solving the four-®eld Stokes problem (38):

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Find …uh ; ph ; rh ; dh † 2 Vh  Qh  Th  Th such that 2…gs ‡ d†……uh †; …v†† …ph ; div v† ‡ …rh ; …v†† 2d…dh ; …v†† …div uh ; q† X ah2 1 K … 2gs div …uh † ‡ rph div rh ; rq†K …rh ; s† ‡ ……uh †; s† 2gP 2gp K2Th ‡ 2d…dh ; e†

2d……uh †; e† ˆ …f 1 ; v†

…f2 ; q†

…f 3 ; s†

X

K2Th

ah2K 2gp

…39†

…f 1 ; rq†K

for all …v; q; s; e† 2 Vh  Qh  Th  Th : Note that in (39) a stabilizing term corresponding to the ®rst equation of (38) has been added, aiming at avoiding spurious oscillations in the pressure by circumventing (8), as in GLS methods for the two-®eld Stokes problem. This is the main di€erence between method (39) and the EVSS method of [1,2], which must still satisfy (8) in the ®nite element subspaces. According to [27], (39) can also be seen as a continuous, piecewise linear approximation of (38) with bubble stabilization and static condensation. Note also that the stabilizing terms are not added in a fully consistent way, since the term 2ddiv dh is missing. Since no stabilization terms are added explicitly for the velocity-stress discretization, method (39) can be considered as a modi®ed EVSS method, in the sense that attention is paid only to the velocity±pressure discretization and not to the velocity-stress discretization. We show here that the modi®ed EVSS formulation (39) is equivalent to the reduced GLS method of Section 4 in the sense of the following lemma. Lemma 10. Assume that f 3 2 Th . Let …uh ; ph ; rh † be the solution of (33) and let dh 2 Th be defined by …dh

…uh †; e† ˆ 0

…40†

for all e in Th . Then …uh ; ph ; rh ; dh † is also the solution of (39), provided we set d ˆ gp b. Conversely, let …uh ; ph ; rh ; dh † be the solution of (39) with d ˆ gp b. Then, …uh ; ph ; rh ; † is solution of (33). Proof of Lemma 10. Let …uh ; ph ; rh † be the solution of (33). If we choose v ˆ 0; q ˆ 0 in (33), we obtain ! 1 rh …uh †; s ˆ …f 3 ; s† 2gp for all s in Th . Using the de®nition of dh , Eq. (40), we clearly have dh ˆ

1 rh 2gp

f3:

…41†

Now, since …uh ; ph ; rh † is the solution of (33), we have …2gs ‡ 2gp b†……uh †; …v††

…ph ; div v† ‡ …1

1 …rh ; s† ‡ ……uh †; s† ˆ …f 1 ; v† 2gp

b†…rh ; …v††

…f2 ; q†

…f 3 ; s†

…div uh ; q†

X ah2 K …rph 2g p K2Th

div rh ; rq†K

X ah2 K …f 1 ; rq†K ‡ 2gp b…f 3 ; …v†† 2g p K2sh

for all …v; q; s† 2 Wh . Using relation (41), we replace the term b…rh ; …v†† by 2gp b…dh ‡ f 3 ; …v†† in the above equation, and add (40). Then …uh ; ph ; rh ; dh † satis®es …2gs ‡ 2gp b†……uh †; …v†† …ph ; div v† ‡ …rh ; …v†† 2gp b…dh ; …v†† …div uh ; q† X ah2 1 K …rph div rh ; rq†K …rh ; s† ‡ ……uh †; s† ‡ 2gp b…dh ; e† 2gp b……uh †; e† 2gp 2gp K2Th ˆ …f 1 ; v†

…f2 ; q†

…f 3 ; s†

X ah2 K …f 1 ; rq †K 2g p K2Th

for all …v; q; s; e† 2 Vh  Qh  Th  Th . This is nothing but Eq. (39), provided we set d ˆ gp b.

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6. A stable iterative GLS method In this section, a simple iterative algorithm is presented for solving (33), similar to the method presented in [1,2]. The interest of such an algorithm is to de-couple velocity±pressure computations from stress computations, thus leading to smaller linear systems. Let …unh ; phn ; rnh † be the known approximation of …uh ; ph ; rh † after n steps. Roughly speaking, step …n ‡ 1† of n‡1 the algorithm consists in ®rst computing …un‡1 h ; ph † by solving the incompressible Navier±Stokes equations using the previous stress rnh in the right-hand side of the linear system, and then computing rn‡1 by using its h constitutive equation, with the new velocity uhn‡1 in the right-hand side. Thus iteration …n ‡ 1† consists in 1. Seeking …uhn‡1 ; phn‡1 † 2 Vh  Qh such that for all …v; q† 2 Vh  Qh
…2gs ‡ 2gp b† …uhn‡1 †; …v† ˆ …f 1 ; v†

…f2 ; q†

…1

…phn‡1 ; div v† b†…rnh ; …v††

…div uhn‡1 ; q†

X ah2 K …rphn‡1 ; rq†K 2g p K2Th

X ah2 K …f 1 ‡ div rnh ; rq†K ‡ 2gp b…f 3 ; …v††: 2g p K2Th

…42†

2. Seeking rn‡1 2 Th such that for all s 2 Th h 1 n‡1 r ;s 2gp h

! ˆ ……uhn‡1 †; s† ‡ …f 3 ; s†:

…43†

The following stability result holds for this algorithm. n‡1 n‡1 be the solution of (42) and (43) with f 1 ˆ f2 ˆ f 3 ˆ 0; a ˆ CI and Lemma 11 (Stability). Let un‡1 h ; ph ; rh b ˆ 1; where CI be the largest positive constant that satisfies (18). Then for all n P 0, we have 2

…2gs ‡ gp †k…uhn‡1 †k ‡

CI X 2 1 1 2 2 2 hK krphn‡1 kK ‡ krhn‡1 k 6 krn k : 4gp 4gp h 4gp K2Th

…44†

Remark 12. From (44) we deduce that the sequence …unh ; phn ; rnh †n P 0 is a lower bounded decreasing sequence. Thus the solution of (42) and (43) converges towards the unique solution of (33) when f 1 ˆ f2 ˆ f 3 ˆ 0, which is 0. Remark 13. This lemma ensures stability of the decoupling algorithm hereabove for the optimal values of stabilization parameters a and b discussed in Remark 9. As will be seen in Section 7, numerical experiments con®rm these predictions. Remark 14. The usual mass lumping procedure can be used to integrate numerically the mass matrix in Eq. (43). In this case, the scheme remains stable and (43) reduces to an explicit formula to compute the stress at the nodes of the ®nite element mesh.
Proof of Lemma 11. We set a ˆ CI and b ˆ 1 in (42) and (43) and choose …v; q; s† ˆ uhn‡1 ; phn‡1 ; rhn‡1 to obtain 2

…2gs ‡ 2gp †k…uhn‡1 †k ‡ ˆ

CI X 2 1 2 2 hK krphn‡1 kK ‡ krn‡1 k 2gp h 2gp K2Th


CI X 2 hK div rnh ; rphn‡1 K : 2gp K2Th

n‡1 …un‡1 h †; rh




…45†

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Young's inequality yields
div rn ; rpn‡1 6 c1 krpn‡1 k2 ‡ 1 kdiv rn k2 ; h h h h K K K 2c1 2
…un‡1 †; rn‡1 6 2gp c2 k…un‡1 †k2 ‡ 1 krn‡1 k2 : h h h 2gp 2c2 h 2 By introducing these inequalities into (45) we obtain 

 2gs ‡ 2gp 1 6

c2  CI  2 †k ‡ 1 k…un‡1 h 2 2gp

 c1  X 2 1 2 hK krphn‡1 kK ‡ 1 2gp 2 K2Th

 1 2 krhn‡1 k 2c2

1 1 2 krn k : 2gp 2c1 h

Then, setting c1 ˆ c2 ˆ 1 yields (44).

7. Numerical results Numerical results of computations of the three-®eld Stokes problem (1) in the 4:1 abrupt contraction ¯ow case are presented and compared to [1]. The symmetry of the geometry is used to reduce the computation domain by half, as shown in Fig. 1. The computations were performed on three meshes: the rather coarse, structured mesh M1 shown with the geometry de®nitions in Fig. 1, the coarse, unstructured mesh M2 and the re®ned, unstructured mesh M3, both shown in Fig. 2. The authors of [2] found coarse, structured meshes like M1 to be more likely to reveal stability problems, whereas unstructured meshes like M2 or more re®ned, unstructured meshes like M3 tended to mask the instabilities. In every computation, zero Dirichlet boundary conditions are imposed on the walls, the Poiseuille ve2 locity pro®le ux …y† ˆ Vu …1 …y=Ru † † is imposed at the inlet, natural boundary conditions on the symmetry axis and at the outlet of the domain. We choose Vu ˆ 0:15 so the ¯ow rate is D ˆ 0:1, all the physical parameters being expressed in SI units. The results corresponding to the GLS formulations (14) and (33) are presented ®rst, in a case where the stabilization of the third equation is of importance, that is when gs is small compared to gp . Then the results for the iterative algorithm (42) and (43) are presented.

Fig. 1. Geometry and mesh M1 used for the 4:1 abrupt contraction ¯ow simulations, with the values Ru ˆ 1, Rd ˆ 0:25 and Lu ˆ Ld ˆ 2.

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Fig. 2. Meshes M2 (top) and M3 (bottom) used for the 4:1 abrupt contraction ¯ow simulations.

Fig. 3. Isolines and cross-section at y ˆ 0:25 of the vertical velocity component uy for the coupled resolution of the three-®eld Stokes problem with gs ˆ 0:01, gp ˆ 1, a ˆ 0:01, on mesh M1 and with the reduced GLS method (33): (a) b ˆ 0, and (b) b ˆ 1.

7.1. Coupled resolution For the computations presented in this section, the linear system corresponding to the GLS schemes (14) and (33) was solved in one block, yielding the velocity, pressure and stress in one step. The viscosity constants were set to gs ˆ 0:01 and gp ˆ 1, which are realistic values in the case of nonnewtonian ¯uids. Both stabilization schemes are used, with parameter a ˆ 0:01 in all the coupled computations presented here. The reasons for this choice are the following. Many computations on other test cases (e.g., quadratic exact solution on unit square) showed the optimal values of a to be approximately

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Fig. 4. Isolines and cross-section at y ˆ 0:25 of the vertical velocity component uy for the coupled resolution of the three-®eld Stokes problem with gs ˆ 0:01, gp ˆ 1, a ˆ 0:01, on mesh M2 and with the reduced GLS method (33): (a) b ˆ 0, and (b) b ˆ 1.

0:01 6 a 6 0:02, whereas a ˆ 0 always yielded a singular linear system (a zero pivot was encountered). Remember that, according to Remarks 4 and 9, the optimal value for the reduced scheme (33) should be a ˆ CI , where CI is the (unknown) constant involved in (18). This interval can be compared to the value of the inverse inequality constant for elasticity problems on triangular meshes, which was analytically computed in [28] for straight-edged triangles, assuming a speci®c de®nition of hK : the authors found a value of 1=42  0:023809 for the stress inverse inequality constant, which is of the same order as our choice for a. Figs. 3±5 show the isolines and cross-sections at abscissa y ˆ 0:25 of the vertical component of the velocity uy for the three-®eld Stokes problem, on meshes M1, M2 and M3, respectively. In each ®gure, the results are those obtained when using the reduced GLS method (33) with (a) b ˆ 0, and (b) b ˆ 1. The part of the domain represented is de®ned by 1 6 x 6 2:25, 0 6 y 6 1. The numerical results corresponding to the GLS method (14) with b ˆ 0 and b ˆ 0:5 are not represented here since they are almost identical to those of Figs. 3±5. Note that these results compare well to those of [1] in the frame of the modi®ed EVSS method. These results agree with the theoretical predictions of Lemma 8: b > 0 ensures stabilization when gs is nought or too small. If b ˆ 0 with small or nought values of gs , spurious oscillations appear in the stress rh , the pressure ph and the velocity uh , especially near the reentrant corner. When setting b ˆ 2 with the reduced GLS method, the linear system was singular (a zero pivot was encountered; also note that the constant C in (35) becomes large when b approaches 2). When a or b are out of the ranges predicted by Lemma 8, spurious oscillations appear. The same observations hold for the GLS method (14). However, there is no upper bound to a for this method but the error clearly increases when a increases. Fig. 4 shows that a coarse, unstructured mesh like M2 does not mask the instabilities, nor does a re®ned mesh like M3. Actually, making b ! 0 or a ! 0 caused spurious modes to appear, no matter the mesh. The only di€erence we have observed is that, as these modes appear at element level, re®ned meshes give a better global impression than coarse meshes. Finally, let us stress that the reduced GLS scheme (33) behaves as well as the full GLS scheme (14), even though only a selected number of stabilization terms have been added.

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Fig. 5. Isolines and cross-section at y ˆ 0:25 of the vertical velocity component uy for the coupled resolution of the three-®eld Stokes problem with gs ˆ 0:01, gp ˆ 1, a ˆ 0:01, on mesh M3 and with the reduced GLS method (33): (a) b ˆ 0, and (b) b ˆ 1.

7.2. Decoupled resolution In order to ensure convergence of the decoupling algorithms (42) and (43) presented in Section 6, the values of parameters a and b have to be tuned, a priori within the intervals given by Lemma 8. Fig. 6 shows the number of iterations n for algorithms (42) and (43) to converge in the sense that the following criteria are satis®ed as from step n: kuhn‡1 unh k < 10 kuhn‡1 k

14

;

kphn‡1 phn k < 10 kphn‡1 k

14

;

krhn‡1 rnh k < 10 krhn‡1 k

14

on mesh M1 and with gs ˆ 0:01, gp ˆ 1. In other words, the discrepancies here above have to be 10 15 , i.e., nearly of the order of the machine precision. Clearly, the fastest convergence properties were found for a 6 0:02 and b ˆ 1, where our convergence criteria were satis®ed as from the eighth iteration. For b 6 0:485, the algorithm was always divergent. Note that, contrary to what happens with the iterative algorithm presented in [1,2], our algorithm does not converge in one step when b ˆ 1. This is due to the presence of the term hK …div rnh ; rq†K among the stabilization terms. Indeed, even if Fig. 6 seems to show that when a ! 0 the number of iterations for convergence tends towards 1, it is not so. In fact, since the velocity±pressure linear system is singular for a ˆ 0, too small values of a cause the rounding errors to become dominant and hinder the algorithm to converge, while allowing spurious modes for the pressure to appear. This is the case as from a 6 0:001. Hence choosing 0:01 6 a 6 0:02, guarantees an optimal compromise between the control on the pressure and the convergence speed. When other values of the viscosities are set, the minimum value of b to ensure convergence changes, whereas the optimal a interval remains unchanged. For instance, with gs ˆ 0:5 and gp ˆ 1, convergence was

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Fig. 6. Dependence on parameters a and b of the number of iterations n for convergence of the decoupled algorithms (42) and (43) with the reduced GLS stabilization, on mesh M1 and with gs ˆ 0:01, gp ˆ 1.

obtained for b P 0:25, whereas with gs P gp ˆ 1, convergence was obtained for b P 0. Actually, the following empirical convergence criterion on b was observed: ! gs ‡ gp 1 gs 1 gs ‡ g p b P ; i:e:; b P : 2 2 gp When gs ˆ 0, this criterion reduces to that of [1,2]. Moreover, in every case the optimal value remained b ˆ 1. Computations have been performed on meshes coarser and ®ner than M1, the minimum mesh spacing varying from 0.075 to 0.006. With a and b within the optimal range it resulted that the number of iterations does not depend on the mesh size. As a last remark, let us mention that in the computations presented in this paper the mass matrices issued from terms of the form …rh ; s† were always lumped. However, when these mass matrices were not lumped, the results were very similar to those presented in Figs. 3±6, with the same optimal values for parameters a and b. 8. Conclusions Throughout this paper, we have presented two continuous, piecewise linear ®nite element methods for solving the three-®eld Stokes problem (1). The reduced GLS method is put into relation with the modi®ed EVSS method. Stability and convergence of the methods have been established, as well as the convergence of a simple decoupling algorithm. The method is designed to work even when the solvent viscosity gs is much smaller that the polymer viscosity gp . Optimal values of stabilization parameters a and b have been proposed and justi®ed. This approach with continuous, piecewise linears for all ®nite element spaces (velocity, pressure and stress), stabilized through the addition of GLS-like terms, opens large perspectives for developing codes easy to implement, in order to compute viscoelastic ¯ows. Moreover, it should be recalled that triangular elements o€er many advantages, such as meshing of complicated domains or mesh adapting. We have applied the methods of this paper to the Oldroyd-B model (2), as well as to molecular models for polymeric liquids. In such models, the dynamics of the polymer chains are modelled using dumbbells (see [29,30), that is two beads connected by a spring of length Q, with a linear (Hookean dumbbells) or nonlinear (e.g., FENE dumbbells) force F(Q). The evolution of the connection vectors Q obeys a set of coupled

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stochastic partial di€erential equations. Their interaction with the solvent induces an extra-stress r de®ned by r ˆ nkT …E…Q F…Q††

I†;

see [29±31] for modelling details. Numerical methods inspired from this theory have been developed, see [32±37]. The iterative algorithm presented in Section 5 is particularly well suited for these microscopic models, since it allows the extra-stress to be computed separately from the velocity and the pressure. Actually, the two-dimensional computations presented by the authors in [36] were performed on the basis of that stabilized decoupled scheme. References [1] M. Fortin, R. Guenette, R. Pierre, Numerical analysis of the modi®ed EVSS method, Comput. Methods Appl. Mech. Engrg. 143 (1997) 79±95. [2] R. Guenette, M. Fortin, A new mixed ®nite element method for computing viscoelastic ¯ows, J. Non-Newtonian Fluid Mech. 60 (1995) 27±52. [3] J.M. Marchal, M.J. Crochet, A new mixed ®nite element for calculating viscoelastic ¯ows, J. Non-Newtonian Fluid Mech. 26 (1987) 77±114. [4] M. Behr, L. Franca, T. Tezduyar, Stabilized ®nite element methods for the velocity±pressure-stress formulation of incompressible ¯ows, Comput. Methods Appl. Mech. Engrg. 104 (1993) 31±48. [5] Y. Fan, R.I. Tanner, N. Phan-Thien, Galerkin/least-square ®nite element methods for steady viscoelastic ¯ows, J. Non-Newtonian Fluid Mech. 84 (1999) 233±256. [6] M. Fortin, A. Fortin, A new approach for the FEM simulations of viscoelastic ¯ows, J. Non-Newtonian Fluid Mech. 32 (1989) 295±310. [7] F.P.T. Baaijens, S.H.A. Selen, H.P.W. Baaijens, G.W.M. Peters, H.E.H. Meijer, Viscoelastic ¯ow past a con®ned cylinder of a LDPE melt, J. Non-Newtonian Fluid Mech. 60 (1997) 173±203. [8] J. Baranger, D. Sandri, Finite element approximation of viscoelastic ¯uid ¯ow, Numer. Math. 63 (1992) 13±27. [9] M. Fortin, D. Esselaoui, A ®nite element procedure for viscoelastic ¯ows, Int. J. Numer. Methods Fluids 7 (1987) 1035±1052. [10] J. Baranger, C. Guillope, J.-C. Saut, Mathematical analysis of di€erential models for viscoelastic ¯uids, in: J.-M. Piau, J.-F. Agassant (Eds.), Rheology for Polymer Melt Processing, Elsevier, Amsterdam, 1996, pp. 199±236. [11] M. Fortin, R. Pierre, On the convergence of the mixed method of Crochet and Marchal for viscoelastic ¯ows, Comput. Methods Appl. Mech. Engrg. 73 (1989) 341±350. [12] L. Franca, R. Stenberg, Error analysis of some GLS methods for elasticity equations, SIAM J. Numer. Anal. 28 (6) (1991) 1680± 1697. [13] D. Rajagopalan, R.C. Armstrong, R.A. Brown, Finite element methods for calculation of steady, viscoelastic ¯ow using constitutive equations with a newtonian viscosity, J. Non-Newtonian Fluid Mech. 36 (1990) 159±192. [14] F. Yurun, A comparative study of the discontinuous Galerkin and continuous SUPG ®nite element methods for computation of viscoelastic ¯ows, Comput. Methods Appl. Mech. Engrg. 141 (1997) 47±65. [15] F.P.T. Baaijens, Mixed ®nite element methods for viscoelastic ¯ow analysis: A review, J. Non-Newtonian Fluid Mech. 79 (1998) 361±385. [16] J. Sun, N. Phan-Thien, R.I. Tanner, An adaptive viscoelastic stress splitting scheme and its applications: AVSS/SI and AVSS/ SUPG, J. Non-Newtonian Fluid Mech. 65 (1996) 75±91. [17] L.P. Franca, T.J.R. Hughes, Convergence analyses of Galerkin least-squares methods for symmetric advective±di€usive forms of the stokes and incompressible Navier±Stokes equations, Comput. Methods Appl. Mech. Engrg. 105 (1993) 285±298. [18] T.J.R. Hughes, L. Franca, M. Balestra, A new ®nite element formulation for computational ¯uid dynamics, V. Circumventing the Babuska±Brezzi condition: A stable Petrov±Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg. 59 (1986) 85±99. [19] A. Quarteroni, A. Valli, Numerical Approximation of Partial Di€erential Equations, Springer, Berlin, 1994. [20] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, Berlin, 1991. [21] L. Franca, S. Frey, T.J.R. Hughes, Stabilized ®nite element methods: Application to the advective±di€usive model, Comput. Methods Appl. Mech. Engrg. 95 (1992) 253±276. [22] V. Ruas, Finite element methods for the three-®eld stokes system, RAIRO Modell. Math. Anal. Numer. 30 (1996) 489±525. [23] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Academic Press, London, 1990. [24] L.P. Franca, J.R. Hughes, R. Stenberg, Stabilized ®nite element methods, in: M.D. Gunzburger, R.A. Nicolaides (Eds.), Incompressible Computational Fluid Dynamics ± Trends and Advances, Cambridge University Press, Cambridge, 1993, pp. 87± 107. [25] P. Clement, Approximation by ®nite elements using local regularization, RAIRO Serie Rouge 8 (1975) 77±84. [26] C. Bernardi, Optimal ®nite element interpolation on curved domains, SIAM J. Numer. Anal. 26 (5) (1989) 1212±1240. [27] R. Pierre, Simple C0 approximations for the computation of incompressible ¯ows, Comput. Methods Appl. Mech. Engrg. 68 (1988) 205±227.

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