penalty finite element method for stokes problem with nonlinear slip boundary conditions

Applied Mathematics and Computation 204 (2008) 216–226

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Penalty finite element method for Stokes problem with nonlinearslip boundary conditions q

Yuan Li *, Kaitai LiDepartment of Mathematics, College of Science, Xi’an Jiaotong University, Xianning West Road 28#, 710049 Xi’an, Shaanxi Province, China

a r t i c l e i n f o

Keywords:Stokes problemNonlinear slip boundary
Variational inequalityPenalty finite element approximationError estimate
0096-3003/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.amc.2008.06.035

q Supported by the Nature Science Foundation of* Corresponding author.

E-mail address: [email protected] (Y.

a b s t r a c t

The penalty finite element method for Stokes problem with nonlinear slip boundary con-ditions, based on the finite element subspace ðVh;MhÞ which satisfies the discrete inf–sup condition, is investigated in this paper. Since this class of nonlinear slip boundary con-ditions include the subdifferential property, the weak variational formulation associatedwith Stokes problem is variational inequality. Under some regularity assumptions, weobtain the optimal H1 and L2 error estimates between u and uh, and between u and ue

h,where the error orders are eþ h for H1 error and eþ h2 for L2 error.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

Numerical simulation for the incompressible flow is the fundamental and significant problem in computational mathe-matics and computational fluid mechanics. It is well known that the mathematical model of viscous incompressible fluidwith homogeneous boundary conditions is Navier–Stokes equations which can be written as

ouot � lDuþ ðu � rÞuþrp ¼ f in Q T ;

divu ¼ 0 in Q T ;

uð0Þ ¼ u0 in X;

u ¼ 0 on oX� ð0; T�;

8>>><>>>: ð1Þ

where Q T ¼ ð0; T� �X, 0 < T 6 þ1, X � Rn;n ¼ 2;3; is a bounded convex domain. uðt; xÞ and f ðt; xÞ are vector functions anddenote the flow velocity and the external force, respectively. pðt; xÞ is a scalar function and denote the pressure. The viscouscoefficient l > 0 is a positive constant. The solenoidal condition means that the fluid is incompressible.

Note that the velocity u and the pressure p are coupled by the solenoidal condition divu ¼ 0 which makes that solving theabove Navier–Stokes equations (1) is a nontrivial task. the popular technique to overcome this difficulty is to relax thesolenoidal condition in an appropriate method and to result a pseudo-compressible system, such as the penalty method,the artificial compressible method, the pressure stabilized method and the projection method (for example see [1,3,4,12–15,19–25]). In this work, we will apply the penalty method to the Stokes problem with nonlinear slip boundary conditions.The penalty method is firstly introduced to the above Navier–Stokes equations (1) by Temam in [22,23]. Recently, based onthe backward Euler scheme, He in [12] obtains some optimal error estimates of the full discrete penalty finite element

. All rights reserved.

China (Nos.10571142 and 10701061).

Li).

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Y. Li, K. Li / Applied Mathematics and Computation 204 (2008) 216–226 217

approximation for two dimensional time-dependent Navier–Stokes equations. Subsequently, He in [13] applies the two-levelpenalty finite element approximation to the steady Navier–Stokes equations and obtain some error estimates.

In this paper, we will consider Stokes problem

�lDuþrp ¼ f in X;

divu ¼ 0 in X;

�ð2Þ

with the following nonlinear slip boundary conditions

u ¼ 0; on C;

un ¼ 0; �rsðuÞ 2 gojusj on S;

�ð3Þ

where X � R2, is a bounded convex domain. C \ S ¼ C [ S ¼ oX and g are the scalar functions; un ¼ u � n and us ¼ u� unn arethe normal and tangential components of the velocity, where n stands for the unit vector of the external normal to S;rsðuÞ ¼ r� rnn, independent of p, is the tangential components of the stress vector r which is defined byri ¼ riðu; pÞ ¼ ðleijðuÞ � pdijÞnj; where eijðuÞ ¼ oui

oxj þouj

oxi ; i; j ¼ 1;2. The set owðaÞ denotes a subdifferential of the function wat the point a, whose definition will be given in next section.

This class of boundary conditions (3) are introduced by Fujita in [6], where he investigated some hydrodynamics prob-lems under nonlinear boundary conditions, such as leak and slip boundary involving subdifferential property. These typesof boundary conditions appear in the modeling of blood flow in a vein of an arterial sclerosis patient and in that of avalancheof water and rocks. Fujita in [7] showed the existence and uniqueness of weak solution to Stokes problem with slip boundaryconditions (3). Subsequently, Saito in [18] showed the regularity of the weak solution by the Yosida’s regularized methodand finite difference quotients method. Other theory results about steady and time-dependent Stokes problems with non-linear subdifferential boundary conditions can be found in [8–10,17]. We remark that the steady homogeneous and inhomo-geneous Stokes system with linear slip boundary conditions without subdifferential property have recently been studiedfrom the theoretical view point by Veiga in [26–28].

The penalty method applied to the problem (2),(3) is to approximate the solution ðu; pÞ by ðue; peÞ satisfying the followingpenalty Stokes problem with nonlinear slip boundary conditions

�lDue þrpe ¼ f in X;

epe þ divue ¼ 0 in X;

ue ¼ 0; on C;

uen ¼ 0; �rsðueÞ 2 gojue

sj on S;

8>>><>>>: ð4Þ

where 0 < e < 1 is the penalty parameter. We show that the solutions ðu; pÞ of (2) and (3) and ðue; peÞ of (4) satisfy the fol-lowing approximate property:

ku� uekV þ kp� pek 6 ce; ð5Þ

where c > 0 is independent of e. Based on the finite element subspace ðVh;MhÞ, which satisfies the discrete inf–sup condition,we also obtain the following optimal error estimates:

kue � uehkV þ kpe � pe

hk 6 ch

kue � uehk 6 ch2 if ue 2 H2ðXÞ2 \ H2ðoXÞ2;pe 2 H1ðXÞ;

ku� uhkV þ kp� phk 6 ch

ku� uhk 6 ch2 if u 2 H2ðXÞ2 \ H2ðoXÞ2; p 2 H1ðXÞ;

8>>>><>>>>: ð6Þ

where c > 0 is independent of e and h. Hence, by triangular inequality, one has

ku� uehkV þ kp� pe

hk 6 cðeþ hÞ;ku� ue

hk 6 cðeþ h2Þ;

(ð7Þ

where c > 0 is independent of e and h.This paper is organized as follows: in next section, we will introduce some function spaces and describe the theorem

about existence, uniqueness and regularity of the weak solution to the problem (2) and (3); in Section 3, we mainly discussthe penalty problem (4) and show the approximate property (5); we will describe the penalty finite element approximationin Section 4 and show the error estimates (6) and (7) under different assumptions in last two sections.

2. Preliminary

Firstly, we give the definition of the subdifferential property (for example see [5]). Let w : R2 ! R ¼ ð�1;þ1� be a givenfunction possessing the properties of convexity and weak semi-continuity from below ðw is not identical with þ1Þ. We saythat the set owðaÞ is a subdifferential of the function w at the point a if

218 Y. Li, K. Li / Applied Mathematics and Computation 204 (2008) 216–226

owðaÞ ¼ fb 2 R2 : wðhÞ � wðaÞP b � ðh� aÞ;8 h 2 R2g:

We introduce some spaces which are usually used in this paper. Denote

V ¼ fu 2 H1ðXÞ2;ujC ¼ 0;u � njS ¼ 0g; V0 ¼ H10ðXÞ

2;

H ¼ fu 2 L2ðXÞ2;divu ¼ 0; u � njoX ¼ 0g; M ¼ L20ðXÞ ¼ q 2 L2ðXÞ;

ZX

qdx ¼ 0� �

:

Let k � kk be the norm in Hilbert space HkðXÞ2. Let ð�; �Þ and k � k be the inner product and the norm in L2ðXÞ2. Then we canequip the inner product and the norm in V by ðr�;r�Þ and k � kV ¼ kr � k, respectively, because kr � k is equivalent tok � k1. Let X be the Banach space. Denote X0 the dual space of X and < �; � > be the dual pairing in X�X0. Introduce the fol-lowing bilinear forms

aðu; vÞ ¼ lðru;rvÞ 8 u; v 2 V ;bðu; pÞ ¼ ðp;divuÞ 8 u 2 V ; p 2 M:

�
The weak formulation associated with the problem (2) and (3) are the following variational inequality:
Find ðu;pÞ 2 V �M such thataðu; v� uÞ þ jðvsÞ � jðusÞ � bðv� u; pÞP ðf ; v� uÞ 8 v 2 V ;

bðu; qÞ ¼ 0 8 q 2 M;

8><>: ð8Þ

where jðgÞ ¼R

S gjgjds.The following theorem about the existence, uniqueness and regularity of weak solution to (8) is showed by Fujita in [7]

and Saito in [18].

Theorem 2.1. Let f 2 H and g 2 L2ðSÞ. Then (8) admits a unique solution ðu; pÞ 2 V �M such that

kukV þ kpk 6 cðkfk þ kgkL2ðSÞÞ;

where c > 0 depends only on X. Furthermore, if oX is sufficiently smooth, then the weak solution is also regular solution, i.e.,u 2 H2ðXÞ2 and p 2 H1ðXÞ, such that

kuk2 þ kpk1 6 cðkfk þ kgkL2ðSÞÞ:

3. Penalty Stokes problem

In this section, we will discuss the penalty Stokes problem (4) and show the existence and uniqueness of weak solutionðue; peÞ 2 V �M to (4). In addition, we give the approximation property between the weak solutions ðu; pÞ and ðue; peÞ withrespect to the penalty parameter 0 < e < 1.

The weak formulation associated with the penalty problem of (4) is

Find ðue; peÞ 2 V �M such thataðue; v� ueÞ � bðv� ue; peÞ þ jðvsÞ � jðue

sÞP ðf ; v� ueÞ 8 v 2 V ;

ecðpe; qÞ þ bðue; qÞ ¼ 0 8 q 2 M;

8><>: ð9Þ

where

cðpe; qÞ ¼Z

Xpeqdx:

Define the operators B : V ! M0 and C : M ! M0 associated with the bilinear forms b : ðV ;MÞ ! R and c : ðM;MÞ ! R, respec-tively, by

< Bu; p >¼ bðu; pÞ 8 u 2 V ; p 2 M

and

< Cp; q >¼ cðp; qÞ 8 p; q 2 M:

Then the second equation in (9) can be rewritten as the following operator equation

eCpe þ Bue ¼ 0:

On the existence of the weak solution to (9), we have

Theorem 3.1. Assume that f 2 H and g 2 L2ðSÞ, then the penalty problem (9) admits a unique weak solution ðue; peÞ 2 V �M suchthat

kjuekj þ e12kpek 6 cðkfk þ kgkL2ðSÞÞ;


where c > 0 is independent of e, and

kjukj ¼ fkuk2V þ

1ekdivuk2g

12:

Proof. From the second equation of (4), one has

pe ¼ �1e

divue: ð10Þ

Substituting above into (9) yields

aðue; v� ueÞ þ 1eðdivðv� ueÞ;divueÞ þ jðvsÞ � jðue

sÞP ðf ; v� ueÞ 8 v 2 V : ð11Þ

Denote the bilinear form

~aðw; vÞ ¼ aðw; vÞ þ 1eðdivw;divvÞ 8 v;w 2 V :

Then ~aðv; vÞ is coercive in V with norm jjj�jjj. By the standard procedure about the second order elliptic variational inequalityof the second kind (for example see [Theorem I.4.1, [11]]), there exists a unique solution ue 2 V of (11). Setting v ¼ 2ue andv ¼ 0 in (11), respectively, yields

aðue;ueÞ þ 1eðdivue;divueÞ þ jðue

sÞ ¼ ðf ;ueÞ;

which shows that

kjuekj 6 cðkfk þ kgkL2ðSÞÞ:

According to (10), we conclude that pe 2 M exists and is unique. h

On the difference of the solutions of the original problem (8) and the penalty problem (9), we have

Theorem 3.2. Let ðu; pÞ 2 V �M and ðue; peÞ 2 V �M be the solutions of (8) and (9), respectively, then for sufficiently small e > 0,we have

ku� uekV þ kp� pek 6 ceðkfk þ kgkL2ðSÞÞ; ð12Þ

where c > 0 is independent of e.

Proof. For all w 2 V0, let v ¼ uþw in (8), one has

aðu;wÞ � bðw; pÞP ðf ;wÞ 8 w 2 V0: ð13Þ

Let v ¼ ue �w in (9), one has

�aðue;wÞ þ bðw; peÞP ðf ;�wÞ 8 w 2 V0: ð14Þ

Adding (13), (14) yields

aðu� ue;wÞP bðw;p� peÞ 8 w 2 V0:

Hence

aðu� ue;wÞ ¼ bðw; p� peÞ 8 w 2 V0:

From the inf–sup condition, we obtain

kp� pek 6 cku� uekV ; ð15Þ

where c > 0 is independent of e. Next, we will show that, for sufficiently small e > 0, one has

ku� uekV 6 ce: ð16Þ

From the second equation in (8) and (9), we have

ecðpe; qÞ þ bðue � u; qÞ ¼ 0 8 q 2 M:

On the other hand, setting v ¼ ue in (8) and v ¼ u in (9), and adding them yields

aðue � u;ue � uÞ 6 bðu� ue; p� peÞ ¼ ecðpe;p� peÞ ¼ ecðp;p� peÞ � ecðp� pe;p� peÞ 6 ecðp; p� peÞ 6 ekpkkp� pek6 cekpkku� uekV ;

which gives (16). h


4. Finite element approximation

Let Th be a family of regular triangular partition of X into triangles of diameter not great than 0 < h < 1. Let Vh � V andMh � M be the conforming finite element subspaces, which satisfy the discrete inf–sup condition, i.e., there exists a positiveconstant b > 0 independent of h such that

bkphk 6 supvh2Vh

bðvh;phÞkvhkV

8 vh 6¼ 0: ð17Þ

The following interpolation approximation is elementary (for example see [2,16]):

infvh2Vh

ku� vhkV þ h�1ku� vhkL2ðSÞ 6 chðkuk2 þ kukH2ðoXÞÞ8 u 2 V \ H2ðXÞ2 \ H2ðoXÞ2;

infqh2Mh

kp� qhk 6 chkpk1 8 p 2 M \ H1ðXÞ:

8><>:
Then the finite element approximation formulations of (8) and (9) are, respectively,
Find ðuh;phÞ 2 Vh �Mh such thataðuh; vh � uhÞ � bðvh � uh;phÞ þ jðvhsÞ � jðuhsÞP ðf ; vh � uhÞ 8 vh 2 Vh;

bðuh; qhÞ ¼ 0 8 qh 2 Mh

8><>: ð18Þ

and

Find ðueh;p

ehÞ 2 Vh �Mh such that

aðueh; vh � ue

hÞ � bðvh � ueh;p

ehÞ þ jðvhsÞ � jðue

hsÞP ðf ; vh � uehÞ 8 vh 2 Vh;

ecðpeh; qhÞ þ bðue

h; qhÞ ¼ 0 8 qh 2 Mh:

8><>: ð19Þ

Since bðvh; qhÞ satisfies the discrete inf–sup condition, then according to Theorems 2.1 and 3.2, we have

Theorem 4.1. Assume that f 2 H and g 2 L2ðSÞ, then the discrete problem (18) admits a unique solution ðuh; phÞ 2 Vh �Mh suchthat

kuhkV þ kphk 6 cðkfk þ kgkL2ðSÞÞ;

where c > 0 is independent of h. Moreover, if ðueh; p

ehÞ 2 Vh �Mh is the solution of (19), then the following approximation property

holds:

kuh � uehkV þ kph � pe

hk 6 ce; ð20Þ

where c > 0 is independent of e and h.

Next, we will show the existence of solution to the discrete penalty problem (19). Introduce the bilinear formsB : ðV ;MÞ � ðV ;MÞ ! R and Be : ðV ;MÞ � ðV ;MÞ ! R defined, respectively, by

Bðu; p; v; qÞ ¼ aðu; vÞ � bðv; pÞ þ bðu; qÞ 8 ðu;pÞ; ðv; qÞ 2 ðV ;MÞ
and
Beðu;p; v; qÞ ¼ Bðu;p; v; qÞ þ ecðp; qÞ 8 ðu;pÞ; ðv; qÞ 2 ðV ;MÞ:
Define the operators J : ðV ;MÞ ! R and F : ðV ;MÞ ! R by
Jðu; pÞ ¼ jðuÞ; ðF; ðv; qÞÞ ¼ ðf ; vÞ:

Then discrete penalty problem (19) is equivalent to

Findðueh;p

ehÞ 2 Vh �Mh such that for all ðvh; qhÞ 2 Vh �Mh;

e e e e e e e e

�ð21Þ

Beðuh;ph; vh � uh; qh � phÞ þ Jðvhs; qhÞ � Jðuhs; phÞP ðF; ðvh � uhÞ � ðqh � phÞÞ:

Theorem 4.2. Assume that f 2 H and g 2 L2ðSÞ, then the discrete penalty problem (19) or (21) admits a unique solutionðue

h; pehÞ 2 Vh �Mh.

Proof. By the definition, we have

Beðvh; qh; vh; qhÞ ¼ aðvh; vhÞ þ ecðqh; qhÞ ¼ lkvhk2V þ ekqhk

2:


Hence, Beðvh; qh; vh; qhÞ is coercive in ðVh;MhÞ. Then by the existence theorem of the solution to elliptic variational inequalityof the second kind in finite dimensional space (for example see [Theorem I.6.1, [11]]), we conclude that the discrete penaltyproblem (19) or (21) admits a unique solution ðue

h; pehÞ 2 Vh �Mh. h

5. H1 error estimate

In this section, we will show the error estimates (6) and (7) under some regularity assumptions.

Theorem 5.1. Let ðu; pÞ 2 V �M and ðuh; phÞ 2 Vh �Mh be the weak solution of (8) and (18), respectively. Furthermore, ifu 2 H2ðXÞ2 \ H2ðoXÞ2 and p 2 H1ðXÞ, then we have the following error estimate

ku� uhkV þ kp� phk 6 ch; ð22Þ

where c > 0 is independent of h.

Proof. By the definition of the bilinear form B, for all vh 2 Vh and qh 2 Mh, one has

lkuh � vhk2V ¼ Bðuh � vh; ph � qh; uh � vh;ph � qhÞ ¼ Bðuh;ph; uh � vh;ph � qhÞ �Bðvh; qh; uh � vh;ph � qhÞ¼ aðuh;uh � vhÞ � bðuh � vh;phÞ þ bðuh;ph � qhÞ �Bðvh; qh; uh � vh;ph � qhÞ6 ðf ;uh � vhÞ þ jðvhsÞ � jðuhsÞ �Bðvh; qh; uh � vh;ph � qhÞ: ð23Þ

Let v ¼ uh and v ¼ 2u� vh in (8), one has

aðu;uh � uÞ þ jðuhsÞ � jðusÞ � bðuh � u;pÞP ðf ; uh � uÞ

and

aðu;u� vhÞ þ jð2us � vhs Þ � jðusÞ � bðu� vh;pÞP ðf ;u� vhÞ:

Summing above gives

aðu;uh � vhÞ þ jðuhs Þ � 2jðusÞ þ jð2us � vhs Þ � bðuh � vh;pÞP f ðuh � vhÞ:


lkuh � vhk2V 6 aðu;uh � vhÞ � bðuh � vh;pÞ þ jðvhsÞ þ jð2us � vhsÞ � 2jðusÞ � ½aðvh;uh � vhÞ � bðuh � vh; qhÞþ bðvh;ph � qhÞ�

¼ aðu� vh;uh � vhÞ � bðuh � vh;p� qhÞ þ bðu� vh;ph � qhÞ þ jðvhsÞ þ jð2us � vhsÞ � 2jðusÞ6 lku� vhkVkuh � vhkV þ kuh � vhkVkp� qhk þ ku� vhkVkph � qhk þ cku� vhkL2ðSÞ

6l2kuh � vhk2

V þ cðku� vhk2V þ kp� qhk

2 þ ku� vhkL2ðSÞÞ þ akph � qhk2; ð24Þ

where a > 0 is a sufficiently small constant. Denote eV h � V0 be the finite element subspace of V0, then ðeV h;MhÞ also satisfiesthe discrete inf–sup conditions. Let wh 2 eV h � Vh. Setting v ¼ u�wh in (8) yields

aðu;whÞ � bðwh; pÞ ¼ ðf ;whÞ 8 wh 2 eV h:

Similarly, from (18) one has

aðuh;whÞ � bðwh;phÞ ¼ ðf ;whÞ 8 wh 2 eV h:

Hence

aðu� uh;whÞ ¼ bðwh; p� phÞ 8 wh 2 eV h:

So, for all wh 2 eV h,

bðwh; ph � qhÞ ¼ bðwh;ph � pÞ þ bðwh; p� qhÞ ¼ aðuh � u;whÞ þ bðwh;p� qhÞ:

In addition, for all qh 2 Mh, in terms of the discrete inf–sup condition, we have

bkph � qhk 6 supwh2eV h

bðwh;ph � qhÞkwhkV

6 lku� uhkV þ kp� qhk: ð25Þ

Substituting (25) into (24),

kuh � vhk2V 6 cðku� vhk2

V þ kp� qhk2 þ ku� vhkL2ðSÞÞ þ aku� uhk2

V :


Hence

kuh � vhkV 6 cðku� vhkV þ kp� qhk þ ku� vhk12

L2ðSÞÞ þ aku� uhkV :

Thus, from the triangular inequality, we have

ku� uhkV 6 ku� vhkV þ kuh � vhkV 6 cðku� vhkV þ kp� qhk þ ku� vhk12

L2ðSÞÞ þ aku� uhkV :

Then for sufficiently small a > 0, one has

ku� uhkV 6 cðku� vhkV þ kp� qhk þ ku� vhk12

L2ðSÞÞ 6 ch:

From (25) and triangular inequality, we conclude that

kp� phk 6 kp� qhk þ kph � qhk 6 kp� qhk þ cku� uhkV 6 ch: �

Next, we will show the error estimate ku� uehkV .

Theorem 5.2. Let ðue; peÞ 2 V �M and ðueh; p

ehÞ 2 Vh �Mh be the solution of (9) and (19), respectively. Furthermore, if

ue 2 H2ðXÞ2 \ H2ðoXÞ2 and pe 2 H1ðXÞ, then we have

kue � uehkV þ kpe � pe

hk 6 ch;


Proof. For all vh 2 Vh and qh 2 Mh, one has

lkue � uehk

2V þ ekpe � pe

hk26 2lkue � vhk2

V þ 2lkueh � vhk2

V þ 2ekpe � qhk2 þ 2ekpe

h � qhk2: ð26Þ

By the definition of the bilinear form Be, one has

lkueh � vhk2

V þ ekpeh � qhk

2 ¼ Beðueh � vh;pe

h � qh; ueh � vh;pe

h � qhÞ¼ Beðue

h; peh; ue

h � vh; peh � qhÞ �Beðvh; qh; ue

h � vh;peh � qhÞ

¼ aðueh;u

eh � vhÞ � bðue

h � vh;pehÞ þ bðue

h;peh � qhÞ

þ ecðpeh;p

eh � qhÞ �Beðvh; qh; ue

h � vh;peh � qhÞ

6 ðf ; ueh � vhÞ þ jðvhsÞ � jðue

hsÞ �Beðvh; qh; ueh � vh; pe

h � qhÞ: ð27Þ

Let v ¼ ueh and v ¼ 2ue � vh in (9), one has

aðue;ueh � ueÞ þ jðue

hsÞ � jðuesÞ � bðue

h � ue;peÞP ðf ; ueh � ueÞ

and

aðue;ue � vhÞ þ jð2ues � vhs Þ � jðue

sÞ � bðue � vh;peÞP ðf ;ue � vhÞ:

Summing above gives

aðue;ueh � vhÞ þ jðue

hsÞ � 2jðue

sÞ þ jð2ues � vhs Þ � bðue

h � vh; peÞP f ðueh � vhÞ:


lkueh � vhk2

V þ ekpeh � qhk

26 aðue;ue

h � vhÞ þ jðvhsÞ � 2jðuesÞ þ jð2ue

s � vhsÞ � bðueh � vh;peÞ � aðvh;ue

h � vhÞþ bðue

h � vh; qhÞ � bðvh;peh � qhÞ � ecðqh;p

eh � qhÞ ¼ aðue � vh;ue

h � vhÞ� bðue

h � vh;pe � qhÞ þ bðue � vh;peh � qhÞ

þ ecðpe � qh;peh � qhÞ þ jðvhsÞ � 2jðue

sÞ þ jð2ues � vhsÞ

6 lkue � vhkVkueh � vhkV þ kue

h � vhkVkpe � qhk þ kue � vhkVkpeh � qhk

þ ekpe � qhkkpeh � qhk þ 2kue � vhkL2ðSÞ

6l2kue

h � vhk2V þ

e2kpe

h � qhk2 þ lkue � vhk2

V þ1lkpe � qhk

2 þ 14akue

� vhk2V þ

e2kpe � qhk

2 þ 2kue � vhkL2ðSÞ þ akpeh � qhk

2;

where a > 0 is a sufficiently small constant. Hence,

l2kue

h � vhk2V þ

e2kpe

h � qhk26 lkue � vhk2

V þ1lkpe � qhk

2 þ 14akue � vhk2

V þe2kpe � qhk

2 þ 2kue � vhkL2ðSÞ þ akpeh � qhk

2:



lkue � uehk

2V þ ekpe � pe

hk26 6lkue � vhk2

V þ4lkpe � qhk

2 þ 1akue � vhk2

V þ 4ekpe � qhk2 þ 8kue � vhkL2ðSÞ þ 8akpe

h � qhk2:

ð28Þ

On the other hand, with similar method for (25), for all qh 2 Mh, one has

kpeh � qhk 6 cðkue � ue

hkV þ kpe � qhkÞ:

Thus

kue � uehkV 6 cðkue � vhkV þ kpe � qhk þ kue � vhk

12

L2ðSÞÞ þ cakue � uehkV :

Then for sufficiently small a, one has

kue � uehkV 6 cðkue � vhkV þ kpe � qhk þ kue � vhk

12

L2ðSÞÞ 6 ch:

Thus, for all qh 2 Mh, one has

kpe � pehk 6 kpe � qhk þ kqh � pe

hk 6 cðkue � uehkV þ kpe � qhkÞ 6 ch: �

Combining Theorems 3.2 and 5.2, we have

Theorem 5.3. Let ðu; pÞ 2 V �M and ðueh; p

ehÞ 2 Vh �Mh be the solution of (8) and (19), respectively. Furthermore, if the solution

ue 2 H2ðXÞ2 \ H2ðoXÞ2 and pe 2 H1ðXÞ of (9), then we have


hk 6 cðeþ hÞ; ð29Þ


6. L2 error estimate

In this section, we will give the L2 error estimates ku� uhk and ku� uehk by the Aubin–Nitsche’s technique. To do this, we

need the following regular assumptions about the homogeneous and penalty Stokes problems with linear slip boundarycondition.

(A1) Given u and uh the solutions of (8) and (18), respectively. We assume that the following linear Stokes problem:

�lDwþrp ¼ u� uh in X;

divw ¼ 0 in X;

w ¼ 0; on C;

wn ¼ 0; �rsðwÞ ¼ 0 on S

8>>><>>>: ð30Þ

admits a unique solution ðw;pÞ 2 H2ðXÞ2 \ V � H1ðXÞ \M such that

kwk2 þ kpk1 6 cku� uhk;

where c > 0 depends on l and X.(A2) Given ue and ue

h the solutions of (9) and (19), respectively. We assume that the following linear penalty Stokesproblem:

�lDwe þrpe ¼ ue � ueh in X;

epe þ divwe ¼ 0 in X;

we ¼ 0; on C;

wen ¼ 0; �rsðweÞ ¼ 0 on S

8>>><>>>: ð31Þ

admits a unique solution ðwe;peÞ 2 H2ðXÞ2 \ V � H1ðXÞ \M such that

kwek2 þ e12kpek1 6 ckue � ue

hk;

where c > 0 depends on l and X.About the problem (30), we refer the reader to [26,28]. The weak variational formulation associated with (30) is

Find ðw;pÞ 2 V �M such thataðw; vÞ � bðv;pÞ ¼ ðu� uh; vÞ 8 v 2 V ;

bðw; qÞ ¼ 0 8 q 2 M:

8><>: ð32Þ


Let wh 2 eV h � V0 and ph 2 Mh be the finite element approximation solution of (32), then it is well-known that

kw�whkV þ kp� phk 6 chku� uhk; ð33Þ

where c > 0 is independent of h.The weak variational formulation associated with (31) is

Findðwe;peÞ 2 V �M such thataðwe; vÞ � bðv;peÞ ¼ ðue � ue

h; vÞ 8 v 2 V ;

bðwe; qÞ þ ecðpe; qÞ ¼ 0 8 q 2 M:

8><>: ð34Þ

Let weh 2 eV h � V0 and pe

h 2 Mh be the finite element approximation solution of (34), then we can easily show

kwe �wehkV þ kpe � pe

hk 6 chkue � uehk; ð35Þ

where c > 0 is independent of h.Firstly, we describe the theorem about the L2 error estimate ku� uhk.

Theorem 6.1. Under the assumption (A1), let ðu; pÞ 2 V �M and ðuh; phÞ 2 Vh �Mh be the weak solution of (8) and (18),respectively. Furthermore, if u 2 H2ðXÞ2 \ H2ðoXÞ2 and p 2 H1ðXÞ, then we have the following L2 error estimate

ku� uhk 6 ch2;

where c > 0 is independent of h.

Proof. Setting v ¼ u� uh in (32) yields

ku� uhk2 ¼ aðw;u� uhÞ � bðu� uh;pÞ ¼ aðw�wh;u� uhÞ þ aðwh;u� uhÞ � bðu� uh;p� phÞ ð36Þ

since bðu� uh; qhÞ ¼ 0 for all qh 2 Mh. Setting v ¼ u�wh in (8) yields

aðu;whÞ � bðwh; pÞ ¼ ðf ;whÞ 8 wh 2 eV h:

Similarly, setting vh ¼ uh �wh in (18) yields

aðuh;whÞ � bðwh;phÞ ¼ ðf ;whÞ 8 wh 2 eV h:

Hence

aðu� uh;whÞ ¼ bðwh;p� phÞ 8 wh 2 eV h:

Substituting above into (36) gives

ku� uhk2 ¼ aðw;u� uhÞ � bðu� uh;pÞ ¼ aðw�wh;u� uhÞ þ bðwh;p� phÞ � bðu� uh;p� phÞ¼ aðw�wh; u� uhÞ þ bðwh �w;p� phÞ � bðu� uh;p� phÞ

since bðw; p� phÞ ¼ 0. Thus, we have

ku� uhk26 lkw�whkVku� uhkV þ kw�whkVkp� phk þ ku� uhkVkp� phk 6 chðkw�whkV þ kp� phkÞ

6 ch2ku� uhk: �

Next theorem gives the L2 error estimate ku� uehk.

Theorem 6.2. Under the assumption (A2), let ðu; pÞ 2 V �M and ðueh; p

ehÞ 2 Vh �Mh be the weak solution of (8) and (19),

respectively. Furthermore, if ue 2 H2ðXÞ2 \ H2ðoXÞ2 and pe 2 H1ðXÞ, then we have the following L2 error estimate

kue � uehk 6 ch2

;

where c > 0 is independent of h. Hence,

ku� uehk 6 cðeþ h2Þ:

Proof. Setting v ¼ ue � ueh in (34) yields

kue � uehk

2 ¼ aðwe;ue � uehÞ � bðue � ue

h;peÞ ¼ aðwe �we

h;ue � ue

hÞ þ aðweh;u

e � uehÞ � bðue � ue

h;peÞ

¼ aðwe �weh;u

e � uehÞ þ aðwe

h;ue � ue

hÞ � bðue � ueh;p

e � pehÞ � bðue � ue

h;pehÞ

¼ aðwe �weh;u

e � uehÞ þ aðwe

h;ue � ue

hÞ � bðue � ueh;p

e � pehÞ þ ecðpe � pe

h;pehÞ; ð37Þ


where we use

bðue � ueh; qhÞ þ ecðpe � pe

h; qhÞ ¼ 0 8 qh 2 Mh:

On the other hand, for weh 2 eV h, one has

aðue � ueh;w

ehÞ ¼ bðwe

h;pe � pe

hÞ ¼ bðweh �we;pe � pe

hÞ þ bðwe; pe � pehÞ ¼ bðwe

h �we; pe � pehÞ � ecðpe;pe � pe

hÞ:

Substituting above into (37), one has

kue � uehk

2 ¼ aðwe �weh;u

e � uehÞ þ bðwe

h �we;pe � pehÞ � bðue � ue

h;pe � pe

hÞ þ ecðpe � peh;p

eh � peÞ

6 lkwe �wehkVkue � ue

hkV þ kweh �wekVkpe � pe

hk þ kue � uehkVkpe � pe

hk þ ekpe � pehkkpe � pe

hk

6 chðkwe �wehkV þ kpe � pe

hkÞ 6 ch2kue � uehk; ð38Þ

which shows that

kue � uehk 6 ch2

: �

7. Conclusion

In this paper, based on the finite element subspace ðVh;MhÞ which satisfies the discrete inf-sup condition, we discuss thepenalty finite element approximation for the Stokes problem with nonlinear slip boundary conditions and obtain the follow-ing optimal H1 error estimates


hk 6 cðeþ hÞ;ku� uhkV þ kp� phk 6 ch

�
and the optimal L2 error estimates
ku� uehk 6 cðeþ h2Þ;

ku� uhk 6 ch2:

(
If the penalty parameter e ¼ OðhÞ, then we conclude that the error order of the H1 error estimates for penalty finite ele-
ment approximation is the same as that for general finite element approximation and the error order of the L2 error esti-mates for penalty finite element approximation is less than that for general finite element. If the penalty parametere ¼ Oðh2Þ, the error orders of the H1 error estimates and the L2 error estimates for penalty finite element approximation isthe same as that for general finite element approximation. Hence, the penalty method can overcome the computational dif-ficulty without loss of accuracy if e ¼ Oðh2Þ.

Acknowledgement

The authors would like to thank the editor and the reviewers for their valuable comments and suggestions to improve theresults of this paper.

References

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penalty finite element method for stokes problem with nonlinear slip boundary conditions

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