A domain decomposition method for the Oseen-viscoelastic flow equations

A doman decomposton method for the Oseen-vscoelastc flow equatons Eleanor Jenkns Hyesuk Lee Abstract We study a non-overlappng doman decomposton method for the Oseen-vscoelastc flow problem. The data on the nterface are transported through Newmann and Drchlet boundary condtons for the momentum and consttutve equatons, respectvely. The dscrete varatonal formulatons of subproblems are presented and nvestgated for the exstence of solutons. We show convergence of the doman decomposton soluton to a soluton of the one-doman problem. Convergence of an teratve algorthm and some numercal results are also presented. Key words doman decomposton, vscoelastc flows, fnte element method 1 Introducton In ths paper we nvestgate a doman decomposton method for the Oseen-vscoelastc flud flow of Johnson-Segalman type. Vscoelastc flow equatons consst of a consttutve equaton for the materal, a momentum equaton, and a mass equaton. For Newtonan fluds, the equatons for statonary, ncompressble, creepng flow can be smplfed n terms of velocty and pressure because of the smplstc relatonshp between stress and velocty. For non-newtonan fluds, however, there s an extra stress component that accounts for the forces that the materal develops under deformaton. Thus, no smplfcaton takes place, and the complete system conssts of three equatons n the unknowns pressure (scalar), velocty (vector), and stress (symmetrc tensor). One of the dffcultes n smulatng vscoelastc flows arses from the hyperbolc nature of the consttutve equaton for whch one needs to use a stablzaton technque such as the streamlne upwndng Petrov-Galerkn (SUPG) method and the dscontnuous Galerkn method 1. Here we consder the dscontnuous Galerkn method for the fnte element approxmaton of stress. When dscretzed usng a dscontnuous space to approxmate the stress tensor, the sze of the dscrete system s ncreased consderably ncreased. In order to handle the large number of unknowns assocated wth the vscoelastc systems, we propose a doman decomposton algorthm for a two-subdoman problem. An extenson to a mut-doman problem s straghtforward. In each subdoman we mpose a Neumann condton for the momentum equaton and a Drchlet condton for the consttutve equaton on the nterface boundary n order to transport data. The dea of usng a Neumann type condton s based on the methods found n 15, 17, 18. For the fnte element approxmaton, we use the standard Taylor-Hood element for velocty and pressure and the dscontnuous Galerkn method for stress. Doman decomposton methods have been used extensvely to solve ellptc partal dfferental equatons (see 22 and references theren), along wth the Stokes and Naver-Stokes equatons 8, 10, 13, 15, 17. Some doman decomposton methods for vscoelastc fluds are found n 4, 16, 23. In 4 the authors use the addtve Schwarz method n conjuncton wth the DEVSS-G operator Department of Mathematcal Scences, Clemson Unversty, Clemson, SC 29634-0975, USA, lea@clemson.edu hklee@clemson.edu. Ths work was partally supported by the NSF under grant no. DMS-0410792. 1

splttng method. The two-level Schwarz method s appled to a Stokes-lke problem that results from the splttng. A doman decomposton spectral collocaton (DDSC) method s ntroduced n 23 for an Oldroyd-B flud n model porous geometres. The method dscussed n 16 s based on the concepts of streamlnes and local transformaton functons, where the local functons are the prmary unknowns. The functons map the physcal subdomans to transformed domans where the mapped streamlnes are parallel straght lnes. The remander of the document s organzed as follows. In Secton 2, we ntroduce the model equatons for the Oseen-vscoelastc flow problem, and n Secton 3, we descrbe the model equatons for the problem decomposed nto two subdomans. Convergence of a doman decomposton soluton to a soluton of the one-doman problem s dscussed n Secton 4. In Secton 5, we present teratve algorthms and prove convergence of the algorthms. We provde numercal results n Secton 6, and conclusons and future work are gven n Secton 7. 2 Model equatons and fnte element approxmaton Let Ω be a bounded doman n RI d wth the Lpschtz contnuous boundary Ω. Johnson-Segalman problem Consder the σ + λ(u )σ + λg a (σ, u) 2 α D(u) = 0 n Ω, (2.1) σ 2(1 α) D(u) + p = f n Ω, (2.2) dv u = 0 n Ω, (2.3) where σ denotes the polymerc stress tensor, u the velocty vector, p the pressure of flud, and λ s the Wessenberg number defned as the product of the relaxaton tme and a characterstc stran rate. Assume that p has zero mean value over Ω. In (2.1) and (2.2), D(u) := ( u + u T )/2 s the rate of the stran tensor, α a number such that 0 < α < 1 whch may be consdered as the fracton of vscoelastc vscosty, and f the body force. In (2.1), g a (σ, u) s defned by g a (σ, u) := 1 a 2 (σ u + ut σ) 1 + a 2 ( u σ + σ ut ) (2.4) for a 1, 1. We use the Sobolev spaces W m,p (D) wth norms m,p,d f p <, m,,d f p =. We denote the Sobolev space W m,2 by H m wth the norm m. The correspondng space of vector-valued or tensor-valued functons s denoted by H m. If D = Ω, D s omtted,.e., (, ) = (, ) Ω and = Ω. Exstence of a soluton to the problem (2.1)-(2.3) wth the homogeneous boundary condton for u has been documented by Renardy (19) wth the small data condton: f f s suffcently regular and small, the problem admts a unque bounded soluton (u, p, σ) H 3 (Ω) H 2 (Ω) H 2 (Ω). In ths paper, the consttutve equaton (2.1) s smplfed to defne our doman decomposton problem. We wll consder the lnear Oseen problem wth the gven velocty b(x): We make the followng assumpton for b: σ + λ(b )σ + λg a (σ, b) 2 α D(u) = 0 n Ω, (2.5) σ 2(1 α) D(u) + p = f n Ω, (2.6) dv u = 0 n Ω, (2.7) u = 0 on Ω. (2.8) b H 1 0(Ω), b = 0, b M, b M <. 2

Defne the functon spaces for the velocty u, the pressure p and the stress σ, respectvely: X := H 1 0(Ω), S := L 2 0(Ω) = {q L 2 (Ω) : q dω = 0}, Ω Σ := {τ L 2 (Ω) : τ j = τ j, (b )τ L 2 (Ω)}. Note that the velocty and pressure spaces, X and S, satsfy the nf-sup condton The correspondng weak formulaton s then gven by: nf sup (q, v) C. (2.9) q S v X v 1 q 0 (σ, τ ) + λ((b )σ, τ ) + λ(g a (σ, b), τ ) 2α(D(u), τ ) = 0 τ Σ, (2.10) (σ, D(v)) + 2(1 α)(d(u), D(v)) (p, v) = (f, v) v X, (2.11) (q, u) = 0 q S. (2.) In the weak dvergence free space V := {v X : the weak formulaton (2.10)-(2.) s equvalent to: Ω q dv v dω = 0 q L 2 0(Ω)}, (σ, τ ) + λ((b )σ, τ ) + λ(g a (σ, b), τ ) 2α(D(u), τ ) = 0 τ Σ, (2.13) (σ, d(v)) + 2(1 α)(d(u), D(v)) = (f, v) v V. (2.14) We now consder the fnte element approxmaton of the problem (2.10)-(2.). Suppose T h s a trangulaton of Ω such that Ω = { K : K T h }. Assume that there exst postve constants c 1, c 2 such that c 1 h h K c 2 ρ K, where h K s the dameter of K, ρ K s the dameter of the greatest ball ncluded n K, and h = max K T h h K. Due to the hyperbolc nature of the consttutve equaton, a stablzaton technque s needed for the fnte element smulaton of vscoelastc flows. Streamlne upwndng (14, 20) and the dscontnuous Galerkn method (2, 7) are the commonly used dscretzaton technques to handle ths problem. We use the dscontnuous Galerkn method for approxmatng the stress. Let P k (K) denote the space of polynomals of degree less than or equal to k on K T h. Then we defne fnte element spaces for the approxmaton of (u, p): X h := {v X (C 0 (Ω)) d : v K P 2 (K) d, K T h }, S h := {q S C 0 (Ω) : q K P 1 (K), K T h }, V h := {v X h : (q, v) = 0, q S h }. The stress σ s approxmated n the dscontnuous fnte element space of pecewse lnears: Σ h := {τ Σ : τ K P 1 (K) d d, K T h }. 3

We ntroduce some notaton below n order to analyze an approxmate soluton by the dscontnuous Galerkn method. We defne K (b) := {x K, b n < 0}, where K s the boundary of K and n s outward unt normal, and We also defne for σ, τ < σ ±, τ ± > h,b := K T h (L 2 (K)) d d, and τ ± (b) := lm ɛ 0 τ (x ± ɛb(x)). (σ, τ ) h := K T h K T h (σ, τ ) K, K (b) τ 0,Γ h := τ 2 0, K K T h (σ ± (b), τ ± (b)) n b ds. τ m,h := 1/2 τ 2 m,k K T h for τ (W m,2 (K)) d d, f m <. K T h The dscontnuous Galerkn fnte element approxmaton of (2.10) (2.) s then as follows: fnd u h X h, p h S h, σ h Σ h such that (σ h, τ h ) + λ ((b )σ h, τ h ) h + < σ h+ σ h, τ h+ > h,b (2.15) 1/2 +λ(g a (σ h, b h ), τ h ) 2α(d(u h ), τ h ) = 0 τ h Σ h, (σ h, d(v h )) + 2(1 α) (d(u h ), d(v h )) (p h, v h ) = (f, v h ) v h X h, (2.16) (q h, u h ) = 0 q h S h. (2.17) It was shown n 6 that the dscrete problem (2.15)-(2.17) has a unque soluton f 1 2λMd > 0, and the soluton satsfes the error estmate. σ σ h 0 + u u h 1 Ch. (2.18) 3 Descrpton of the doman decomposton method We descrbe the doman decomposton method usng two subdomans, and the analyss that follows s based on ths problem. However, there are no dffcultes assocated wth extendng the analyss to handle as many subdomans as necessary. Let Ω be dvded nto two dsjont subdomans Ω 1 and Ω 2 such that Ω = Ω 1 Ω 2. The nterface between the two domans s denoted by Γ 0 so that Γ 0 = Ω 1 Ω 2. Defne Γ := {x Γ 0 : b n 1 < 0}, Γ := {x Γ 0 : b n 2 < 0}, 4

where n s the outward unt normal vector to Ω for = 1, 2. Let Γ 1 = Ω 1 Ω and Γ 2 = Ω 2 Ω. We consder the followng par of Oseen-vscoelastc systems wth mxed boundary condtons: and σ 1 + λ(b )σ 1 + λg a (σ 1, b) 2 α D(u 1 ) = 0 n Ω 1, (3.1) σ 1 2(1 α) D(u 1 ) + p 1 = f n Ω 1, (3.2) dv u 1 = 0 n Ω 1, (3.3) u 1 = 0 on Γ 1, (3.4) (σ 1 + 2(1 α) D(u 1 ) p 1 I) n 1 = 1 ɛ (u 1 u 2 ) on Γ 0, (3.5) σ 1 = σ 2 on Γ, (3.6) σ 2 + λ(b )σ 2 + λg a (σ 2, b) 2 α D(u 2 ) = 0 n Ω 2, (3.7) σ 1 2(1 α) D(u 2 ) + p 2 = f n Ω 2, (3.8) dv u 2 = 0 n Ω 2, (3.9) u 2 = 0 on Γ 2, (3.10) (σ 2 + 2(1 α) D(u 2 ) p 2 I) n 2 = 1 ɛ (u 1 u 2 ) on Γ 0, (3.11) σ 2 = σ 1 on Γ. (3.) Note that (3.6), the Drchlet boundary condtons for σ 1, s mposed on a part of the nterface Γ 0 where b n 1 < 0. Smlarly, (3.) s mposed on the nflow boundary of Ω 2 assocated wth the gven velocty feld b. Defne the functon spaces for the velocty u, the pressure p and the stress σ, respectvely, for = 1, 2: X := {v H 1 (Ω ) : v = 0 on Γ }, and the dv free space S := L 2 (Ω ), Σ := (L 2 (Ω )) 2 2 {τ = (τ j ) : τ j = τ j, b τ (L 2 (Ω )) 2 2 } V := {v X : q dv v dω = 0 Ω q S }. The correspondng weak formulaton of (3.1)-(3.) s then gven by (σ 1, τ 1 ) + λ (b )σ 1, τ 1 ) + λ (g a (σ 1, b), τ 1 ) 2α(D(u 1 ), τ 1 ) = 0 τ 1 Σ 1, (3.13) (σ 1, D(v 1 )) + 2(1 α)(d(u 1 ), D(v 1 )) (p 1, v 1 ) + 1 ɛ (u 1 u 2, v 1 ) Γ0 = (f, v 1 ) v 1 X 1, (3.14) (q 1, u 1 ) = 0 q 1 S 1, (3.15) σ 1 = σ 2 on Γ, (3.16) 5

and (σ 2, τ 2 ) + λ (b )σ 2, τ 2 ) + λ (g a (σ 2, b), τ 2 ) 2α(D(u 2 ), τ 2 ) = 0 τ 2 Σ 2, (3.17) (σ 2, D(v 2 )) + 2(1 α)(d(u 2 ), D(v 2 )) (p 2, v 2 ) 1 ɛ (u 1 u 2, v 2 ) Γ0 = (f, v 2 ) v 2 X 2, (3.18) (q 2, u 2 ) = 0 q 2 S 2. (3.19) σ 2 = σ 1 on Γ. (3.20) We defne dscrete subspaces for the fnte element approxmaton of the two-doman problem (3.13)-(3.20). Let T h denote a trangulaton of Ω such that Ω = { K : K T h } for = 1, 2. Defne fnte element spaces for the approxmaton of (σ h, uh, ph ) for = 1, 2: Σ h := {τ Σ : τ K P 1 (K) d d, K T h }, X h := {v X (C 0 (Ω )) d : v K P 2 (K) d, K T h }, S h := {q S C 0 (Ω ) : q K P 1 (K), K T h }. We also defne the dscrete dvergence free subspace V h := {v X h : (q, v) = 0, q S h }. The fnte element spaces defned above satsfy the standard approxmaton propertes (see 3 or 9),.e., there exst an nteger k and a constant C such that and nf v v h 1 Ch 2 v 3 v H 3 (Ω ), (3.) v h X h nf q q h 0 Ch 2 q 2 q H 2 (Ω ), (3.22) q h S h nf τ τ h τ 0 Ch 2 τ 2 τ H 2 (Ω ). (3.23) h Σ h It s also well known that the Taylor-Hood par (X h, Sh ) satsfes the nf-sup condton, nf 0 q h S h sup 0 v h X h (q h, v h ) v h 1 q h 0 C, (3.24) where C s a postve constant ndependent of h. In addton to notaton ntroduced n the prevous secton for the dscontnuous Galerkn method, we present more notaton to analyze σ h on the nterface Γ 0. We defne, for σ, τ (L 2 (K)) d d < σ ±, τ ± > h,b := K T h, K Γ 0= K (b) σ ± h,b :=< σ ±, σ± > 1/2 τ 0,Γ h := τ 2 0, K K T h (σ ± (b) : τ ± (b)) n b ds, h,b, 1/2, K T h 6

and τ m,h := τ 2 m,k 1/2 for τ K T h (W m,2 (K)) d d, f m <. Also defne, for, j = 1, 2 K T h < σ ±, τ ± j > h,b,γ := lm K Tl h, K Γ lm K (b) (σ ± (b) : τ ± j (b)) n b ds, σ ± h,b,γ :=< σ ±, σ± > 1/2, lm h,b,γ lm where (l, m) = (1, 2) or (2, 1). The dscrete varatonal formulatons of (3.13)-(3.16) and (3.17)-(3.20) are then gven by (σ h 1, τ h 1) + λ ((b )σ h 1, τ h 1) h + λ (g a (σ h 1, b), τ h 1) 2α(D(u h 1), τ h 1) +λ < σ h 1 + σ h 1, τ h 1 + >h,b +λ < σ h 1 σ h 2, τ h 1 > h,b,γ= 0 τ h 1 Σ h 1, (3.25) (σ h 1, D(v h 1 )) + 2(1 α)(d(u h 1), D(v h 1 )) (p h 1, v h 1 ) + 1 ɛ (uh 1 u h 2, v h 1 ) Γ0 = (f, v h 1 ) v h 1 X h 1, (3.26) and (q h 1, u h 1) = 0 q 1 S h 1 (Ω), (3.27) (σ h 2, τ h 2) + λ ((b )σ h 2, τ h 2) h + λ (g a (σ h 2, b), τ h 2) 2α(D(u h 2), τ h 2) +λ < σ h + 2 σ h 2, τ h+ 2 >h,b +λ < σ h 2 σ h 1, τ h 2 > h,b,γ = 0 τ h 2 Σ h 2, (3.28) (σ h 2, D(v h 2 )) + 2(1 α)(d(u h 2), D(v h 2 )) (p h 2, v h 2 ) 1 ɛ (uh 1 u h 2, v h 2 ) Γ0 = (f, v h 2 ) v h 2 X h 2, (3.29) (q h 2, u h 2) = 0 q 2 S h 2 (Ω). (3.30) Note that the stress boundary condtons (3.16), (3.20) are mposed weakly 11. The next theorem shows the exstence and unqueness of a soluton to the above coupled system (3.25)-(3.30). In provng the theorem we use the blnear forms A defned on Σ h V h for = 1, 2 by A ((σ h, u h ), (τ h, v h )) := (σ h, τ h ) 2α(D(u h ), τ h ) + 2α(σ h, D(v h )) Note that A s contnuous and coercve, f 1 2λMd > 0: snce we have +4α(1 α) (D(u h ), D(v h )) + λ (g a (σ h, b), τ h ). (3.31) (g a (σ h, b), τ h ) 2d b σ h 0 τ h 0 2dM σ h 0 τ h 0, (3.32) A ((σ h, u h ), (τ h, v h )) σ h 0 τ h 0 + 2α D(u h ) 0 τ h 0 +2α σ h 0 D(v h ) 0 + 4α(1 α) D(u h ) 0 D(v h ) 0 +2Mdλ σ h 0 τ h 0 C( σ h 0 + u 1 )(( τ h 0 + v 1 ) C (σ h, u h ) Σ X h (τ h, v h ) Σ X, (3.33) 7

where (τ h, vh ) Σ X s defned as ( τ h 2 0 + v h 2 1) 1/2. Also, usng (3.32), A ((σ h, uh ), (σh, uh )) = σh 2 0 + λ(g a (σ h, b), σh ) + 4α(1 α) D(uh ) 2 0 σ h 2 0 2λMd σ h 2 0 + 4α(1 α) D(u h ) 2 0 C (σ h, uh ) 2 Σ X, (3.34) f 1 2λMd > 0. In the proof of the next theorem we wll also use the nverse estmate 3, 9 the local nverse nequalty, and the trace theorem 9, σ h 0,h Ch 1 σ h 0 for σ h Σ h, (3.35) σ h 0, K Ch 1/2 σ h 0,K for σ h Σ h, (3.36) u 0,Γ Γ 0 C u 1 for u X. (3.37) Theorem 3.1. The system (3.25)-(3.30) has a unque soluton, f 1 2λMd > 0. Proof. Usng the dv free spaces V h = 1, 2, the coupled system (3.25)-(3.30) s equvalent to A ((σ h, u h ), (τ h, v h )) + λ((b )σ h, τ h ) h + λ < σ h + σ h, τ h+ >h,b +λ < σ h 1 σ h 2, τ h 1 > h,b,γ = where F ( ) : Σ h V h R s a functonal defned by It s straght forward to show F s bounded: + < σ h 2 σ h 1, τ h 2 > h,b,γ + 1 ɛ (uh 1 u h 2, v h 1 v h 2 ) Γ0 F ((τ h, v h )) (τ h, v h ) Σ h V h, (3.38) F ((τ h, v h )) := 2α(f, v h ). F ((τ h, v h )) 2α f 1 v h 1 2α f 1 (τ h, v h ) Σ X. (3.39) We wll show the blnear form n (3.38) s contnuous and coercve n 2 (Σh V h ), f 12λMd > 0. Usng (3.35) and (3.36), ((b )σ h, τ h ) h + < σ h + σ h, τ h+ >h,b C b σ h 0,h τ h 0 + b σ h 0,Γ τ h h 0,Γ h C b (h 1 σ h 0 ) τ h 0 + b (h 1/2 σ h 0 )(h 1/2 τ h 0 ) and CMh 1 σ h 0 τ h 0, (3.40) < σ h 1 σ h 2, τ h 1 > h,b,γ + < σ h 2 σ h 1, τ h 2 > h,b,γ C b ( σ h 1 0,Γ0 + σ h 2 0,Γ0 ) τ h 1 0,Γ0 + b ( σ h 2 0,Γ0 + σ h 1 0,Γ0 ) τ h 2 0,Γ0 CM (h 1/2 σ h 1 0 + h 1/2 σ h 2 0 )(h 1/2 τ h 1 0 ) +(h 1/2 σ h 2 0 + h 1/2 σ h 1 0 )(h 1/2 τ h 2 0 ) CMh 1,j=1 σ h 0 τ h j 0. (3.41) 8

Also, by (3.37), Hence, by (3.33) and (3.40)-(3.42), (u h 1 u h 2, v h 1 v h 2 ) Γ0 C,j=1 u h 1 v h j 1. (3.42) A ((σ h, u h ), (τ h, v h )) + λ((b )σ h, τ h ) h + λ < σ h + σ h, τ h+ >h,b +λ < σ h 1 σ h 2, τ h 1 > h,b,γ + < σ h 2 σ h 1, τ h 2 > h,b,γ + 1 ɛ (uh 1 u h 2, v1 h v2 h ) Γ0 C ( σ h 0 + u h 1 )( τ h 0 + v h 1 ) + Mh 1 σ h 0 τ h 0 C h ( σ h 0 + u h 1 )( τ h 0 + v h 1 ),j=1 C h (σ h 1, u h 1, σ h 2, u h 2) Σ1 X 1 Σ 2 X 2 (τ h 1, v h 1, τ h 2, v h 2 ) Σ1 X 1 Σ 2 X 2, (3.43) where C h s a constant dependent on h. We now show the coercvty of the blnear form. Frst note, usng ntegraton by parts, that =,j=1 ((b )σ h, τ h ) h + < σ h + σ h, τ h+ >h,b + < σ h 1 σ h 2, τ h 1 > h,b,γ + < σ h 2 σ h 1, τ h 2 > h,b,γ ((b )τ h, σ h ) h + < σ h, τ h τ h+ >h,b Hence, f τ h = σh, we obtan + < σ h 2, τ h 2 τ h 1 > h,b,γ + < σ h 1, τ h 1 τ h 2 > h,b,γ ((b )σ h, σ h ) h + < σ h + σ h, σ h+ >h,b. (3.44) + < σ h 1 σ h 2, σ h 1 > h,b,γ + < σ h 2 σ h 1, σ h 2 > h,b,γ = 1 2 0. σ h + σ h 2 h,b + σ h 1 σ h 2 2 h,b,γ + σ h 2 σ h 1 2 h,b,γ (3.45) 9

Now the coercvty of the blnear form follows from (3.34) and (3.45): +λ C A ((σ h, u h ), (σ h, u h )) + λ((b )σ h, σ h ) + λ < σ h + σ h, σ h+ >h,b < σ h 1 σ h 2, σ h 1 > h,b,γ + < σ h 2 σ h 1, σ h 2 > h,b,γ + 1 ɛ (uh 1 u h 2, u h 1 u h 2) Γ0 σ h 2 0 2λMd σ h 2 0 + 4α(1 α) D(u h ) 2 0 + λ 2 σh + σ h 2 h,b + λ σ h 1 σ h 2 2 + σ h 2 h,b,γ 2 σ h 1 2 + 1 h,b,γ ɛ (uh 1 u h 2) 2 Γ 0 (1 2λMd) σ h 2 0 + 4α(1 α) D(u h ) 2 0 (σ h, u h ) 2 Σ X C (σ h 1, u h 1, σ h 2, u h 2) 2 Σ 1 X 1 Σ 2 X 2. (3.46) Therefore, by the Lax-Mlgram theorem, there exsts a unque (σ h, uh ) Σh V h, f 12λMd > 0. Fnally, exstence of a unque soluton (σ h, uh, ph ) Σh X h Sh follows from the nf-sup condton (3.24). 4 Convergence Analyss We wll prove the soluton of the two-doman problem (3.25)-(3.30) converges to the soluton (σ ex, u ex, p ex ) satsfyng (2.5)-(2.8). Let σ ex = σ ex Ω Γ 0, u ex = u ex Ω Γ 0 and p ex = p ex Ω Γ 0 for = 1, 2. Defne g ex := (σ ex + 2(1 α)d(u ex ) + p ex I) n 1 on Γ 0. Note that, for = 1, 2, (σ ex (σ ex (σ ex, u ex, p ex ) satsfy, τ ) + λ((b )σ ex, τ ) + λ(g a (σ ex, D(v )) + 2(1 α)(d(u ex ), D(v )) (p ex, b), τ ) 2α(D(u ex ), τ ) = 0 τ Σ, (4.1), v ) = (f, v ) + (1) +1 (g ex, v ) Γ0 v X, (4.2) (q, u ex ) = 0 q S. (4.3) In the dv free spaces V, (4.1)-(4.3) s equvalent to A ((σ ex, u ex ), (τ, v )) + λ((b )σ ex, τ ) = 2α(f, v ) + 2α(g ex, v 1 v 2 ) Γ0. (4.4) We ntroduce some approxmaton propertes (see 3 or 9), whch wll be used n order to prove the next theorem. If σ h Σ h s the orthogonal projecton of σ ex on T h n Σ, and ũ h V h s defned as the nterpolant of u ex n V, then we have the followng standard results. For u ex H 3 (Ω ) and σ ex H 2 (Ω ) σ ex (u ex ũ h ) 0 C h 2 u ex 3, (4.5) σ h 0 + h (σ ex σ h ) 0 Ch 2 σ ex 2. (4.6) 10

Theorem 4.1. If 12λMd > 0 and (σ h, uh ) for = 1, 2, satsfy the coupled decomposton problem (3.25)-(3.30), they converge to the exact soluton (σ ex, u ex ) as ɛ goes to 0. Furthermore, we have the estmate σ ex σ h 0 + u ex u h 1 C(h + ɛ g ex Γ0 ). (4.7) Proof. Subtractng (3.38) from (4.4), and addng and subtractng σ h, ũ h, mply = +λ A (( σ h σ h, ũ h u h ), (τ h, v h )) +λ((b )( σ h σ h ), τ h ) + λ < ( σ h σ h ) + ( σ h σ h ), τ h + >h,b < ( σ h 1 σ h 1) ( σ h 2 σ h 2), τ h 1 > h,b,γ 2α ɛ (uh 1 u h 2, v h 1 v h 2 ) Γ0 A (( σ h σ ex, ũ h u ex ), (τ h, v h )) + < ( σ h 2 σ h 2) ( σ h 1 σ h 1), τ h 2 > h,b,γ + >h,b +λ((b )( σ h σ ex ), τ h ) + λ < ( σ h σ ex ) + ( σ h σ ex ), τ h +λ < ( σ h 1 σ ex 1 ) ( σ h 2 σ ex 2 ), τ h 1 > h,b,γ + < ( σ h 2 σ ex 2 ) ( σ h 1 σ ex 1 ), τ h 2 > h,b,γ +2α(g ex, v h 1 v h 2 ) Γ0 (τ h, v h ) Σ h V h, (4.8) where we used σ ex 1 = σ ex 2 on Γ 0 and that σ ex s a contnuous functon. Let τ h = σ h σ h, v h = ũh uh. We have a lower bound of the left hand sde of (4.8) usng (3.34) and (3.45): LHS (1 2λMd) σ h σ h 2 0 + 4α(1 α) D(ũ h u h ) 2 0 + 2α ɛ uh 1 u h 2 2 Γ 0. (4.9) Estmates for the jump terms n the rght sde of (4.8) are obtaned n the smlar way shown n (3.40) and (3.41): < ( σ h σ ex ) + ( σ h σ ex ), σ h σ h + >h,b C b σ h σ ex 0,Γ h σ h σ h 0,Γ h CMh 1 σ h σ ex 0 σ h σ h 0, (4.10) < ( σ h 1 σ ex 1 ) ( σ h 2 σ ex 2 ), σ h 1 σ h 1 > h,b,γ + < ( σ h 2 σ ex 2 ) ( σ h 1 σ ex 1 ), σ h 2 σ h 2 > h,b,γ C b ( σ h 1 σ ex 1 0,Γ0 + σ h 2 σ ex 2 0,Γ0 )( σ h 1 σ h 1 0,Γ0 + σ h 2 σ h 2 0,Γ0 ) CMh 1 ( σ h 1 σ ex 1 0 + σ h 2 σ ex 2 0 )( σ h 1 σ h 1 0 + σ h 2 σ h 2 0 ). (4.11) 11

Usng (3.33), (4.9)-(4.11) and the Young s nequalty, we have C (1 2λMd) σ h σ h 2 0 + 4α(1 α) D(ũ h u h ) 2 0 + 2α ɛ uh 1 u h 2 2 Γ 0 ( σ h σ ex 0 + D(ũ h u ex ) 0 )( σ h σ h 0 + D(ũ h u h ) 0 ) +λm ( σ h σ ex ) 0 σ h σ h 0 + λmh 1 σ h σ ex 0 σ h σ h 0 +CλMh 1,j=1 σ h σ ex 0 σ h j σ h j 0 + 2α g ex Γ0 u 1 u 2 Γ0 C δ 1 ( σ h σ h 0 + D(ũ h u h ) 0 ) 2 + 1 ( σ h σ ex 0 + D(ũ h u ex ) 0 ) 2 4δ 1 +δ 2 λm σ h σ h 2 0 + λm ( σ h σ ex ) 2 0 + δ 3 λm σ h σ h 2 0 + λmh2 σ h σ ex 2 0 4δ 2 4δ 3 +C δ 4 λm σ h σ h 20 + λmh2 σ h σ ex 2 0 4δ 4 1 +2α u h 1 u h 4δ 2 2 Γ 0 + δ 5 g ex 2 Γ 0, (4.) 5 for arbtrary δ > 0, = 1, 2,..., 5. If we take δ 5 = ɛ 2, C (1 2λMd 2δ 1 C (δ 2 + δ 3 + δ 4 )λmc) σ h σ h 2 0 1 +(4α(1 α) 2Cδ 1 ) D(ũ h u h ) 2 α 0 + ɛ uh 1 u h 2 2 Γ 0 ( σ h σ ex 4δ 1 +( λmh2 4δ 3 0 + D(ũ h u ex ) 0 ) 2 + λm ( σ h σ ex ) 2 0 4δ 2 + λmh2 ) σ h σ ex 2 0 + αɛ g ex 2 Γ 4δ 0. (4.13) 4 Therefore, usng (4.5)-(4.6) and the trangle nequalty, (4.7) follows. Remark 4.2. The pressure on each subdoman, p h, s determned up to a constant. In order to compute a pressure convergent to p ex whch has the zero mean value, construct p h nl 2 0(Ω) by { p h p h (x) = 1 K for x Ω 1 p h, 2 K for x Ω 2 ( where K = ph Ω1 1 dω 1 + ) ph Ω2 2 dω 2 /meas(ω). Then, convergence of p h to p ex can be proved as shown n 17 or 18. Remark 4.3. The estmate (4.7) suggests the optmal scalng ɛ = Ch 2 between ɛ and h. 5 Iteratve Algorthm In ths secton we delete the superscrpt h n (σ h, uh, ph ) to smplfy our notatons. Consder the followng teratve algorthm.

Algorthm 5.1. (σ n+1 1, τ h ) + λ((b )σ n+1 1, τ h ) + λ < σ n+1 + 1 σ n+1 1, τ h + > h,b 2α(D(u n+1 1 ), τ h ) + λ < σ n+1 1, τ h > h,b,γ = λ (g a (σ n 1, b), τ h ) + λ < σ n 2, τ h > h,b,γ τ h Σ h 1, (5.1) and (σ 1 n+1, D(v h )) + 2(1 α)(d(u 1 n+1 ), D(v h )) (p 1 n+1, v h ) + 1 ɛ (u 1 n+1, v h ) Γ0 = (f, v h ) + 1 ɛ (u 2 n, v h ) Γ0 v h X h 1, (5.2) (q h, u 1 n+1 ) = 0 q S h 1, (5.3) (σ n+1 2, τ h ) + λ((b )σ n+1 2, τ h ) + λ < σ n+1 + 2 σ n+1 2, τ h + > h,b 2α(D(u n+1 2 ), τ h ) + λ < σ n+1 2, τ h > h,b,γ = λ (g a (σ n 2, b), τ h ) + λ < σ n 1, τ h > h,b,γ τ h Σ h 2, (5.4) (σ 2 n+1, D(v h )) + 2(1 α)(d(u 2 n+1 ), D(v h )) (p 2 n+1, v h ) + 1 ɛ (u 2 n+1, v h ) Γ0 = (f, v h ) + 1 ɛ (u 1 n, v h ) Γ0 v h X h 2, (5.5) (q h, u h 2 n+1 ) = 0 q S h 2. (5.6) In the proof of the next theorem we wll use the Trace theorem C u 0,Γ0 D(u ) 0. (5.7) Theorem 5.2. The teratve algorthm 5.1 converges for each fxed ɛ and h f λm s suffcently small so that λmd + λ M λm 2h < 1 λmd and α satsfes λmd + 2h 4 C 2 α(1 α) < α ɛ < 1 λmd, where C s the constant n (5.7). Proof. In the dscrete dv free space V h, subtractng (5.1)-(5.2) and (5.4)-(5.5) from (3.25)-(3.26) and (3.28)-(3.29) respectvely, and lettng τ = σ σ n+1, v = u u n+1, we obtan σ 1 σ 1 n+1 2 0 + λ 2 (σ 1 σ n+1 1 ) + (σ 1 σ n+1 1 ) 2 h,b 2α(D(u 1 u n+1 1 ), σ 1 σ n+1 1 ) + λ σ 1 σ n+1 1 2 h,b,γ = λ (g a (σ 1 σ n 1, b), σ 1 σ n+1 1 ) + λ < σ 2 σ n 2, σ 1 σ n+1 1 > h,b,γ, (5.8) (σ 1 σ 1 n+1, D(u 1 u 1 n+1 )) + 2(1 α) D(u 1 u 1 n+1 ) 2 0 + 1 ɛ u 1 u 1 n+1 2 Γ 0 = 1 ɛ (u 2 u 2 n, u 1 u 1 n+1 ) Γ0, (5.9) 13

and σ 2 σ 2 n+1 2 0 + λ 2 (σ 2 σ n+1 2 ) + (σ 2 σ n+1 2 ) 2 h,b 2α(D(u 2 u n+1 2 ), σ 2 σ n+1 2 ) + λ σ 2 σ n+1 2 2 h,b,γ = λ (g a (σ 2 σ n 2, b), σ 2 σ n 2 ) + λ < σ 1 σ n 1, σ 2 σ n+1 2 > h,b,γ, (5.10) (σ 2 σ 2 n+1, D(u 2 u 2 n+1 )) + 2(1 α) D(u 2 u 2 n+1 ) 2 0 + 1 ɛ u 2 u 2 n+1 2 Γ 0 = 1 ɛ (u 1 u 1 n, u 2 u 2 n+1 ) Γ0. (5.11) Multplyng (5.9), (5.11) by 2α and addng all equatons together, we have σ σ n+1 2 0 + λ 2 (σ σ n+1 ) + (σ σ n+1 ) 2 h,b +4α(1 α) D(u u n+1 ) 2 0 + 2α ɛ u u n+1 2 Γ 0 +λ σ 1 σ n+1 1 2 +λ σ h,b,γ 2 σ n+1 2 2 h,b,γ 2λMd σ σ n 0 σ σ n+1 0 +λ < σ 2 σ 2 n, σ 1 σ 1 n+1 > h,b,γ + < σ 1 σ 1 n, σ 2 σ 2 n+1 > h,b,γ + 2α ɛ (u 2 u 2 n, u 1 u 1 n+1 ) Γ0 + 2α ɛ (u 1 u 1 n, u 2 u 2 n+1 ) Γ0 λmd( σ σ n 2 0 + σ σ n+1 2 0) + λ σ 2 σ n 2 2 + σ 2 h,b,γ 1 σ n+1 1 2 h,b,γ + σ 1 σ n 1 h,b,γ + σ 2 σ n+1 2 2 h,b,γ + α u2 u n 2 2 Γ ɛ 0 + u 1 u n+1 1 2 Γ 0 + u 1 u n 1 2 Γ 0 + u 2 u n+1 2 2 Γ 0, (5.) by usng (3.32), (3.34), (3.45), and the Young s nequalty. Usng (5.7), (5.) becomes ( (1 λmd) σ σ n+1 2 0 + 4 C 2 α(1 α) + α ) u u n+1 2 Γ0 ɛ + λ σ 1 σ n+1 1 2 + σ 2 h,b,γ 2 σ n+1 2 2 h,b,γ λmd σ σ n 2 0 + λ 2 σ 2 σ n 2 2 + σ h,b,γ 1 σ n 1 2 + α h,b,γ ɛ u u n 2 Γ 0. As Γ, Γ Γ 0, usng the nverse estmate Ch 1/2 σ h 0,Γ0 σ h Ω (5.13) 14

we obtan ( (1 λmd) σ σ n+1 2 0 + 4 C 2 α(1 α) + α ) u u n+1 2 Γ0 ɛ 1=2 ( λmd + λm ) σ σ n 20 + αɛ 2h u u n 2Γ0. Note that α ɛ 4 Cα(1 α) + α ɛ Hence (σ n+1, u n+1 ) converges, f λmd+ λ M λm 2h < 1λMd and λmd+ 2h 4 C 2 α(1α) < α ɛ < 1λMd. In order to prove convergence of pressure, subtract (5.2), (5.5) from (3.26), (3.29), respectvely. < 1. (σ σ n+1, D(v )) + 2(1 α)(d(u u n+1 ), D(v )) (p p n+1, v ) + 1 ɛ (u u n+1, v ) Γ0 = 1 ɛ (u j u j n, v ) Γ0, (5.14) where (, j) = (1, 2) or (2, 1). Usng the nf-sup condton (3.24), (5.7) and (5.14), we have Ths mples p p n+1 0 C sup 0 v h Xh C sup 0 v h Xh p p n+1 0 C (p p n+1, v h) v h 1 1 v h σ σ n+1 0 + 2(1 α) D(u u n+1 ) 0 1 + 1 Cɛ ( D(u u n+1 ) 0 + D(u j u j n ) 0 ) D(v ). σ σ n+1 0 + D(u u n+1 ) 0 + D(u u n ) 0. Therefore, convergence of p n+1 follows from convergence of (σ n+1, u n+1 ). As an alternate scheme, we may consder the slghtly modfed teratve algorthm. Algorthm 5.3. (σ n+1 1, τ h ) + λ((b )σ n+1 1, τ h ) + λ < σ n+1 + 1 σ n+1 1, τ h + > h,b +λ (g a (σ n+1 1, b), τ h ) 2α(D(u n+1 1 ), τ h ) + λ < σ n+1 1, τ h > h,b,γ = λ < σ n 2, τ h > h,b,γ τ h Σ h 1, (σ 1 n+1, D(v h )) + 2(1 α)(d(u 1 n+1 ), D(v h )) (p 1 n+1, v h ) + 1 ɛ (u 1 n+1, v h ) Γ0 (q h, u 1 n+1 ) = 0 q S h 1, = (f, v h ) + 1 ɛ (u 2 n, v h ) Γ0 v h X h 1, 15

and (σ n+1 2, τ h ) + λ((b )σ n+1 2, τ h ) + λ < σ n+1 + 2 σ n+1 2, τ h + > h,b +λ (g a (σ n+1 2, b), τ h ) 2α(D(u n+1 2 ), τ h ) + λ < σ n+1 2, τ h > h,b,γ = λ < σ n 1, τ h > h,b,γ τ h Σ h 2, (σ 2 n+1, D(v h )) + 2(1 α)(d(u 2 n+1 ), D(v h )) (p 2 n+1, v h ) + 1 ɛ (u 2 n+1, v h ) Γ0 = (f, v h ) + 1 ɛ (u 1 n, v h ) Γ0 v h X h 2, (q h, u h 2 n+1 ) = 0 q S h 2. Note that the dfference between two algorthms les n the g a terms of the consttutve equatons. Convergence of Algorthm 5.3 can be shown under stronger condtons on parameters λm and α; we need the assumptons λm < 2C 2 h(1 2λMd) and λm 2 4 C 2 α(1 α) < α ɛ < C2 h(1 2λMd), where C s the constant appears n the nverse nequalty (5.13). The requrement on α s more restrctve than the condton n the Theorem 5.2 snce ɛ s expected to be Ch 2 as stated n Remark 4.3. However, t turned out that the condtons for convergence are not necessary n numercal tests and the convergence rate of each algorthm s strongly dependent on ɛ. The convergence ssue wll be addressed n the next secton.. 6 Prelmnary Numercal Results In ths secton we present numercal results obtaned usng Algorthm 5.3. The example s a nonphyscal problem wth doman Ω = 0, 1 0, 1 and a specfed soluton. The functon b was chosen to be the exact soluton u. The rght hand sde functons n (2.5)-(2.8) were approprately gven so that the exact soluton was sn(πx)y(y 1) u = sn(x) (x 1)y cos(πy/2) p = cos(2πx) y(y 1) σ = 2α D(u) The parameters λ, α and a n the equatons were chosen as 1, 0.5 and 0, respectvely. We used the Taylor-Hood element for (u, p) and dscontnuous lnear polynomals for σ. Although we assumed that dv u = 0 for analyss, t turns out that the dv free condton s not necessary for the numercal experments. Note that the example we chose has the homogeneous boundary condton on Ω, but dv u 0. The doman Ω was dvded nto two subdoamns Ω 1 = 0, 1/2 0, 1 and Ω 2 = 1/2, 1 0, 1 For comparson of accuracy, the soluton to (4.1)-(4.3) was frst computed n each subdoman usng the exact Neumann and Drchlet boundary data on the nterface Γ 0. See Table 1 for errors and convergence rates. Then we computed solutons by Algorthm 5.1 usng dfferent values of ɛ. Errors for u and σ are presented n Table 2. Frst, note that the rates presented n Table 1 are better than the theoretcal result n (2.18), whch s not unusual n smulaton of vscoelastc flud flows wth a known exact soluton 5, 6,. Although the scalng ɛ = Ch 2 s suggested for an optmal error estmate n (4.7), the actual optmal scalng may be ɛ = Ch 4 based on the computed rates n Table 1, and ths s partally verfed by results n Table 2. For example, for h 0 = 1/4, ɛ 0 should be as small as 1/80 so that errors are smlar to those errors n Table 1. When h s reduced by half (h 1 = 1/8), the ɛ value should be as small as ɛ 1 = 1/1300 ( 1/16 of ɛ 0 ), as shown n Table 2. Therefore, we. 16

expect ɛ needs to be as small as 1/20480 when h = 1/16, whch we could not check as the maxmum number of teratons were exceeded. h u u h 1 1 rate u u h 2 1 rate σ σ h 1 0 rate σ σ h 2 0 rate 1/4 3.337 10 2 3.083 10 2 1.862 10 2 2.206 10 2 1/8 7.629 10 3 2.1 7.232 10 3 2.1 4.570 10 3 2.0 5.065 10 3 2.1 1/16 1.836 10 3 2.1 1.776 10 3 2.0 1.187 10 3 1.9 1.265 10 3 2.0 Table 1: Errors by exact nterface data h ɛ ter. no. u u h 1 1 u u h 2 1 σ σ h 1 0 σ σ h 2 0 1/4 1/20 13 4.778 10 2 4.580 10 2 2.558 10 2 2.401 10 2 1/40 26 3.798 10 2 3.5 10 2 2.177 10 2 2.272 10 2 1/60 38 3.497 10 2 3.200 10 2 2.059 10 2 2.236 10 2 1/80 48 3.391 10 2 3.094 10 2 2.009 10 2 2.2 10 2 1/100 54 3.398 10 2 3.084 10 2 1.985 10 2 2.5 10 2 1/150 77 3.387 10 2 3.087 10 2 1.960 10 2 2.1 10 2 1/200 101 3.392 10 2 3.109 10 2 1.952 10 2 2.2 10 2 1/500 257 3.398 10 2 3.179 10 2 1.946 10 2 2.6 10 2 1/8 1/20 14 4.053 10 2 4.208 10 2 1.999 10 2 1.130 10 2 1/60 34 2.184 10 2 2.090 10 2 1.018 10 2 6.999 10 3 1/100 66 1.531 10 2 1.458 10 2 7.537 10 3 5.994 10 3 1/200 134 1.089 10 2 9.982 10 3 5.675 10 3 5.391 10 3 1/400 298 8.492 10 3 7.796 10 3 4.924 10 3 5.162 10 3 1/500 404 7.996 10 3 7.4 10 3 4.8 10 3 5.7 10 3 1/1000 704 7.997 10 3 7.282 10 3 4.687 10 3 5.103 10 3 1/1300 973 7.696 10 3 7.2 10 3 4.655 10 3 5.090 10 3 1/16 1/10000 8000 2.414 10 3 2.063 10 3 1.238 10 3 1.297 10 3 1/20000 15000 2.830 10 3 2.310 10 3 1.274 10 3 1.322 10 3 Table 2: Errors by varous ɛ values 7 Conclusons and Future Work We nvestgated a non-overlappng doman decomposton algorthm for the lnearzed vscoelastc flud equatons. The doman decomposton algorthm s presented as a dscrete varatonal formulaton for whch we analyzed for the exstence and unqueness of a soluton. Convergence of the doman decomposton soluton both wth respect to the grd sze h and wth respect to the parameter ɛ was also proved. We also presented some prelmnary computatonal results. However, to make the method more practcal, further studes are needed, mostly n the realm of effcent mplementaton. For example, convergence of algorthm 5.1 could be accelerated by ntroducng a relaxaton parameter n the boundary ntegrals of the momentum equatons as shown n 18. A smlar technque may be effectve n handlng stress boundary condtons on the nterface. We wll also study doman decomposton methods desgned for vscoelastc flud flows wth other type of consttutve equatons, for nstance, fluds havng a power law consttutve equaton. References 1 F.P.T. Baajens, Mxed fnte element methods for vscoelastc flow analyss: A revew, J. Non- Newtonan Flud Mech. 79 (1998), pp. 361-385. 17

2 J. Baranger and D. Sandr, Fnte element approxmaton of vscoelastc flud flow: Exstence of approxmate solutons and error bounds, Numer. Math. 63 (1992), pp. 13-27. 3 S. Brenner and L. Scott, The mathematcal theory of fnte element methods, Sprnger-Verlag, 1994. 4 A. E. Caola, Y. L. Joo, R. C. Armstrong and R. A. Brown, Hghly parallel tme ntegraton of vscoelastc flows, J. Non-Newtonan Flud Mech. 100, pp. 191-6, 2001. 5 V.J. Ervn, H. Lee, and L.N. Ntasn, Analyss of the Oseen-vscoelastc flud flow problem, to appear n J. Non-Newtonan Flud Mech. 6 V.J. Ervn and H. Lee, Defect correcton method for vscoelastc flud flows at hgh Wessenberg number, Numercal Methods for PDEs, 22 (2006), pp. 145-164. 7 M. Fortn and A. Fortn, A new fnte approach for the F.E.M. smulaton of vscoelastc flows, J. Non-Newtonan Flud Mech. 32 (1989), pp. 295-310. 8 F. Gastald and L. Gastald and A. Quarteron, Adaptve doman decomposton methods for advecton domnated equaton, East-West J. Numer. Math., 4 (1996), pp. 165 206. 9 V. Grault and P. Ravart, Fnte element methods for Naver-Stokes equatons, Sprnger, Berln, 1986. 10 M. Gunzburger and H. Lee, An optmzaton based doman decomposton method for the Naver-Stokes equatons, SIAM J. Num. Anal. 37 (2000), pp. 1455-1480. 11 P. Houston, C. Schwab and E. Sddotul, Dscontnuous hp-fnte element methods for advectvedffuson-reacton problems, SIAM J. Numer. Anal. 39 (2002), pp. 33-63. H. Lee, A multgrd method for vscoelastc flud flow, SIAM J. Num. Anal. 42 (2004), pp. 109-9. 13 G. Lube and L. Muller and F.C. Otto,A non-overlappng doman decomposton method for stablzed fnte element approxmatons of the Oseen equatons, J. Comput. Appl. Math. 132 (2001), pp. 1 236. 14 J.M. Marchal and M.J. Crochet, A new fnte element for calculatng vscoelastc flow, J. Non- Newtonan Flud Mech. 26 (1987), pp. 77-114. 15 A. Quarteron and A. Vall, Doman Decomposton Methods for Partal Dfferental Equatons, Oxford Scence Publcatons, 1999. 16 D.G. Radu, M. Normandn and J.R. Clearmont, A numercal approach for computng flows by local transformatons and doman decomposton usng an optmzaton algorthm, Comput. Mehtods Appl. Mech. Engrg. 191 (2002), pp. 4401-4419. 17 T.C. Rebollo and E.C. Vera, Study of a non-overlappng doman decomposton method: Posson and Stokes problems, Appl. Numer. Math., 48, (2004), pp. 169 194. 18 T.C. Rebollo and E.C. Vera, Study of a non-overlappng doman decomposton method: Steady Naver-Stokes equatons, Appl. Numer. Math., 55, (2005), pp. 100 4. 19 M. Renardy, Exstence of slow steady flows of vscoelastc fluds wth dfferental consttutve equatons, Z.A.M.M. 65 (1985), pp. 449-451. 18

20 D. Sandr, Fnte element approxmaton of vscoelastc flud flow: Exstence of approxmate solutons and error bounds. Contnuous approxmaton of the stress, SIAM J. Numer. Anal. 31 (1994), pp. 362-377. C. Schwab, p- and hp- Fnte element methods, Theory and applcatons to sold and flud mechancs, Oxford Unversty Press, 1998. 22 B. Smth and P. Bjørstad and W. Gropp, Doman Decomposton: Parallel Multlevel Methods for Ellptc Partal Dfferental Equatons, Cambrdge Unversty Press, 1996. 23 A. Souvalots and A.N. Bers, Spectral colloacaton/doman decomposton method for vscoelastc flow smulatons n model porous geometres, Comput. Mehtods Appl. Mech. Engrg. 9 (1996), pp. 9-28. 19