A domain decomposition method for the Oseen-viscoelastic flow equations

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Applied Mathematics and Computation 195 (2008) 127–141

www.elsevier.com/locate/amc

A domain decomposition method for the Oseen-viscoelasticflow equations q

Eleanor Jenkins, Hyesuk Lee *

Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA

Abstract

We study a non-overlapping domain decomposition method for the Oseen-viscoelastic flow problem. The data on theinterface are transported through Newmann and Dirichlet boundary conditions for the momentum and constitutive equa-tions, respectively. The discrete variational formulations of subproblems are presented and investigated for the existence ofsolutions. We show convergence of the domain decomposition solution to a solution of the one-domain problem. Conver-gence of an iterative algorithm and some numerical results are also presented.� 2007 Elsevier Inc. All rights reserved.

Keywords: Domain decomposition; Viscoelastic flows; Finite element method

1. Introduction

In this paper we investigate a domain decomposition method for the Oseen-viscoelastic fluid flow of John-son–Segalman type. Viscoelastic flow equations consist of a constitutive equation for the material, a momen-tum equation, and a mass equation. For Newtonian fluids, the equations for stationary, incompressible,creeping flow can be simplified in terms of velocity and pressure because of the simplistic relationship betweenstress and velocity. For non-Newtonian fluids, however, there is an ‘‘extra’’ stress component that accounts forthe forces that the material develops under deformation. Thus, no simplification takes place, and the completesystem consists of three equations in the unknowns pressure (scalar), velocity (vector), and stress (symmetrictensor). One of the difficulties in simulating viscoelastic flows arises from the hyperbolic nature of the consti-tutive equation for which one needs to use a stabilization technique such as the streamline upwinding Petrov–Galerkin (SUPG) method and the discontinuous Galerkin method [1].

Here we consider the discontinuous Galerkin method for the finite element approximation of stress. Whendiscretized using a discontinuous space to approximate the stress tensor, the size of the discrete system isincreased considerably. In order to handle the large number of unknowns associated with the viscoelastic

0096-3003/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2007.04.086

q This work was partially supported by the NSF under Grant No. DMS-0410792.* Corresponding author.

E-mail addresses: [email protected] (E. Jenkins), [email protected] (H. Lee).

mailto:[email protected]

mailto:[email protected]

128 E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141

systems, we propose a domain decomposition algorithm for a two-subdomain problem. An extension to amuti-domain problem is straightforward. In each subdomain we impose a Neumann condition for themomentum equation and a Dirichlet condition for the constitutive equation on the interface boundary inorder to transport data. The idea of using a Neumann type condition is based on the methods found in[15,17,18]. For the finite element approximation, we use the standard Taylor–Hood element for velocityand pressure and the discontinuous Galerkin method for stress.

Domain decomposition methods have been used extensively to solve elliptic partial differential equations(see [22] and references therein), along with the Stokes and Navier–Stokes equations [8,10,13,15,17]. Somedomain decomposition methods for viscoelastic fluids are found in [4,16,23]. In [4] the authors use the additiveSchwarz method in conjunction with the DEVSS-G operator splitting method. The two-level Schwarz methodis applied to a Stokes-like problem that results from the splitting. A domain decomposition spectral colloca-tion (DDSC) method is introduced in [23] for an Oldroyd-B fluid in model porous geometries. The methoddiscussed in [16] is based on the concepts of streamlines and local transformation functions, where the localfunctions are the primary unknowns. The functions map the physical subdomains to transformed domainswhere the mapped streamlines are parallel straight lines.

The remainder of the document is organized as follows. In Section 2, we introduce the model equations forthe Oseen-viscoelastic flow problem, and in Section 3, we describe the model equations for the problemdecomposed into two subdomains. Convergence of a domain decomposition solution to a solution of theone-domain problem is discussed in Section 4. In Section 5, we present iterative algorithms and prove conver-gence of the algorithms. We provide numerical results in Section 6, and conclusions and future work are givenin Section 7.

2. Model equations and finite element approximation

Let X be a bounded domain in Rd with the Lipschitz continuous boundary oX. Consider the Johnson–Segalman problem

rþ kðu � rÞrþ kgaðr;ruÞ � 2aDðuÞ ¼ 0 in X; ð2:1Þ� r � r� 2ð1� aÞr � DðuÞ þ rp ¼ f in X; ð2:2Þdivu ¼ 0 in X; ð2:3Þ

where r denotes the polymeric stress tensor, u the velocity vector, p the pressure of fluid, and k is the Weiss-enberg number defined as the product of the relaxation time and a characteristic strain rate. Assume that p haszero mean value over X. In (2.1) and (2.2), DðuÞ :¼ ðruþruTÞ=2 is the rate of the strain tensor, a a numbersuch that 0 < a < 1 which may be considered as the fraction of viscoelastic viscosity, and f the body force. In(2.1), gaðr;ruÞ is defined by

gaðr;ruÞ :¼ 1� a2ðrruþruTrÞ � 1þ a

2ðrurþ rruTÞ ð2:4Þ

for a 2 ½�1; 1�.We use the Sobolev spaces W m;pðDÞ with norms kÆkm,p,D if p <1, kÆkm,1,D if p =1. We denote the Sobolev

space Wm,2 by Hm with the norm kÆkm. The corresponding space of vector-valued or tensor-valued functions isdenoted by Hm. If D = X, D is omitted, i.e., ð�; �Þ ¼ ð�; �ÞX and kÆk = kÆkX. Existence of a solution to the prob-lem (2.1)–(2.3) with the homogeneous boundary condition for u has been documented by Renardy [19] withthe small data condition: if f is sufficiently regular and small, the problem admits a unique bounded solutionðu; p; rÞ 2 H3ðXÞ � H 2ðXÞ �H2ðXÞ.

In this paper, the constitutive equation (2.1) is simplified to define our domain decomposition problem. Wewill consider the linear Oseen problem with the given velocity b(x):

rþ kðb � rÞrþ kgaðr;rbÞ � 2aDðuÞ ¼ 0 in X; ð2:5Þ� r � r� 2ð1� aÞr �DðuÞ þ rp ¼ f in X; ð2:6Þdivu ¼ 0 in X; ð2:7Þu ¼ 0 on oX: ð2:8Þ

E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141 129

We make the following assumption for b:

b 2 H10ðXÞ; r � b ¼ 0; kbk1 6 M ; krbk1 6 M <1:

Define the function spaces for the velocity u, the pressure p and the stress r, respectively:

X :¼ H10ðXÞ;

S :¼ L20ðXÞ ¼ q 2 L2ðXÞ :

ZX

qdX ¼ 0

� �;

R :¼ s 2 L2ðXÞ : sij ¼ sji; ðb � rÞs 2 L2ðXÞ� �

:

Note that the velocity and pressure spaces, X and S, satisfy the inf-sup condition

infq2S

supv2X

ðq;r � vÞkvk1kqk0

P C: ð2:9Þ

The corresponding weak formulation is then given by:

ðr; sÞ þ kððb � rÞr; sÞ þ kðgaðr;rbÞ; sÞ � 2aðDðuÞ; sÞ ¼ 0 8s 2 R; ð2:10Þðr;DðvÞÞ þ 2ð1� aÞðDðuÞ;DðvÞÞ � ðp;r � vÞ ¼ ðf; vÞ 8v 2 X; ð2:11Þðq;r � uÞ ¼ 0 8q 2 S: ð2:12Þ

In the weak divergence free space

V :¼ v 2 X :

ZX

qdivvdX ¼ 0 8q 2 L20ðXÞ

� �;

the weak formulation (2.10)–(2.12) is equivalent to:

ðr; sÞ þ kððb � rÞr; sÞ þ kðgaðr;rbÞ; sÞ � 2aðDðuÞ; sÞ ¼ 0 8s 2 R; ð2:13Þðr; dðvÞÞ þ 2ð1� aÞðDðuÞ;DðvÞÞ ¼ ðf; vÞ 8v 2 V: ð2:14Þ

We now consider the finite element approximation of the problem (2.10)–(2.12). Suppose Th is a triangu-lation of X such that X ¼ f[K : K 2 T hg. Assume that there exist positive constants c1, c2 such that

c1h 6 hK 6 c2qK ;

where hK is the diameter of K, qK is the diameter of the greatest ball included in K, and h ¼ maxK2T h hK .Due to the hyperbolic nature of the constitutive equation, a stabilization technique is needed for the finite

element simulation of viscoelastic flows. Streamline upwinding [14,20] and the discontinuous Galerkin method[2,7] are the commonly used discretization techniques to handle this problem. We use the discontinuous Galer-kin method for approximating the stress. Let Pk(K) denote the space of polynomials of degree less than orequal to k on K 2 Th. Then we define finite element spaces for the approximation of ðu; pÞ:

Xh :¼ v 2 X \ ðC0ðXÞÞd : vjK 2 P 2ðKÞd 8K 2 T hn o

;

Sh :¼ q 2 S \ C0ðXÞ : qjK 2 P 1ðKÞ 8K 2 T h� �

;

Vh :¼ v 2 Xh : ðq;r � vÞ ¼ 0 8q 2 Sh� �

:

The stress r is approximated in the discontinuous finite element space of piecewise linears:

Rh :¼ s 2 R : sjK 2 P 1ðKÞd�d 8K 2 T hn o

:

We introduce some notation below in order to analyze an approximate solution by the discontinuousGalerkin method. We define

oK�ðbÞ :¼ fx 2 oK; b � n < 0g;
where oK is the boundary of K and n is outward unit normal, and
s�ðbÞ :¼ lim�!0

sðx� �bðxÞÞ:


We also define

ðr; sÞh :¼XK2T h

ðr; sÞK ;

hr�; s�ih;b :¼XK2T h

ZoK�ðbÞ

ðr�ðbÞ; s�ðbÞÞjn � bjds;

ksk0;Ch :¼XK2T h

jsj20;oK

!1=2

for r; s 2Q

K2T hðL2ðKÞÞd�d , and

kskm;h :¼XK2T h

jsj2m;K

!1=2

for s 2Q

K2T hðW m;2ðKÞÞd�d , if m <1.The discontinuous Galerkin finite element approximation of (2.10)–(2.12) is then as follows: find uh 2 Xh,

ph 2 Sh, rh 2 Rh such that

ðrh; shÞ þ k ðb � rÞrh; sh� �

hþ rhþ � rh� ; shþD E

h;b

� þ kðgaðrh;rbhÞ; shÞ � 2aðdðuhÞ; shÞ ¼ 0 8sh 2 Rh;

ð2:15Þðrh; dðvhÞÞ þ 2ð1� aÞðdðuhÞ; dðvhÞÞ � ðph;r � vhÞ ¼ ðf; vhÞ 8vh 2 Xh; ð2:16Þðqh;r � uhÞ ¼ 0 8qh 2 Sh: ð2:17Þ

It was shown in [6] that the discrete problem (2.15)–(2.17) has a unique solution if 1 � 2kMd > 0, and the solu-tion satisfies the error estimate

kr� rhk0 þ ku� uhk1 6 Ch: ð2:18Þ

3. Description of the domain decomposition method

We describe the domain decomposition method using two subdomains, and the analysis that follows isbased on this problem. However, there are no difficulties associated with extending the analysis to handleas many subdomains as necessary.

Let X be divided into two disjoint subdomains X1 and X2 such that X ¼ X1 [ X2. The interface between thetwo domains is denoted by C0 so that C0 ¼ X1 \ X2. Define

C�12 :¼ fx 2 C0 : b � n1 < 0g; C�21 :¼ fx 2 C0 : b � n2 < 0g;
where ni is the outward unit normal vector to Xi for i = 1,2. Let C1 ¼ X1 \ oX and C2 ¼ X2 \ oX.
We consider the following pair of Oseen-viscoelastic systems with mixed boundary conditions:

r1 þ kðb � rÞr1 þ kgaðr1;rbÞ � 2aDðu1Þ ¼ 0 in X1; ð3:1Þ� r � r1 � 2ð1� aÞr �Dðu1Þ þ rp1 ¼ f in X1; ð3:2Þdivu1 ¼ 0 in X1; ð3:3Þu1 ¼ 0 on C1; ð3:4Þ

ðr1 þ 2ð1� aÞDðu1Þ � p1IÞ � n1 ¼ �1

�ðu1 � u2Þ on C0; ð3:5Þ

r1 ¼ r2 on C�12; ð3:6Þ

and

r2 þ kðb � rÞr2 þ kgaðr2;rbÞ � 2aDðu2Þ ¼ 0 in X2; ð3:7Þ


�r � r1 � 2ð1� aÞr �Dðu2Þ þ rp2 ¼ f in X2; ð3:8Þdivu2 ¼ 0 in X2; ð3:9Þu2 ¼ 0 on C2; ð3:10Þ

ðr2 þ 2ð1� aÞDðu2Þ � p2IÞ � n2 ¼1

�ðu1 � u2Þ on C0; ð3:11Þ

r2 ¼ r1 on C�21: ð3:12Þ

Note that (3.6), the Dirichlet boundary conditions for r1, is imposed on a part of the interface C0 whereb Æ n1 < 0. Similarly, (3.12) is imposed on the inflow boundary of X2 associated with the given velocity field b.

Define the function spaces for the velocity ui, the pressure pi and the stress ri, respectively, for i = 1,2:

Xi :¼ fv 2 H1ðXiÞ : v ¼ 0 on Cig;Si :¼ L2ðXiÞ;

Ri :¼ ðL2ðXiÞÞ2�2 \ s ¼ ðsijÞ : sij ¼ sji; b � rs 2 ðL2ðXiÞÞ2�2n o

;

and the div free space

Vi :¼ v 2 Xi :

ZXi

q divvdXi ¼ 0 8q 2 Si

� �:

The corresponding weak formulation of (3.1)–(3.12) is then given by

ðr1; s1Þ þ kðb � rÞðr1; s1Þ þ kðgaðr1;rbÞ; s1Þ � 2aðDðu1Þ; s1Þ ¼ 0 8s1 2 R1; ð3:13Þ

ðr1;Dðv1ÞÞ þ 2ð1� aÞðDðu1Þ;Dðv1ÞÞ � ðp1;r � v1Þ þ1

�ðu1 � u2; v1ÞC0

¼ ðf; v1Þ 8v1 2 X1; ð3:14Þ

ðq1;r � u1Þ ¼ 0 8q1 2 S1; ð3:15Þr1 ¼ r2 on C�12; ð3:16Þ

and

ðr2; s2Þ þ kðb � rÞðr2; s2Þ þ kðgaðr2;rbÞ; s2Þ � 2aðDðu2Þ; s2Þ ¼ 0 8s2 2 R2; ð3:17Þ

ðr2;Dðv2ÞÞ þ 2ð1� aÞðDðu2Þ;Dðv2ÞÞ � ðp2;r � v2Þ �1

�ðu1 � u2; v2ÞC0

¼ ðf; v2Þ 8v2 2 X2; ð3:18Þ

ðq2;r � u2Þ ¼ 0 8q2 2 S2; ð3:19Þr2 ¼ r1 on C�21: ð3:20Þ

We define discrete subspaces for the finite element approximation of the two-domain problem (3.13)–(3.20).Let T h

i denote a triangulation of Xi such that Xi ¼ f[K : K 2 T hi g for i = 1,2. Define finite element spaces for

the approximation of ðrhi ; u

hi ; p

hi Þ for i = 1,2:

Rhi :¼ fs 2 Ri : sjK 2 P 1ðKÞd�d 8K 2 T h

i g;Xh

i :¼ fv 2 Xi \ ðC0ðXiÞÞd : vjK 2 P 2ðKÞd 8K 2 T hi g;

Shi :¼ fq 2 Si \ C0ðXiÞ : qjK 2 P 1ðKÞ 8K 2 T h

i g:

We also define the discrete divergence free subspace

Vhi :¼ fv 2 Xh

i : ðq;r � vÞ ¼ 0 8q 2 Shi g:

The finite element spaces defined above satisfy the standard approximation properties (see [3] or [9]), i.e.,there exist an integer k and a constant C such that

infvh2Xh

i

kv� vhk1 6 Ch2kvk3 8v 2 H3ðXiÞ; ð3:21Þ

infqh2Sh

i

kq� qhk0 6 Ch2kqk2 8q 2 H 2ðXiÞ; ð3:22Þ


and

infsh2Rh

i

ks� shk0 6 Ch2ksk2 8s 2 H2ðXiÞ: ð3:23Þ

It is also well known that the Taylor–Hood pair ðXhi ; S

hi Þ satisfies the inf-sup condition:

inf06¼qh2Sh

i

sup06¼vh2Xh

i

ðqh;r � vhÞkvhk1kqhk0

P C; ð3:24Þ

where C is a positive constant independent of h.In addition to notation introduced in the previous section for the discontinuous Galerkin method, we pres-

ent more notation to analyze rhi on the interface C0. We define, for ri; si 2

QK2T h

iðL2ðKÞÞd�d

r�i ; s�i

�h;b

:¼X

K2T hi ;oK\C0¼;

ZoK�ðbÞ

ðr�i ðbÞ : s�i ðbÞÞjn � bjds;

hhr�i iih;b :¼ r�i ; r�i

�1=2

h;b;

ksik0;Chi

:¼XK2T h

i

jsij20;oK

0@ 1A1=2

;

and

ksikm;h :¼XK2T h

i

jsij2m;K

0@ 1A1=2

for si 2Q

K2T hiðW m;2ðKÞÞd�d , if m <1. Also define, for i, j = 1,2

r�i ; s�j

D Eh;b;C�lm

:¼X

K2T hl ;oK\C�lm 6¼;

ZoK�ðbÞ

ðr�i ðbÞ : s�j ðbÞÞjn � bjds;

hhr�i iih;b;C�lm:¼ r�i ;r

�i

�1=2

h;b;C�lm;

where (l,m) = (1, 2) or (2,1). The discrete variational formulations of (3.13)–(3.16), (3.18)–(3.20) are then givenby

ðrh1; s

h1Þ þ kððb � rÞrh

1; sh1Þh þ kðgaðrh

1;rbÞ; sh1Þ � 2aðDðuh

1Þ; sh1Þ

þ k rhþ

1 � rh�

1 ; shþ

1

D Eh;bþ k rh

1 � rh2; s

h1

�h;b;C12�

¼ 0 8sh1 2 Rh

1; ð3:25Þ

ðrh1;Dðvh

1ÞÞ þ 2ð1� aÞðDðuh1Þ;Dðvh

1ÞÞ � ðph1;r � vh

1Þ þ1

�ðuh

1 � uh2; v

h1ÞC0¼ ðf; vh

1Þ 8vh1 2 Xh

1; ð3:26Þ

ðqh1;r � uh

1Þ ¼ 0 8q1 2 Sh1ðXÞ; ð3:27Þ

and

ðrh2; s

h2Þ þ kððb � rÞrh

2; sh2Þh þ kðgaðrh

2;rbÞ; sh2Þ � 2aðDðuh

2Þ; sh2Þ

þ k r2hþ � r2h�; shþ

2

D Eh;bþ k rh

2 � rh1; s

h2

�h;b;C�

21

¼ 0 8sh2 2 Rh

2; ð3:28Þ

ðrh2;Dðvh

2ÞÞ þ 2ð1� aÞðDðuh2Þ;Dðvh

2ÞÞ � ðph2;r � vh

2Þ �1

�ðuh

1 � uh2; v

h2ÞC0¼ ðf; vh

2Þ 8vh2 2 Xh

2; ð3:29Þ

ðqh2;r � uh

2Þ ¼ 0 8q2 2 Sh2ðXÞ: ð3:30Þ

Note that the stress boundary conditions (3.16) and (3.20) are imposed weakly [11].


The next theorem shows the existence and uniqueness of a solution to the above coupled system (3.25)–(3.30). In proving the theorem we use the bilinear forms Ai defined on Rh

i � Vhi for i = 1,2 by

Ai rhi ; u

hi

� �; sh

i ; vhi

� �� :¼ ðrh

i ; shi Þ � 2aðDðuh

i Þ; shi Þ þ 2aðrh

i ;Dðvhi ÞÞ þ 4að1� aÞðDðuh

i Þ;Dðvhi ÞÞ

þ kðgaðrhi ;rbÞ; sh

i Þ: ð3:31Þ
Note that Ai is continuous and coercive, if 1 � 2kMd > 0: since
ðgaðrhi ;rbÞ; sh

i Þ 6 2dkrbk1krhk0kshk0 6 2dMkrhk0kshk0; ð3:32Þ
we have
Ai rhi ; u

hi

� �; sh

i ; vhi

� �� 6 krh

i k0kshi k0 þ 2akDðuh

i Þk0kshi k0 þ 2akrh

i k0kDðvhi Þk0

þ 4að1� aÞkDðuhi Þk0kDðvh

i Þk0 þ 2Mdkkrhi k0ksh

i k0

6 C krhi k0 þ kuk1

� �ksh

i k0 þ kvk1

� �6 Ckðrh

i ; uhi ÞkRi�Xh

ikðsh

i ; vhi ÞkRi�Xi

; ð3:33Þ

where kðshi ; v

hi ÞkRi�Xi

is defined as ðkshi k

20 þ kvh

i k21Þ

1=2. Also, using (3.32),

Ai rhi ; u

hi

� �; rh

i ; uhi

� �� ¼ krh

i k20 þ kðgaðrh

i ;rbÞ; rhi Þ þ 4að1� aÞkDðuh

i Þk20

P krhi k

20 � 2kMdkrh

i k20 þ 4að1� aÞkDðuh

i Þk20 P Ckðrh

i ; uhi Þk

2Ri�Xi

; ð3:34Þ

if 1 � 2kMd > 0. In the proof of the next theorem we will also use the inverse estimate [3,9]

krrhi k0;h 6 Ch�1krh

i k0 for rhi 2 Rh

i ; ð3:35Þ

the local inverse inequality [21],

krhi k0;oK 6 Ch�1=2krh

i k0;K for rhi 2 Rh

i ; ð3:36Þ

and the trace theorem [9],

kuik0;Ci[C06 Ckuik1 for ui 2 Xi: ð3:37Þ

Theorem 3.1. The system (3.25)–(3.30) has a unique solution, if 1 � 2kMd > 0.

Proof. Using the div free spaces Vhi for i ¼ 1; 2, the coupled system (3.25)–(3.30) is equivalent to
X2
i¼1

Ai rhi ; u

hi

� �; sh

i ; vhi

� �� þ k b � rð Þrh

i ; shi

� �hþ k rhþ

i � rh�

i ; shþ

i

D Eh;b

� þ k rh

1 � rh2; s

h1

�h;b;C�

12

þ rh2 � rh

1; sh2

�h;b;C�

21

h iþ 1

�uh

1 � uh2; v

h1 � vh

2

� �C0

¼X2

i¼1

F i shi ; v

hi

� �� 8ðsh

i ; vhi Þ 2 Rh

i � Vhi ; ð3:38Þ

where F ið�Þ : Rhi � Vh

i ! R is a functional defined by

F i shi ; v

hi

� �� :¼ 2aðf; vh

i Þ:
It is straight forward to show Fi is bounded:
F i shi ; v

hi

� �� 6 2akfk�1kvhi k1 6 2akfk�1kðsh

i ; vhi ÞkRi�Xi

: ð3:39Þ

We will show the bilinear form in (3.38) is continuous and coercive inQ2

i¼1ðRhi � Vh

i Þ, if 1 � 2kMd > 0.Using (3.35) and (3.36):

ððb � rÞrhi ; s

hi Þh þ rhþ

i � rh�

i ; shþ

i

D Eh;b6 C kbk1krrh

i k0;hkshi k0 þ kbk1krh

i k0;Chiksh

i k0;Chi

h i6 C kbk1ðh�1krh

i k0Þkshi k0 þ kbk1ðh�1=2krh

i k0Þðh�1=2kshi k0Þ

�6 CMh�1krh

i k0kshi k0; ð3:40Þ


and

rh1 � rh

2; sh1

�h;b;C�

12

þ rh2 � rh

1; sh2

�h;b;C�

21

6 C kbk1 krh1k0;C0

þ krh2k0;C0

� �ksh

1k0;C0þ kbk1 krh

2k0;C0þ krh

1k0;C0

� �ksh

2k0;C0

h i6 CM ðh�1=2krh

1k0 þ h�1=2krh2k0Þðh�1=2ksh

1k0Þ

þðh�1=2krh2k0 þ h�1=2krh

1k0Þðh�1=2ksh2k0Þ

�6 CMh�1

X2

i;j¼1

krhi k0ksh

jk0: ð3:41Þ

Also, by (3.37),

ðuh1 � uh

2; vh1 � vh

2ÞC06 C

X2

i;j¼1

kuhi k1kvh

jk1: ð3:42Þ

Hence, by (3.33) and (3.40)–(3.42),
X2
i¼1

Ai rhi ; u

hi

� �; sh

i ; vhi

� �� þ kððb � rÞrh

i ; shi Þh þ k rhþ

i � rh�

i ; shþ

i

D Eh;b

� þ k rh

1 � rh2; s

h1

�h;b;C�

12

þ rh2 � rh

1; sh2

�h;b;C�

21

h iþ 1

�ðuh

1 � uh2; v

h1 � vh

2ÞC0

6 CX2

i¼1

krhi k0 þ kuh

i k1

� �ksh

i k0 þ kvhi k1

� �þMh�1

X2

i;j¼1

krhi k0ksh

i k0

" #

6 ChX2

i;j¼1

krhi k0 þ kuh

i k1

� �ksh

i k0 þ kvhi k1

� �6 Chkðrh

1; uh1; r

h2; u

h2ÞkR1�X1�R2�X2

kðsh1; v

h1; s

h2; v

h2ÞkR1�X1�R2�X2

; ð3:43Þ

where Ch is a constant dependent on h.We now show the coercivity of the bilinear form. First note, using integration by parts, that
X2
i¼1

ððb � rÞrhi ; s

hi Þh þ rhþ

i � rh�

i ; shþ

i

D Eh;b

� þ rh

1 � rh2; s

h1

�h;b;C�

12

þ rh2 � rh

1; sh2

�h;b;C�

21

¼X2

i¼1

� ðb � rÞshi ; r

hi

� �hþ rh�

i ; sh�

i � shþ

i

D Eh;b

� þ rh

2; sh2 � sh

1

�h;b;C�

12

þ rh1; s

h1 � sh

2

�h;b;C�

21

: ð3:44Þ

Hence, if shi ¼ rh

i , we obtain
X2
i¼1

ððb � rÞrhi ; r

hi Þh þ rhþ

i � rh�

i ; rhþ

i

D Eh;b

� þ rh

1 � rh2; r

h1

�h;b;C�

12

þ rh2 � rh

1; rh2

�h;b;C�

21

¼ 1

2

X2

i¼1

hhrhþ

i � rh�

i ii2h;b þ hhrh

1 � rh2ii

2h;b;C�

12þ hhrh

2 � rh1ii

2h;b;C�

21

" #P 0: ð3:45Þ

Now the coercivity of the bilinear form follows from (3.34) and (3.45):
X2
i¼1

Ai rhi ; u

hi

� �; rh

i ; uhi

� �� þ kððb � rÞrh

i ; rhi Þ þ k rhþ

i � rh�

i ; rhþ

i

D Eh;b

� þ k rh

1 � rh2; r

h1

�h;b;C�

12

þ rh2 � rh

1;rh2

�h;b;C�

21

h iþ 1

�ðuh

1 � uh2; u

h1 � uh

2ÞC0

PX2

i¼1

krhi k

20 � 2kMdkrh

i k20 þ 4að1� aÞkDðuh

i Þk20 þ

k2hhrhþ

i � rh�

i ii2h;b

� þ k

2hhrh

1 � rh2ii

2h;b;C�

12þ hhrh

2 � rh1ii

2h;b;C�

21

h iþ 1

�ðuh

1 � uh2Þ

2C0


PX2

i¼1

ð1� 2kMdÞkrhi k

20 þ 4að1� aÞkDðuh

i Þk20

h iP C

X2

i¼1

kðrhi ; u

hi Þk

2Ri�Xi

P Ckðrh1; u

h1; r

h2; u

h2Þk

2R1�X1�R2�X2

: ð3:46Þ

Therefore, by the Lax-Milgram theorem, there exists a unique ðrhi ; u

hi Þ 2 Rh

i � Vhi , if 1 � 2kMd > 0. Finally,

existence of a unique solution ðrhi ; u

hi ; p

hi Þ 2 Rh

i � Xhi � Sh

i follows from the inf-sup condition (3.24). h

4. Convergence analysis

We will prove the solution of the two-domain problem (3.25)–(3.30) converges to the solution ðrex; uex; pexÞsatisfying (2.5)–(2.8). Let rex

i ¼ rexjXi[C0, uex

i ¼ uexjXi[C0and pex

i ¼ pexjXi[C0for i = 1,2. Define

gex :¼ ðrex þ 2ð1� aÞDðuexÞ þ pexIÞ � n1 on C0:

Note that, for i = 1,2, ðrexi ; u

exi ; p

exi Þ satisfy

ðrexi ; siÞ þ kððb � rÞrex

i ; siÞ þ kðgaðrexi ;rbÞ; siÞ � 2aðDðuex

i Þ; siÞ ¼ 0 8si 2 Ri; ð4:1Þðrex

i ;DðviÞÞ þ 2ð1� aÞðDðuexi Þ;DðviÞÞ � ðpex

i ;r � viÞ ¼ ðf; viÞ þ ð�1Þiþ1ðgex; viÞC08vi 2 Xi; ð4:2Þ

ðqi;r � uexi Þ ¼ 0 8qi 2 Si: ð4:3Þ

In the div free spaces Vi, (4.1)–(4.3) is equivalent to
X2
i¼1

Ai rexi ; u

exi

� �; si; við Þ

� �þ kððb � rÞrex

i ; siÞ �

¼ 2aðf; viÞ þ 2aðgex; v1 � v2ÞC0: ð4:4Þ

We introduce some approximation properties (see [3] or [9]), which will be used in order to prove the next the-orem. If ~rh

i 2 Rhi is the orthogonal projection of rex

i on T hi in Ri, and ~uh

i 2 Vhi is defined as the interpolant of uex

i

in Vi, then we have the following standard results. For uex 2 H3(Xi) and rex 2 H2(Xi)

krðuexi � ~uh

i Þk0 6 Ch2kuexi k3; ð4:5Þ

krexi � ~rh

i k0 þ hkrðrexi � ~rh

i Þk0 6 Ch2krexi k2: ð4:6Þ

Theorem 4.1. If 1 � 2kMd > 0 and ðrhi ; u

hi Þ for i = 1,2, satisfy the coupled decomposition problem (3.25)–(3.30),

they converge to the exact solution ðrex; uexÞ as � goes to 0. Furthermore, we have the estimate
X2
i¼1

krexi � rh

i k0 þ kuexi � uh

i k1

�6 Cðhþ

ffiffi�pkgexkC0

Þ: ð4:7Þ

Proof. Subtracting (3.38) from (4.4), and adding and subtracting ~rhi , ~uh

i , imply
X2
i¼1

Ai ~rhi � rh

i ; ~uhi � uh

i

� �; sh

i ; vhi

� �� þ k ðb � rÞ ~rh

i � rhi

� �; sh

i

� �þ k ~rh

i � rhi

� �þ � ~rhi � rh

i

� ��; shþ

i

D Eh;b

þ k ~rh

1 � rh1

� �� ~rh

2 � rh2

� �; sh

1

�h;b;C�

12

þ ~rh2 � rh

2

� �� ~rh

1 � rh1

� �; sh

2

�h;b;C�

21

h i� 2a

�ðuh

1 � uh2; v

h1 � vh

2ÞC0

¼X2

i¼1

Ai ~rhi � rex

i ; ~uhi � uex

i

� �; sh

i ; vhi

� �� þkððb � rÞ ~rh

i � rexi

� �; sh

i Þ þ k ~rhi � rex

i

� �þ � ~rhi � rex

i

� ��; shþ

i

D Eh;b

þ k ~rh

1 � rex1

� �� ~rh

2 � rex2

� �; sh

1

�h;b;C�

12

þ ~rh2 � rex

2

� �� ~rh

1 � rex1

� �; sh

2

�h;b;C�

21

h iþ 2aðgex; vh

1 � vh2ÞC0

8ðshi ; v

hi Þ 2 Rh

i � Vhi ; ð4:8Þ


where we used rex1 ¼ rex

2 on C0 and that rexi is a continuous function. Let sh

i ¼ ~rhi � rh

i , vhi ¼ ~uh

i � uhi . We have a

lower bound of the left-hand side of (4.8) using (3.34) and (3.45):

LHS PX2

i¼1

ð1� 2kMdÞk~rhi � rh

i k20 þ 4að1� aÞ D ~uh

i � uhi

� �� 2

0

h iþ 2a

�kuh

1 � uh2k

2C0: ð4:9Þ

Estimates for the jump terms in the right side of (4.8) are obtained in the similar way shown in (3.40) and(3.41):

~rhi � rex

i

� �þ � ~rhi � rex

i

� ��; ~rh

i � rhþ

i

D Eh;b6 Ckbk1k~rh

i � rexi k0;Ch

ik~rh

i � rhi k0;Ch

i

6 CMh�1k~rhi � rex

i k0k~rhi � rh

i k0; ð4:10Þ

~rh1 � rex

1

� �� ~rh

2 � rex2

� �; ~rh

1 � rh1

�h;b;C�

12

þ ~rh2 � rex

2

� �� ~rh

1 � rex1

� �; ~rh

2 � rh2

�h;b;C�

21

6 Ckbk1 k~rh1 � rex

1 k0;C0þ k~rh

2 � rex2 k0;C0

� �k~rh

1 � rh1k0;C0

þ k~rh2 � rh

2k0;C0

� �6 CMh�1 k~rh

1 � rex1 k0 þ k~rh

2 � rex2 k0

� �k~rh

1 � rh1k0 þ k~rh

2 � rh2k0

� �: ð4:11Þ

Using (3.33) and (4.9)–(4.11) and the Young’s inequality, we have

X2

i¼1

ð1� 2kMdÞk~rhi � rh

i k20 þ 4að1� aÞ D ~uh

i � uhi

� �� 2

0

h iþ 2a

�kuh

1 � uh2k

2C0

6 CX2

i¼1

k~rhi � rex

i k0 þ D ~uhi � uex

i

� �� 0

� �k~rh

i � rhi k0 þ D ~uh

i � uhi

� �� 0

� �hþkM r ~rh

i � rexi

� �� 0k~rh

i � rhi k0 þ kMh�1k~rh

i � rexi k0k~rh

i � rhi k0

iþ CkMh�1

X2

i;j¼1

k~rhi � rex

i k0k~rhj � rh

jk0 þ 2akgexkC0ku1 � u2kC0

6 CX2

i¼1

d1 k~rhi � rh

i k0 þ D ~uhi � uh

i

� �� 0

� �2

þ 1

4d1

k~rhi � rex


i

� �� 0

� �2�

þd2kMk~rhi � rh

i k20 þ

kM4d2

r ~rhi � rex

i

� �� 2

0þ d3kMk~rh

i � rhi k

20 þ

kMh�2

4d3

k~rhi � rex

i k20

þ C

X2

i

d4kMk~rhi � rh

i k20 þ

kMh�2

4d4

k~rhi � rex

i k20

� þ 2a

1

4d5

kuh1 � uh

2k2C0þ d5kgexk2

C0

� ; ð4:12Þ

for arbitrary di > 0, i = 1,2, . . ., 5. If we take d5 ¼ �2,

X2

i¼1

ð1� 2kMd � 2d1C � ðd2 þ d3 þ d4ÞkMCÞk~rhi � rh

i k20

hþð4að1� aÞ � 2Cd1Þ D ~uh

i � uhi

� �� 2

0

iþ a�kuh

1 � uh2k

2C0

6 CX2

i¼1

1

4d1

k~rhi � rex


i

� �� 0

� �2

þ kM4d2

r ~rhi � rex

i

� �� 2

0

�þ kMh�2

4d3

þ kMh�2

4d4

� �k~rh

i � rexi k

20

þ a�kgexk2

C0: ð4:13Þ

Therefore, using (4.5) and (4.6) and the triangle inequality, (4.7) follows. h


Remark 4.2. The pressure on each subdomain, phi , is determined up to a constant. In order to compute a

pressure convergent to pex which has the zero mean value, construct ph in L20ðXÞ by

phðxÞ ¼ ph1 � K for x 2 X1;

ph2 � K for x 2 X2;

(
where K ¼
RX1

ph1 dX1 þ

RX2

ph2 dX2

� �=measðXÞ. Then, convergence of ph to pex can be proved as shown in [17]

or [18].

Remark 4.3. The estimate (4.7) suggests the optimal scaling � = Ch2 between � and h.

5. Iterative algorithm

In this section we delete the superscript h in ðrhi ; u

hi ; p

hi Þ to simplify our notations. Consider the following

iterative algorithm.

Algorithm 5.1

ðrnþ11 ; shÞ þ kððb � rÞrnþ1

1 ; shÞ þ k rnþ1þ

1 � rnþ1�

1 ; shþD E

h;b� 2aðDðunþ1

1 Þ; shÞ þ k rnþ11 ; sh

�h;b;C�

12

¼ �kðgaðrn1;rbÞ; shÞ þ k rn

2; sh

�h;b;C�

12

8sh 2 Rh1; ð5:1Þ

ðrnþ11 ;DðvhÞÞ þ 2ð1� aÞðDðunþ1

1 Þ;DðvhÞÞ � ðpnþ11 ;r � vhÞ þ 1

�ðunþ1

1 ; vhÞC0

¼ ðf; vhÞ þ 1

�ðun

2; vhÞC0

8vh 2 Xh1; ð5:2Þ

ðqh;r � unþ11 Þ ¼ 0 8q 2 Sh

1; ð5:3Þ
and


2 � rnþ1�

2 ; shþD E

h;b� 2aðDðunþ1

2 Þ; shÞ þ k rnþ12 ; sh

�h;b;C�

21

¼ �kðgaðrn2;rbÞ; shÞ þ k rn

1; sh

�h;b;C�

21

8sh 2 Rh2; ð5:4Þ



�ðunþ1

2 ; vhÞC0

¼ ðf; vhÞ þ 1

�ðun

1; vhÞC0

8vh 2 Xh2; ð5:5Þ

ðqh;r � uh2

nþ1Þ ¼ 0 8q 2 Sh2: ð5:6Þ

In the proof of the next theorem we will use the Trace theorem

eCkuik0;C06 kDðuiÞk0: ð5:7Þ

Theorem 5.2. The iterative Algorithm 5.1 converges for each fixed � and h if kM is sufficiently small so that

kMd þ kM2h < 1� kMd and a satisfies kMd þ kM

2h � 4eC2að1� aÞ < a� < 1� kMd, where eC is the constant in (5.7).

Proof. In the discrete div free space V hi , subtracting (5.1), (5.2), (5.4) and (5.5) from (3.25), (3.26), (3.28) and

(3.29), respectively, and letting si ¼ ri � rnþ1i , vi ¼ ui � unþ1

i , we obtain

kr1 � rnþ11 k

20 þ

k2hhðr1 � rnþ1

1 Þþ � ðr1 � rnþ1

1 Þ�ii2h;b � 2aðDðu1 � unþ1

1 Þ; r1 � rnþ11 Þ þ khhr1 � rnþ1

1 ii2h;b;C�

12

¼ �kðgaðr1 � rn1;rbÞ; r1 � rnþ1

1 Þ þ k r2 � rn2; r1 � rnþ1

1

�h;b;C�

12

; ð5:8Þ


ðr1 � rnþ11 ;Dðu1 � unþ1

1 ÞÞ þ 2ð1� aÞkDðu1 � unþ11 Þk

20 þ

1

�ku1 � unþ1

1 k2C0¼ 1

�ðu2 � un

2; u1 � unþ11 ÞC0

; ð5:9Þ

and

kr2 � rnþ12 k

20 þ

k2hhðr2 � rnþ1

2 Þþ � ðr2 � rnþ1

2 Þ�ii2h;b � 2aðDðu2 � unþ1

2 Þ; r2 � rnþ12 Þ þ khhr2 � rnþ1

2 ii2h;b;C�

21

¼ �kðgaðr2 � rn2;rbÞ; r2 � rn

2Þ þ k r1 � rn1; r2 � rnþ1

2

�h;b;C�

21

; ð5:10Þ

ðr2 � rnþ12 ;Dðu2 � unþ1

2 ÞÞ þ 2ð1� aÞkDðu2 � unþ12 Þk

20 þ

1

�ku2 � unþ1

2 k2C0¼ 1

�ðu1 � un

1; u2 � unþ12 ÞC0

: ð5:11Þ

Multiplying (5.9) and (5.11) by 2a and adding all equations together, we have
X2
i¼1

kri � rnþ1i k

20 þ

k2hhðri � rnþ1

i Þþ � ðri � rnþ1

i Þ�ii2h;b

�þ4að1� aÞkDðui � unþ1

i Þk20 þ

2a�kui � unþ1

i k2C0

þ khhr1 � rnþ1

1 ii2h;b;C�

12þ khhr2 � rnþ1

2 ii2h;b;C�

21

6

X2

i¼1

2kMdkri � rni k0kri � rnþ1

i k0 þ k r2 � rn2; r1 � rnþ1

1

�h;b;C�

12

þ r1 � rn1; r2 � rnþ1

2

�h;b;C�

21

h iþ 2a

�ðu2 � un

2; u1 � unþ11 ÞC0

þ 2a�ðu1 � un

1; u2 � unþ12 ÞC0

6

X2

i¼1

kMd kri � rni k

20 þ kri � rnþ1

i k20

� �þ k

2hhr2 � rn

2ii2h;b;C�

12þ hhr1 � rnþ1

1 ii2h;b;C�

12

hþhhr1 � rn

1iih;b;C�21þ hhr2 � rnþ1

2 ii2h;b;C�

21

iþ a�ku2 � un

2k2C0þ ku1 � unþ1

1 k2C0þ ku1 � un

1k2C0þ ku2 � unþ1

2 k2C0

h i; ð5:12Þ

by using (3.32), (3.34), and (3.45), and the Young’s inequality. Using (5.7) and (5.12) becomes
X2
i¼1

ð1� kMdÞkri � rnþ1i k

20 þ 4eC2að1� aÞ þ a

�

� �kui � unþ1

i k2C0

h iþ k

2hhr1 � rnþ1

1 ii2h;b;C�

12þ hhr2 � rnþ1

2 ii2h;b;C�

21

h i6

X2

i¼1

kMdkri � rni k

20 þ

k2hhr2 � rn

2ii2h;b;C�

12þ hhr1 � rn

1ii2h;b;C�

21

h iþ a�

X2

i¼1

kui � uni k

2C0:

As C�12, C�21 � C0, using the inverse estimate

Ch1=2krhi k0;C0

6 krhi kXi

; ð5:13Þ

we obtain
X2
i¼1

ð1� kMdÞkri � rnþ1i k

20 þ 4eC2að1� aÞ þ a

�

� �kui � unþ1

i k2C0

h i6

X2

1¼2

kMd þ kM2h

� �kri � rn

i k20 þ

a�kui � un

i k2C0

� :

Note that
a�
4eCað1� aÞ þ a�

< 1:

Hence ðrnþ1; unþ1Þ converges, if kMd þ kM2h < 1� kMd and kMd þ kM

2h � 4eC2að1� aÞ < a�< 1� kMd.


In order to prove convergence of pressure, subtract (5.2) and (5.5) from (3.26) and (3.29), respectively.

ðri � rnþ1i ;DðviÞÞ þ 2ð1� aÞðDðui � unþ1

i Þ;DðviÞÞ � ðpi � pnþ1i ;r � viÞ þ

1

�ðui � unþ1

i ; viÞC0

¼ 1

�ðuj � un

j ; viÞC0; ð5:14Þ

where (i, j) = (1, 2) or (2,1). Using the inf-sup condition (3.24), (5.7) and (5.14), we have

kpi � pnþ1i k0 6 C sup

06¼vhi 2Xh

i

ðpi � pnþ1i ;r � vh

i Þkvh

i k1

6 C sup06¼vh

i 2Xhi

1

kvhi k1

kri � rnþ1i k0 þ 2ð1� aÞkDðui � unþ1

i Þk0

þ 1eC� kDðui � unþ1

i Þk0 þ kDðuj � unj Þk0

� �ikDðviÞk:

This implies

X2

i¼1

kpi � pnþ1i k0 6 C

X2

i¼1

kri � rnþ1i k0 þ kDðui � unþ1

i Þk0 þ kDðui � uni Þk0

�:

Therefore, convergence of pn+1 follows from convergence of ðrnþ1; unþ1Þ. h

As an alternate scheme, we may consider the slightly modified iterative algorithm.

Algorithm 5.3



1 � rnþ1�

1 ; shþD E

h;bþ kðgaðrnþ1

1 ;rbÞ; shÞ � 2aðDðunþ11 Þ; shÞ

þ k rnþ11 ; sh

�h;b;C�

12

¼ k rn2; s

h �

h;b;C�12

8sh 2 Rh1;



�ðunþ1

1 ; vhÞC0¼ ðf; vhÞ þ 1

�ðun

2; vhÞC0

8vh 2 Xh1;

ðqh;r � unþ11 Þ ¼ 0 8q 2 Sh

1;

and



2 � rnþ1�

2 ; shþD E

h;bþ kðgaðrnþ1

2 ;rbÞ; shÞ � 2aðDðunþ12 Þ; shÞ þ k rnþ1

2 ; sh �

h;b;C�21

¼ k rn1; s

h �

h;b;C�21

8sh 2 Rh2;



�ðunþ1

2 ; vhÞC0¼ ðf; vhÞ þ 1

�ðun

1; vhÞC0

8vh 2 Xh2;

qh;r � uh2

nþ1� �

¼ 0 8q 2 Sh2:

Note that the difference between two algorithms lies in the ga terms of the constitutive equations. Conver-gence of Algorithm 5.3 can be shown under stronger conditions on parameters kM and a; we need the assump-tions kM < 2C2hð1� 2kMdÞ and kM

2� 4eC2að1� aÞ < a

�< C2hð1� 2kMdÞ, where C is the constant appears in

the inverse inequality (5.13). The requirement on a is more restrictive than the condition in Theorem 5.2 since �is expected to be Ch2 as stated in Remark 4.3. However, it turned out that the conditions for convergence arenot necessary in numerical tests and the convergence rate of each algorithm is strongly dependent on �. Theconvergence issue will be addressed in the next section.


6. Preliminary numerical results

In this section we present numerical results obtained using Algorithm 5.3. The example is a non-physicalproblem with domain X ¼ ½0; 1� � ½0; 1� and a specified solution. The function b was chosen to be the exactsolution u. The right-hand side functions in (2.5)–(2.8) were appropriately given so that the exact solution was

TableErrors

h

1/41/81/16

TableErrors

h

1/4

1/8

1/16

u ¼sinðpxÞyðy � 1Þ

sinðxÞðx� 1Þy cosðpy=2Þ

� ;

p ¼ cosð2pxÞyðy � 1Þ;r ¼ 2aDðuÞ:

8>>><>>>:
The parameters k, a and a in the equations were chosen as 1, 0.5 and 0, respectively. We used the Taylor–Hoodelement for ðu; pÞ and discontinuous linear polynomials for r. Although we assumed that divu = 0 for anal-ysis, it turns out that the div free condition is not necessary for the numerical experiments. Note that the exam-ple we chose has the homogeneous boundary condition on oX, but divu 5 0.
The domain X was divided into two subdoamins X1 ¼ ½0; 1=2� � ½0; 1� and X2 ¼ ½1=2; 1� � ½0; 1� For com-parison of accuracy, the solution to (4.1)–(4.3) was first computed in each subdomain using the exact Neu-mann and Dirichlet boundary data on the interface C0. See Table 1 for errors and convergence rates. Thenwe computed solutions by Algorithm 5.1 using different values of �. Errors for u and r are presented in Table2. First, note that the rates presented in Table 1 are better than the theoretical result in (2.18), which is notunusual in simulation of viscoelastic fluid flows with a known exact solution [5,6,12]. Although the scaling� = Ch2 is suggested for an optimal error estimate in (4.7), the actual optimal scaling may be � = Ch4 basedon the computed rates in Table 1, and this is partially verified by results in Table 2. For example, forh0 = 1/4, �0 should be as small as 1/80 so that errors are similar to those errors in Table 1. When h is reducedby half (h1 = 1/8), the � value should be as small as �1 = 1/1300 (�1/16 of �0), as shown in Table 2. Therefore,

1by exact interface data

ku� uh1k1 Rate ku� uh

2k1 Rate kr� rh1k0 Rate kr� rh

2k0 Rate

3.337 · 10�2 3.083 · 10�2 1.862 · 10�2 2.206 · 10�2

7.629 · 10�3 2.1 7.232 · 10�3 2.1 4.570 · 10�3 2.0 5.065 · 10�3 2.11.836 · 10�3 2.1 1.776 · 10�3 2.0 1.187 · 10�3 1.9 1.265 · 10�3 2.0

2by various � values

� Iteration No. ku� uh1k1 ku� uh

2k1 kr� rh1k0 kr� rh

2k0

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 3.512 · 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 2.221 · 10�2

1/100 54 3.398 · 10�2 3.084 · 10�2 1.985 · 10�2 2.215 · 10�2

1/150 77 3.387 · 10�2 3.087 · 10�2 1.960 · 10�2 2.211 · 10�2

1/200 101 3.392 · 10�2 3.109 · 10�2 1.952 · 10�2 2.221 · 10�2

1/500 257 3.398 · 10�2 3.179 · 10�2 1.946 · 10�2 2.216 · 10�2

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 7.412 · 10�3 4.812 · 10�3 5.127 · 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 7.122 · 10�3 4.655 · 10�3 5.090 · 10�3

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


we expect � needs to be as small as 1/20480 when h = 1/16, which we could not check as the maximum numberof iterations were exceeded.

7. Conclusions and future work

We investigated a non-overlapping domain decomposition algorithm for the linearized viscoelastic fluidequations. The domain decomposition algorithm is presented as a discrete variational formulation for whichwe analyzed for the existence and uniqueness of a solution. Convergence of the domain decomposition solu-tion with respect to both the grid size h and the parameter � was also proved. We also presented some preli-minary computational results. However, to make the method more practical, further studies are needed,mostly in the realm of efficient implementation. For example, convergence of Algorithm 5.1 could be acceler-ated by introducing a relaxation parameter in the boundary integrals of the momentum equations as shown in[18]. A similar technique may be effective in handling stress boundary conditions on the interface. We will alsostudy domain decomposition methods designed for viscoelastic fluid flows with other type of constitutiveequations, for instance, fluids having a power law constitutive equation.

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A domain decomposition method for the Oseen-viscoelastic flow equations

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