A domain decomposition method for the Oseen-viscoelastic flow equations
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Transcript of 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).
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, ands�ðbÞ :¼ lim�!0
sðx� �bðxÞÞ:
130 E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141
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
E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141 131
�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
132 E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141
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].
E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141 133
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: sinceðgaðrhi ;rbÞ; sh
i Þ 6 2dkrbk1krhk0kshk0 6 2dMkrhk0kshk0; ð3:32Þ
we haveAi 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
X2i¼1
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
134 E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141
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),
X2i¼1
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
X2i¼1
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
X2i¼1
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):
X2i¼1
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
E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141 135
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
X2i¼1
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
X2i¼1
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
X2i¼1
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
136 E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141
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 k0 þ D ~uhi � uex
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 k0 þ D ~uhi � uex
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
E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141 137
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 ¼RX1
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
andðrnþ12 ; shÞ þ kððb � rÞrnþ1
2 ; shÞ þ k rnþ1þ
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
ðrnþ12 ;DðvhÞÞ þ 2ð1� aÞðDðunþ1
2 Þ;DðvhÞÞ � ðpnþ12 ;r � vhÞ þ 1
�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
138 E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141
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
X2i¼1
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
X2i¼1
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
X2i¼1
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.
E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141 139
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
ðrnþ11 ; shÞ þ kððb � rÞrnþ1
1 ; shÞ þ k rnþ1þ
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;
ðrnþ11 ;DðvhÞÞ þ 2ð1� aÞðDðunþ1
1 Þ;DðvhÞÞ � ðpnþ11 ;r � vhÞ þ 1
�ðunþ1
1 ; vhÞC0¼ ðf; vhÞ þ 1
�ðun
2; vhÞC0
8vh 2 Xh1;
ðqh;r � unþ11 Þ ¼ 0 8q 2 Sh
1;
and
ðrnþ12 ; shÞ þ kððb � rÞrnþ1
2 ; shÞ þ k rnþ1þ
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;
ðrnþ12 ;DðvhÞÞ þ 2ð1� aÞðDðunþ1
2 Þ;DðvhÞÞ � ðpnþ12 ;r � vhÞ þ 1
�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.
140 E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141
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
E. Jenkins, H. Lee / Applied Mathematics and Computation 195 (2008) 127–141 141
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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