A consistent and stabilized continuous/discontinuous Galerkin method for fourth-order incompressible flow problems

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A consistent and stabilized continuous/discontinuous Galerkin method

for fourth-order incompressible ﬂow problems

A.G.B. Cruz

a,⇑

, E.G. Dutra do Carmo

b

, F.P. Duda

a a

Mechanical Engineering Department-PEM/COPPE, Federal University of Rio de Janeiro, Ilha do Fundão, 21945-970, P.B. 68503, Rio de Janeiro, RJ, Brazil b

Nuclear Engineering Department-PEN/COPPE, Federal University of Rio de Janeiro, Ilha do Fundão, 21945-970, P.B. 68509, Rio de Janeiro, RJ, Brazil

a r t i c l e

i n f o

Article history: Received 27 March 2011 Accepted 3 May 2012 Available online 15 May 2012 Keywords:

Discontinuous Galerkin methods Fourth-order problems GLS stability Second gradient

a b s t r a c t

This paper presents a new consistent and stabilized finite-element formulation for fourth-order incompressible flow problems. The formulation is based on the C0_{-interior penalty} method, the Galerkin least-square (GLS) scheme, which assures that the formulation is weakly coercive for spaces that fail to satisfy the inf-sup condition, and considers discon-tinuous pressure interpolations. A stability analysis through a lemma establishes that the proposed formulation satisfies the inf-sup condition, thus confirming the robustness of the method. This lemma indicates that, at the element level, there exists an optimal or quasi-optimal GLS stability parameter that depends on the polynomial degree used to interpolate the velocity and pressure fields, the geometry of the finite element, and the fluid viscosity term. Numerical experiments are carried out to illustrate the ability of the formulation to deal with arbitrary interpolations for velocity and pressure, and to stabilize large pressure gradients.

1. Introduction

Finite-element formulations for fourth-order differential operators require the introduction of C1_{-ﬁnite element spaces, a}

feature that gives rise to difﬁculties in terms of computational implementation (see, for instance,[1–5]). This motivated En-gel et al.[1]to develop the C0_{-interior penalty methods (also known as continuous/discontinuous Galerkin methods) for}

fourth-order elliptic boundary value problems. These methods were subsequently investigated in many works, such as

[2,3,6–8].

On the other hand, standard finite-element formulations for incompressible flow problems require, due to their mixed nature (cf.[9]), the choice of interpolation spaces, for velocity and pressure, satisfying the inf-sup condition or Banach–Ne-cas–Babuska condition[10](cf.[9–12]). This forbids, for instance, the use of equal order approximations for velocity and pressure (cf.[12,13]). However, as it is well known (see, for instance, Gresho and Sani[12]), the violation of the inf-sup con-dition potentially introduces, among other pathologies, spurious oscillations of the pressure and locking of the velocity. An efficient approach to circumvent the difficulties imposed by the inf-sup condition consists in using the Galerkin least square (GLS) method, in which least square residuals of the governing equations are accounted for (cf.[14–19]). In particular, the use of the GLS method enables arbitrary choices of velocity and pressure interpolation spaces, which is generally desirable from the computational point of view.

http://dx.doi.org/10.1016/j.jcp.2012.05.002 ⇑Corresponding author.

E-mail address:aguicruz@mecanica.ufrj.br(A.G.B. Cruz).

Contents lists available atSciVerse ScienceDirect

Journal of Computational Physics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j c p

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In this paper, a consistent and stabilized finite-element formulation for fourth-order incompressible flow problems is pro-posed. Motivated by the aforementioned discussion, the formulation is based on the continuous/discontinuous Galerking and Least Square methods. Further, discontinuous pressure interpolations are adopted, which are, in comparison with con-tinuous pressure interpolations, physically more appropriate for enforcing the incompressibility constraint at the element level[11,12]and, in addition, enables a partial condensation of the degrees of freedom. A stability analysis using a lemma encompassing the inf-sup condition is also presented, showing that the proposed formulation is weakly coercive and, there-fore, robust. This lemma also suggests that there exists an optimal or quasi optimal LS (least square) stabilization parameter, which is not necessarily the same for all elements of the mesh. Numerical experiments are provided to illustrate the efficacy of the formulation to deal with any combination of velocity–pressure elements as well as to stabilize high gradients of pressure.

A ﬁnite-element formulation for the problem considered here was advanced in Kim et al.[20]. As in the present paper, Kim et al.[20]based their formulation on the C0_{-interior penalty method proposed in Engel et al.}_[1]_{. However, their}

treat-ment of the incompressibility does not take into account the inf-sup condition. Therefore, it is not clear whether they could prevent the well known pathologies associated with the violation of the inf-sup condition.

The remainder of this paper is organized as follows. In Section2, we introduce the fourth-order model problem and the associated variational formulation. Section3presents the discontinuous Galerkin least square formulation for this model problem and the corresponding stability issue examined in Section4. Numerical examples are presented in Section5. Con-cluding and ﬁnal remarks are provided in Section6.

2. Model problem

LetX Rn_{ðn ¼ 2 or 3Þ be a domain with boundary}_C_{Lipschitz continuous. Let}_C

DandCNbe such that

C

¼

C

D[

C

N and measð

C

D\

C

NÞ ¼ 0; ð1Þ

where meas() denotes the positive Lebesgue measure. We consider the functions f 2 L2ðXÞn;g02 H

3

2_ð_C_D_Þn_{\ C}0_ð_C_D_Þn_;_g

12 H

1 2_ð_C_D_Þn_;

h02 L2ðCNÞn and h12 L2(CN)n. Recall that the spaces L2ðXÞn¼ fs ¼ ðs1; . . . ;snÞ; si2 L2ðXÞg; H

t 2_ð_C_D_Þn_{¼ g ¼ ðg}₁; . . . ;g_n_{Þ; g}_i_{2 H} t 2_ð_C_D_Þ n ð

t

¼ 1; 2; 3 . . .Þg; HtðXÞ and Ht(X)n_{= {w = (w}

1, . . . , wn); wi2 Ht(X);

t

= 1, 2, 3 . . .} are Sobolev spaces as deﬁned in[21].

We now introduce the fourth-order boundary value problem of concern here. It consists of ﬁnding the pair (u, p), belong-ing to H2₍_X₎n_L2₍_X_{) if}

_g

_{> 0 or to H}1₍_X₎n_L2₍_X_{) if}

_g

_{= 0, that satisﬁes}

g

D

ð

D

uÞ

l

D

u þ

r

p ¼ f in

X

; ð2Þ

r

u ¼ 0 in

X

; ð3Þ u ¼ g0on

C

D; ð4Þ @u @n¼ ð

r

uÞn ¼ g1on

C

Dif

g

>0; ð5Þ

l

@u @n pn

g

@ð

D

uÞ @n ¼ h0on

C

N; ð6Þ

g

D

u ¼ h1on

C

Nif

g

>0; ð7Þ

whereDu denotes the Laplacian of the vector u,r u denotes the divergence of u,ru denotes the gradient of u, n denotes the outward normal unit vector deﬁned almost everywhere onC, the dot ‘‘’’ denotes the usual inner product in Rn_{, and}

g

P0 and

l

> 0 are constants.

It should be noted that the Laplacian is considered in the weak sense as follows: we say that u 2 L2₍

X)n_{has weak}

Lapla-cian in L2₍

X)n_{, if and only if, there exists}_D_{u 2 L}2₍

X)n_{such that} Z X

D

u

g

d

X

¼ Z X u ð

D

g

Þ d

X

8 g

2 C10ð

X

Þ n ð8Þ where C1 0ðXÞ n_{is as deﬁned in}_[21]_.

Notice that for

g

= 0 the boundary value problem given by(2)–(7)is reduced to the classical Stokes ﬂow problem. Other-wise, for

g

> 0, it is similar to the governing equations for second-gradient and incompressible fluid, cf.[22,23]. In this case, the length scale ‘ :¼pffiffiffiffiffiffiffiffiffi

g

=

l

is introduced, which opens a way to account for small scale effects[22](see also[20]).

We now proceed the wide form of boundary value problem deﬁned. Toward this end, we begin by noteworthy the appro-priate functional spaces.

Since the conditions associated with g0and g1are essential, we deﬁne the solution set for the velocity u as follows:

Su¼ u 2 H2 ð

X

Þn;u ¼ g0and @u@n¼ g1on

C

D n o if

g

>0 u 2 H1ð

X

Þn;u ¼ g0on

C

D n o if

g

¼ 0 8 > < > : ð9Þ

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and we deﬁne the corresponding space of the admissible variations as follows: Vu¼

v

2 H2ð

X

Þn;

v

¼ 0 and @v @n¼ 0 on

C

D n o if

g

>0 f

v

2 H1ð

X

Þn;

v

¼ 0 on

C

Dg if

g

¼ 0 8 < : ð10Þ

The solution set for the pressure p and the corresponding space of the admissible variations are all given as follows:

Sp¼ L2ð

X

Þ and measð

C

NÞ > 0 q 2 L2 ð

X

Þ;RXq d

X

¼ 0 n o if measð

C

NÞ ¼ 0 8 < : ð11Þ Vp¼ Sp: ð12Þ

Deﬁning the multilinear form

A ðu; p;

v

;qÞ ¼ Z X ð

g

D

w

D

v

þ

l

r

u :

r

v

Þ d

X

þ Z X qð

r

uÞ d

X

Z X pð

r

v

Þ d

X

; ð13Þ

and the linear functional

lð

v

Þ ¼ R Xf

v

d

C

þ R CNh0

v

d

C

þ R CNh1 @v @n d

C

if

g

>0 R Xf

v

d

C

þ R CNh0

v

d

C

if

g

¼ 0 ( ð14Þ

the weak form of the boundary value problem deﬁned by(2)–(7)is introduced as: ﬁnd the pair (u, p) 2 Su Spsatisfying

A

ðu; p;

v

;qÞ ¼ lð

v

Þ

8

ð

v

;qÞ 2 Vu Vp: ð15Þ

3. Finite element approximation 3.1. Discontinuous Galerkin formulation

Finite-element methods for the boundary value problem given in the Section2require ﬁnite-element spaces with C1

-con-tinuity[3–5,24]. Here, however, we adopt the interior penalty method proposed in[1], which uses the standard C0_-Lagrange

ﬁnite-element and enforces the continuity of derivatives across element boundaries in a weak sense. Let Mh_{= {}

X1, . . . ,Xne} be a regular partition ofXinto ne non-degenerate ﬁnite elementsXe. The partition Mhis such that:

Xe can be isoparametrically mapped into standard elements and Xe\Xe0¼ ; if e – e0; X[C¼ [ne_e¼1ðXe[CeÞ and

Cee0¼C_e\C_e0, whereC_edenotes the boundary ofX_e. We deﬁne the broken Sobolev spaces Ht,b(Mh) and Ht,b,1(Mh) on the

partition Mh_{as follows:}

Ht;bðMhÞ ¼ f

u

:

X

#R;

u

e2 H t_ð

_X

eÞ

8X

e2 Mhg;

Ht;b;1ðMhÞ ¼ f

u

2 Ht;b\ H1ð

X

Þg;

where

ue

is the restriction of

u

toXe.

Let k P 1, k P l P 0 and r P 0 be integers and consider Pr_ð_X

eÞ the space of polynomials of degree less than or equal to r

restricted to the elementXe. Introducing the ﬁnite-dimensional spaces

Hh;k¼ f

u

2 H2;b;1ðMhÞ;

u

e2 P k_ð

_X

eÞg; Hh;k;n¼ f

u

¼ ð

u

iÞ16 i 6 n;

u

i2 H h;k g; Lh;l¼ f

u

2 H0;bðMhÞ;

u

e2 P l_ð

_X

eÞg;

we consider then the spaces Sh;ku ;V h;k u ;S h p and V h pdeﬁned as follows: Sh;k u ¼ fuh2 H h;k;n ;uh_{¼ g}h 0on

C

Dg; Vh;k u ¼ f

v

h2 H h;k;n_;

_v

h_{¼ 0 on}

_C

Dg; Sh p¼ V h p¼ Sp\ Lh;l; where gh

0is the usual interpolating of g0.

Note that we envisage functions which are continuous on the entire domain but discontinuous in ﬁrst and higher-order derivatives on interior boundaries. Thus Sh;k

u and V h;k u are C

0_{-ﬁnite-element spaces. We assume specially that S}h p and V

h pare

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The continuous/discontinuous Galerkin method for(15)is to ﬁnd ðuh_;_ph_{Þ 2 S}h;k u S h psuch that Aðuh_;_ph_;

_v

h_;_qh_{Þ ¼ lð}

_v

h_Þ

₈

_ð

_v

h_;_qh_{Þ 2 V}h;k u V h p; ð16Þ where Aðuh_;_ph_;

_v

h_;_qh_{Þ ¼}X ne e¼1 Z Xe

l

r

uh e:

r

v

h ed

X

þ Z Xe

l

‘2

D

uh e

D

v

h e d

X

Z Xe ph e ð

r

v

h eÞd

X

þ Z Xe qh e ð

r

u h eÞd

X

þX e0>e Z Cee0 1 2

l

‘ 2

D

uh eþ

l

‘ 2

D

uh e0 @

v

h e @neþ @

v

h_e0 @ne0 d

C

þ Z Cee0 1 2 @uh e @neþ @uh e0 @ne0

l

‘2

D

v

heþ

l

‘ 2

D

v

he0 d

C

þ Z Cee0

su

@u h e @ne þ@u h e0 @ne0 @

v

h e @ne þ@

v

h e0 @ne0 d

C

! Z Ce\CD

l

‘2

D

uh e @

v

h e @ne d

C

þ Z Ce\CD @uh e @ne

l

‘2

D

v

hed

C

þ Z Ce\CD

sD

@u h e @ne @

v

h e @ne d

C

) ; lð

v

h_{Þ ¼}X ne e¼1 Z Xe fe

v

hed

C

þ Z Ce\CN he;0

v

hed

C

þ Z Ce\CN he;1 @

v

h e @ne d

C

þ Z Ce\CD ge;1

l

‘2

D

v

he d

C

þ Z Ce\CD

sD

ge;1 @

v

h e @ne d

C

; ð18Þ

and

su

and

sD

are penalty parameters to be ﬁxed.

From dimensional analysis we ﬁnd that

su

¼

sD

¼ C

l

‘2=hee0, where h_ee0is a characteristic element edge length and C is a

positive constant independent of the mesh parameters. This constant have to be chosen carefully to guarantee sufﬁciently accuracy and good convergence of the method[1,20,8,25]. For the purpose of this paper, the constant C is at the moment determined by numerical experiments. It is noteworthy that a large penalty parameter adversely affects the accuracy of the C0-interior penalty method[3].

On the other hand, the restrictions imposed by the inf-sup condition need to be veriﬁed for each particular choice of the ﬁnite-element spaces Sh;k u ;S h p;V h;k u and V h

pto reach stabilized and convergent solutions. As is well known, quite few

combina-tions of finite element pairs are able to fulfil the inf-sup condition within the Galerkin formulation (see, e.g.,[12,13]for some stable mixed elements). However, these stable finite element methods for second order problems can not be stable for fourth-order problems (i.e, larger length scales ‘), which was circumvented in cf.[20]using an additional stabilization of the pressure.

3.2. Consistency

The Euler–Lagrange equations associated with the partition Mhof the domainXare given by

r

peþ

l

D

ue

l

‘2

DD

ueþ fe¼ 0 in

X

e; ð19Þ

r

ue¼ 0 in

X

e; ð20Þ ue¼ g0 on

C

e\

C

D if measð

C

e\

C

DÞ > 0; ð21Þ @ue @ne ¼ ð

r

ueÞn ¼ g1 on

C

e\

C

D if

g

>0 end measð

C

e\

C

DÞ > 0; ð22Þ

l

@ue @ne pene @ð

l

‘ 2

D

ueÞ @ne ¼ he;0 if measð

C

e\

C

NÞ > 0; ð23Þ

l

‘2

D

ue¼ he;1 on

C

N; if measð

C

e\

C

NÞ > 0; ð24Þ @ue @neþ @ue0 @ne0 ¼ 0 on

C

ee0; ð25Þ

l

@ue @ne pene @ð

l

‘ 2

D

ueÞ @ne " # þ

l

@ue0 @ne0 pe 0ne0 @ð

l

‘ 2

D

ue0Þ @ne0 " # ¼ 0 on

C

ee0; ð26Þ

l

‘2

D

ue

l

‘2

D

ue0¼ 0 on

C

_ee0; ð27Þ

where ueand peare restrictions toXe.

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0 ¼ Aðu; p;

v

h_;_qh Þ lð

v

hÞ ¼X ne e¼1 Z Xe

l

D

ue

l

‘2

DD

ue

r

peþ fe

v

he d

X

þ Z Xe qh e ð

r

ueÞd

X

þX e0>e Z Cee0 1 2 @ue @neþ @ue0 @ne0

l

‘2

D

v

heþ

l

‘ 2

D

v

he0 d

C

þ Z Cee0 1 2

l

‘ 2

D

ue

l

‘2

D

ue0 @

v

h e @neþ @

v

h e0 @ne0 d

C

þ Z Cee0

su

@ue @ne þ@ue0 @ne0 @

v

h e @ne þ@

v

h e0 @ne0 d

C

! Z Ce\CN 1 2

l

@ue @ne pene @ð

l

‘ 2

D

ueÞ @ne " # þ

l

@ue0 @ne0 pe 0ne0 @ð

l

‘ 2

D

ue0Þ @ne0 " #! ð

v

h eþ

v

h e0Þd

C

Z Ce\CN

l

@ue @ne pene @ð

l

‘ 2

D

ueÞ @ne " # he;0 !

v

h ed

C

þ Z Ce\CN

l

‘2

D

ue he;1 @

v

h e @ne d

C

Z Ce\CD @ue @ne ge;1

l

‘2

D

v

hed

C

þ Z Ce\CD

sD

@ue @ne ge;1 @

v

h e @ne d

C

: ð28Þ

Since this equation holds for all ð

v

h_;_qh_{Þ 2 V}h;k u V

h

p, the continuous/discontinuous Galerkin formulation(16)is therefore

con-sistent in the sense that

Aðu; p;

v

h_;_qh

Þ ¼ lð

v

hÞ ð29Þ

3.3. Galerkin least square stabilization

To alleviate the need of satisfying the inf-sup condition, and to take advantage of using the discontinuous pressure inter-polation as well, we adopted the GLS stabilization method. The technique consists in including additional terms in the Galer-kin form, so as to enhance its stability. These terms are obtained by minimization of the square of the L2_{-norm of the discrete}

residual within each element[26], multiplied by adequate regularization (or stabilization) parameters that will control the contribution of the least square part in the variational sentence (cf.[14–19], for instance).

The continuous/discontinuous Galerkin formulation(16)in the least square sense (hereafter referred to as CDGLS formu-lation) is to ﬁnd ðuh_;_ph_{Þ 2 S}h;k u S h psuch that Ahðuh_;_ph_;

_v

h_;_qh_{Þ ¼ l} totð

v

h;qhÞ

8

ð

v

h;qhÞ 2 Vh;ku V h p; ð30Þ with Ahðuh_;_ph_;

_v

h_;_qh_{Þ ¼ Aðu}h_;_ph_;

_v

h_;_qh_{Þ þ}X ne e¼1 AeLSðu h_;_ph_;

_v

h_;_qh_Þ; _ð31Þ ltotð

v

h;qhÞ ¼ lð

v

h;qhÞ þ Xne e¼1 leLSð

v

h_;_qh_Þ; _ð32Þ

where the forth terms are the GLS terms

AeLSðw; q; dw; dqÞ ¼ Z Xe dGLSðheÞ Xn i¼1 Riðw; qÞRiðdw; dqÞ ! d

X

; ð33Þ Riðw; qÞ ¼ @p @xi

l

D

wiþ

l

‘2

DD

wi; ð34Þ leLSð

v

h_;_qh_{Þ ¼}Z Xe dGLSðheÞ Xn i¼1 fiRið

v

h;qhÞ ! d

X

; ð35Þ

with (w, q) 2 H4,b(Mh)n H1,b(Mh) and (dw, dq) 2 H4,b(Mh)n H 1,b(Mh), and dGLS(he) is the element stabilization parameter,

which here will be deﬁned by

dGLSðheÞ ¼

a

h2e

l

; ð36Þ

where

a

is a dimensionless parameter to be ﬁxed and he is a characteristic length of the element usually given by

he¼ R_X_edX

1=n

; cf., e.g.,[17]. Note that the addition of these terms does not affect the consistency proof presented earlier. Note also that for triangular and tetrahedral elements up to order three, as well as for quadratic quadrilateral and hexahedral elements the fourth-order differential operator vanishes, which implies that any calculation involving derivatives of order greater than two is trivial for these elements. All these elements are widely used in practice.

As will be shown later, the CDGLS formulation(30)is stable, consistent and satisfy the inf-sup condition, thereby allowing and enables a wider choice of velocity and pressure spaces. Moreover, it enables the stabilization of large gradients of pres-sure, as clearly shown by the numerical results discussed in Section5.

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The dimensionless parameter

a

in(36)needs to be carefully determined to ensure sufﬁcient accuracy and good conver-gency of GLS-stabilized formulations (cf.[27]). A methodology that allows to determine, at element level, optimal or quasi-optimal parameters of the GLS-stabilized formulations for second-order incompressible problems has been investigated in Carmo et al.[28,29]. However, the extension of these results to fourth-order problems with incompressibility constraint is not trivial and require a careful analysis of stability. This issue will discussed in the next section.

4. Stability analysis of the fourth-order incompressible problem

We provide a stability analysis of the CDGLS-stabilized method(30)when applied to the fourth-order boundary value problem given in Section2. The analysis suggests the existence of an appropriate stabilization parameter

a

that is dependent of the degree of the polynomial used to interpolate the velocity u and pressure p ﬁelds, the geometry of the elements and the ﬂuid viscosity term

l

. The result that enables to develop a methodology appropriate to determine this parameter is pre-sented in the Lemma1, which extends the results obtained in Carmo et al.[28,29]for second order incompressible problems. We consider that the ﬁnite elements spaces have been constructed using meshes satisfying a regularity condition MC1, wherein the element distortion is controlled: there exist real constants C1,distor> C0,distor> 0 such that

C0;distor<

q

e

he

<C1;distor

8X

e; ð37Þ

where

qe

= sup{diam (S), S is a hypersphere contained inXe[Ce} (cf.[10]). Owing to this control of mesh distortion, we ﬁnd

c0 0>0 and c01>0 to infer c00he<hee0<c0 1he; c00he0<h_ee0<c0 1he; c00he<he0<c0 1he and c00he0<h_e<c0 1he0, where hee0¼ minfh_e;h_e0g.

It follows from the well known theorem that says that all norms on a ﬁnite dimensional space are equivalent and under the regularity condition MC1 that the inverse inequalities hold: there are real constants CLap,1> 0,CLap,2> 0 and Cbound,1> 0

independent of the mesh parameter, depending only on the polynomial degree and the mesh distortion which is controlled by the constants C0,distorand C1,distorsuch that

Z Ce hew2d

X

6Cbound;1 Z Xe w2d

X

8

w2 Pk_ð

_X

eÞðk P 1Þ; ð38Þ Z Xe ‘2j

D

u

j2d

X

6 ‘ 2 ðheÞ2 CLap;1 Z Xe j

r

u

j2d

X

8 u

2 Pk_ð

_X

eÞðk P 1Þ; ð39Þ Z Xe

l

2_‘4 j

DD

u

j2d

X

6 ‘ 2 ðheÞ 2 CLap;2l ðheÞ 2 Z Xe

l

‘2j

D

u

j2d

X

8 u

2 Pkð

X

eÞðk P 1Þ: ð40Þ

We assume that for all we2 H1(Xe)nand qe2 L2(Xe) the following decomposition at element level

we¼ weþ ^we0 and q_e¼ qeþ ^qe; ð41Þ where we¼ R X_Rewed

X

Xe d

X

and qe¼ R Xeqe d

X

R Xe d

X

: ð42Þ

Next, we consider the following Poincaré inequalities

8

we2 H1ð

X

eÞn k ^wekL2_ðX eÞn6C e Poinc hej ^wejH1_ðX eÞn; ð43Þ

8

qe2 H 1 ð

X

eÞn k^qekL2_ðX eÞn6C e Poinc hej^qejH1_ðX eÞ: ð44Þ where j jH1 ðXeÞ and j jH1 ðXeÞn denote seminorms of H 1

(Xe) and H1(Xe)n, respectively, and the positive real constant CePoinc

depending only on the usual aspect ratio which is controlled (MC1 condition) in ﬁnite elements meshes.

The result that follows is strongly inspired in[28,29]and represents an extension of the Lemma (4.38) of reference[10]. However, besides considering the two fundamental differences presented in[28,29], namely, (i) the parameter of stabiliza-tion now depends onXeand, consequently, is not necessarily the same for all of the elements, (ii) the analysis is derived with

a different and more appropriate norm to determine the stabilization parameter; we also extend these two differences to the fourth-order problems with internal incompressibility constraint.

To formalize this idea, for 0 < b < 1,

ae

>

amin

> 0"Xeand k P 1, we introduce the following norm on Vh;ku V h p: kjð

v

h_;_qh_Þjk2 b;h;X¼ Xne e¼1 b ð1 þ bÞjq h_j2 ae;he;l;H1_ðX eÞþ ð1 bÞ j

v

h_j2 l;H1_ðX eÞnþ j

v

h_j2 l;‘;Lapþ X e0_>e Z Cee0

s

_v @

v

e @ne þ@

v

e0 @ne0 2 d

C

( þX n i¼1 Z Xe

ae

ðheÞ2

l

jRið

v

h;qhÞj2d

X

! þ q h e

l

1=2 2 L2_ðX eÞ ) ; ð45Þ

(7)

j

v

h_j2 l;H1 ðXeÞn¼ Z Xe Xn i¼1

l

j

r

v

h ij 2 ! d

X

; ð46Þ j

v

h_j2 l;‘;Lap¼ Z Xe Xn i¼1

l

‘2j

D

_v

h ij 2 ! d

X

; ð47Þ jqh_j2 ae;he;l;H1_ðX eÞ¼ Z Xe

ae

ðheÞ2

l

j

r

q h_j2_d

X

; ð48Þ

which was extended here to the fourth-order problems.

Thus we summarize the discussion above into the following lemma.

Lemma 1. Under assumptions (MC1) and(38)–(40), if the parameter

ae

is such that

ae

>

amin

>0

8X

e; ð49Þ 2 Z Xe

ae

ðheÞ2

l

ð

l

D

uiÞ2d

X

6b Z Xe

l

j

r

uij2d

X

8X

e and

8

i; ð50Þ 2 Z Xe

ae

ðheÞ 2

l

ð

l

‘ 2

_DD

_u iÞ2d

X

6b Z Xe

l

‘2j

D

uij2d

X

8X

e and

8

i; ð51Þ

then there is CInfSup> 0 depending on (‘/he) such that

inf ðuh_;ph_Þ2Vh;k uVhp sup ðvh_;qh_Þ2Vh;k uVhp Ahðuh_;_ph_;

_v

h_;_qh_Þ kjðuh_;_ph_Þjk h;Xkjð

v

h;qhÞjkb;h;X PCInfSup: ð52Þ

Proof. SeeAppendix A.

Remark 1. To proof Lemma1we assume discontinuous pressure interpolation. Formulations with discontinuous pressure interpolations may enforce the incompressibility condition strongly at the element level. However, one can prove, without additional difﬁculties, the stability for continuous pressure, cf.[29].

It can be observed from(52)and the proof of Lemma1that the CDGLS-stabilized formulation(30)is weak coercive with respect to the norm(45)for the discrete problem. Moreover, the results in Lemma1indicate a way to obtain a procedure to determine or bound optimal stability parameters.

5. Numerical results

The following numerical simulations illustrate the performance of the CDGLS formulation. 5.1. Case 1: Plane Poiseuille ﬂow

We consider the steady pressure driven laminar ﬂow between two stationary parallel plates that are separated by a ﬂuid with viscosity

l

and distance d. It is the most common type of ﬂows observed in long, narrow channels (e.g, microﬂuidic

(8)

devices)[30]. An analytical solution to this ﬂow problem governed by the incompressible Eqs.(2) and (3)with no forcing term was developed by Fried and Gurtin in reference[22]. The pressure ﬁeld is only known up to an arbitrary additive con-stant with gradient

Fig. 2. Plane channel flow: stabilized pressure field by CDGLS scheme,a= 0.05, and Lagrangian Q2=Q1finite element with discontinuous pressure.

Fig. 3. Plane channel flow: stabilized pressure field by CDGLS scheme,a= 0.005, Lagrangian Q2=Q1finite element with discontinuous pressure.

0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

dimensionless velocity 2μu/πd2

dimensionless distance y/d

CDGLS,_α=1.5 CDGLS,_α=0.5 CDGLS,_α=0.05 CDGLS,_α=0.005 Exact Classical

Fig. 4. Plane channel flow: exact and numerical velocity profiles across the centerline x/d = 0.5 for different stabilization parameteravalues. All numerical results were obtained with a mesh made up of 100 Lagrangian Q2=Q1finite elements with discontinuous pressure.

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r

p ¼

p

ex ð53Þ

and

p

= constant, whereas the velocity solution u = u(y)exis given by

uðyÞ ¼

p

d 2 2

l

y d 1 y d bl‘ d sinhðd=‘Þ sinh d ‘ sinh y ‘ sinh d y ‘ ; ð54Þ where bl¼ ð2‘=dÞ þ ðl=‘Þ 1 þ ðl=‘Þ tanhðd=2‘Þ ð55Þ

is a nonnegative dimensionless number, and l > 0 is a adherence length scale; cf.[22]

As in reference[20], we use this exact solution which is essentially one-dimensional to construct a two-dimensional boundary value problem to verify the simulation results obtained with the CDGLS-stabilized scheme(30). We solved this ﬂow problem in the rectangular channel domainX= [0, 10d] [0, d] with d = 0.5 mm, prescribing the input boundary con-ditions u = u(y)ex, (ru)n = 0 agreeing with exact velocity(54)on {x = 0, 0 < y < d} and output boundary conditions u free,v = 0,

(ru)n = 0 on {x = 10d, 0 < y < d}. Fixed wall boundary conditions u = 0, (ru)n = g1were speciﬁed on both {y = 0, 0 < x < 10d},

{y = d, 0 < x < 10d}.

We begin by examining the numerical and exact solutions, and the sensitivity of the discrete solution for the stabilization parameter dGLStrying several values of the parameter

a

. For this, we set the material characteristic length ‘equal to d/4 and

length scale l = 0 are considered for the gradient theory. The spatial mesh is made up of 100 elements. The discrete solutions of the pressure ﬁeld are shown in theFigs. 1–3for Q2=Q1ﬁnite element with discontinuous pressure.Fig. 4shows the

veloc-0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

dimensionless velocity 2μu/π d2

dimensionless distance y/d

0.1 1.0 3.0 Classical Exact Numerical

Fig. 5. Plane channel ﬂow: exact and numerical proﬁles across x/d = 0.5 for ratios l/‘equal to 0.1, 1.0 and 3.0. All numerical results were obtained with a mesh made up of 100 Lagrangian Q2=Q1elements with discontinuous pressure.

(10)

ity proﬁles across the center line x/d = 0.5, together with exact and classical reference solution for this problem (cf.[22]), and are similar those obtained by Kim et al.[20]. Note that stability is reached for different

a

values. However, we may observe that for small values of

a

the pressure exhibits small perturbation in the corners of the domain. In the other extreme, if larger

Fig. 7. Plane channel ﬂow: oscillated pressure ﬁeld by CDGLS scheme,a= 0.000015, and Lagrangian Q2=Q2element with discontinuous pressure.

Fig. 8. Plane channel ﬂow: stabilized pressure ﬁeld by CDGLS scheme,a= 0.05, and Lagrangian Q2=Q2element with discontinuous pressure.

0 0.5 1 x 10 −3 0 0.5 1 x 10−3 −1.5 −1 −0.5 0 0.5 1 1.5 x−coordinate y−coordinate Pressure

Fig. 9. Driven cavity ﬂow: oscillated pressure ﬁeld obtained with the Lagrangian Q2=Q1element with continuous pressure; and stabilization parameter

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values of

a

are used the velocity proﬁles present oscillations and do not reproduce the exact solution, while accompanied by reasonable pressure approximation.

Fig. 5compares numerical and exact solutions across the channel center line x/d = 0.5 for the case of generalized

adher-ence conditions using three different ratios l/‘of adheradher-ence length to material length scale, compared with the classical ref-erence solution for this problem. The results show a satisfactory match between the numerical and exact velocity ﬁelds, and are also similar those obtained in reference[20], It is expected that the velocity proﬁles at small length scales are smaller than those of conventional theory, as discussed in reference[22,20].

Figs. 6 and 7show the pressure ﬁeld obtained with Q2=Q2(biquadratic) element with continuous and discontinuous

pres-sure interpolations, respectively, and they clearly depict the adverse effects of violating the inf-sup condition.Fig. 8depicts that the pressure is stabilized, however, presenting slight perturbations in the corners of the computational domain. 5.2. Case 2: Lid driven cavity ﬂow problem

In this second case, we solve the lid driven cavity ﬂow problem in the square domainX= [0, d]2_{at small length scale}

(d = 1 mm) in which the velocity is set to zero and (ru)n = g1 are enforced on all boundary except for the top lid

{y = d, 0 < x < d}, which moves with velocity u = Udexwith speed Udconstant; (ru)n = 0. We also set the material length scale

‘equal to 1.5d, comparable to the geometric length scale d. and the Reynolds number was ﬁxed igual to 2 103_.

We begin by considering the spatial mesh made up of 20 20 elements.Figs. 9 and 10show oscillated pressure ﬁelds for the Q2=Q1ﬁnite element by the CDGLS method with the pressure continuously and discontinuously interpolated,

respec-0 0.5 1 x 10 −3 0 0.5 1 x 10−3 −1.5 −1 −0.5 0 0.5 1 1.5 x−coordinate y−coordinate Pressure

Fig. 10. Driven cavity ﬂow: oscillated pressure ﬁeld obtained with the Lagrangian Q2=Q1element with discontinuous pressure; and stabilization parameter

a= 0.0005 for the CDGLS method.

Fig. 11. Driven cavity ﬂow: stabilized pressure ﬁeld obtained with the Lagrangian Q2=Q1element with discontinuous pressure; and stabilization parameter

(12)

Fig. 12. Driven cavity ﬂow: stabilized pressure ﬁeld obtained with the Lagrangian Q2=Q1element with discontinuous pressure; and stabilization parameter

a= 0.1 for the CDGLS method.

0 0.2 0.4 0.6 0.8 1 x 10−3 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 x−coordinate Pressure CDGLS,α=0.0005 CDGLS,_α=0.01 CDGLS,α=0.1

Fig. 13. Driven cavity ﬂow: pressure distribution across the centerline y = 0.0005; mesh made up of 20 20 elements.

−0.40 −0.2 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

dimensionless velocity u/U

d

dimensionless heigth y/d CDGLS,_α=0.0005

CDGLS,_α=0.01

CDGLS,α=0.1

Classical

(13)

0 0.2 0.4 0.6 0.8 1 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3

dimensionless velocity v/U

d dimensionless width x/d CDGLS,_α=0.0005 CDGLS,_α=0.01 CDGLS,α=0.1 Classical

Fig. 15. Driven cavity ﬂow: normalized velocity proﬁles across x/d = 0.5; mesh made up of 20 20 elements.

0 0.5 1 x 10 −3 0 0.5 1 x 10−3 −0.2 −0.1 0 0.1 0.2 0.3 x−coordinate y−coordinate Pressure

Fig. 16. Driven cavity ﬂow: oscillated pressure ﬁeld obtained with the Lagrangian Q2=Q2element with continuous pressure, and stabilization parameter

a= 0.00025 for the CDGLS method; mesh made up of 10 10 elements.

Fig. 17. Driven cavity ﬂow: oscillated pressure ﬁeld obtained with the Lagrangian Q2=Q2element with discontinuous pressure, and stabilization parameter

(14)

tively.Figs. 11 and 12show that the stability of the large pressure gradients can be reached gradually by using appropriate

a

values. It is important to observe that for a sufﬁciently small value of

a

the pressure is polluted by oscillations even for stable mixed interpolations. However, for large values of

a

there is the risk of over-stabilization by damping out discrete results of the pressure ﬁeld changing the physics of the problem and the pressure on the corners of the domain might not be correctly captured (we recall that the pressure is singular at the top corners for this problem). Observations on this point agree quite well with those of the reference[27]. The pressure distribution along the horizontal line y = 0.0005 is shown inFig. 13, and accompanied by reasonable velocity approximations as shown by the velocity proﬁles across the centerlines x/d = 0.5 and y/d = 0.5 ofFigs. 14 and 15.

We note that these results clearly indicate that the accuracy of the pressure ﬁeld is sensitive to the stabilization parameter.

We conclude this section considering equal order approximations of velocity and pressure which are Galerkin unstable for second order problems; that is, do not satisfy the inf-sup condition. For this, we consider the spatial mesh made up of 10 10 elements. Pressure ﬁelds are shown inFigs. 16 and 17for the biquadratic Q2=Q2element with continuous and

dis-continuous pressure respectively. The pressure ﬁeld obtained with Q2=Q2element is polluted by spurious oscillations. These

adverse effects are caused by violation of the inf-sup condition. The stability of the discrete pressure ﬁeld can be reached gradually by using a suitable stabilization parameter

a

as shown inFigs. 18–20. We note once again that for a large

a

the discrete solutions of the pressure ﬁeld are very diffusive owing to over-stabilization of the large pressure gradients. Conse-quently, optimal or quasi optimal stabilization parameters have to be identiﬁed when using the CDGLS-stabilized method

Fig. 18. Driven cavity ﬂow: stabilized pressure ﬁeld obtained with the Lagrangian Q2=Q2element with discontinuous pressure, and stabilization parameter

a= 0.005 for the CDGLS method; mesh made up of 10 10 elements.

Fig. 19. Driven cavity ﬂow: stabilized oscillated pressure ﬁeld obtained with the Lagrangian Q2=Q2element with discontinuous pressure, and stabilization parametera= 0.035 for the CDGLS method; mesh made up of 10 10 elements.

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Fig. 20. Driven cavity ﬂow: stabilized oscillated pressure ﬁeld obtained with the Lagrangian Q2=Q2element with discontinuous pressure, and stabilization parametera= 0.1 for the CDGLS method; mesh made up of 10 10 elements.

0 0.2 0.4 0.6 0.8 1 x 10−3 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 x−coordinate Pressure CDGLS,α=0.00025 CDGLS,α=0.005 CDGLS,α=0.035 CDGLS,α=0.1

Fig. 21. Driven cavity ﬂow: pressure distribution across the centerline y = 0.0005; mesh made up of 10 10 elements.

−0.40 −0.2 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

dimensionless velocity u/U

d

dimensionless heigth y/d

CDGLS,_α=0.00025

CDGLS,α=0.005

CDGLS,α=0.035

CDGLS,_α=0.1

Classical

(16)

Fig. 21shows the pressure distribution across the line y/d = 0.5, without any signiﬁcant loss for the velocity as shown by the velocity proﬁles across the cavity centerlines x/d = 0.5 and y/d = 0.5 ofFigs. 22 and 23respectively.

All these numerical results reinforce the importance of the GLS stabilization method in the proposed formulation. It can be observed that an optimum stabilization parameter needs to be obtained and that this parameter is not necessarily the same for all the elements in the mesh.

6. Concluding remarks

We have obtained an efficient and stabilized finite-element formulation for fourth-order incompressible flow problems. Our formulation is based on the C0_{-interior penalty and the Galerkin least-square methods, and adopts discontinuous}

pres-sure interpolations. Furthermore, it is weakly coercive for discrete spaces that fail to satisfy the restrictions imposed by the inf-sup condition. Numerical results show that the proposed formulation can effectively stabilize large gradients of pressure, and also indicate that the choice of the discrete pressure and velocity spaces can be made freely. While this approach yields satisfactory results, the accuracy of the discrete results of the pressure is extremely sensitive to the stability parameter used in the GLS method. Therefore an optimum stabilization parameter has to be obtained to ensure accuracy and good conver-gency of the method.

A methodology to determine optimal or quasi-optimal stability parameters of GLS-stabilized finite element methods for second-order incompressible problems has been investigated in Carmo et al.[28,29]. However, the extension of these results to fourth-order problems with incompressibility constraint is not trivial. However the Lemma1indicates that there exists an optimal or quasi-optimal least square parameter that is not necessarily the same for all the elements in the mesh. It also asserts that this parameter depends on the degree of the polynomial used to interpolate the velocity and pressure fields, the geometry of the elements of the mesh and the fluid viscosity term.

From this analysis, we intend to extend the procedure proposed in[28,29]to identify optimal stability parameters of GLS method applied to fourth-order problems with internal constraint. We also hope to be able to obtain a stability analysis with the inf-sup constant independent of the parameter (‘/he), together with a robust error estimate. It will be pursued in the near

future.

Acknowledgement

The support from the Brazilian agency CAPES is gratefully acknowledged. Appendix A

The following elementary inequality will be used in the proof of Lemma1below

ab 61 2

c

a 2_þ1

c

b 2

8

a; b 2 R and

8 c

2 R ð

c

>0Þ: ðA:1Þ

Let k be deﬁned as follows

0 0.2 0.4 0.6 0.8 1 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3

dimensionless velocity v/U

d dimensionless width x/d CDGLS,_α=0.00025 CDGLS,_α=0.005 CDGLS,_α=0.035 CDGLS,α=0.1 Classical

(17)

k¼ kð‘; MhÞ ¼ max ‘ he 2 ; e ¼ 1; . . . ; ne ( ) : ðA:2Þ

Proof of Lemma 1. Let ðuh_;_ph_{Þ 2 V}h;k u V

h

p. The proof proceeds in three steps.

(1) A straightforward calculation yields

Ahðuh_;_ph_;_uh_;_ph_{Þ ¼}X ne e¼1 ð1 bÞ juh_j2 l;H1_ðX eÞnþ ju h_j2 l;‘;Lapþ X e0_>e Z Cee0

su

@ue @ne þ@ue0 @ne0 2 d

C

þX n i¼1 Z Xe

ae

ðheÞ2

l

jRiðuh;phÞj2d

X

! ( þb juh_j2 l;H1_ðX eÞnþ juhj2_l_;‘;Lapþ X e0>e Z Cee0

su

@ue @neþ @ue0 @ne0 2 d

C

þX n i¼1 Z Xe

ae

ðheÞ2

l

jRiðuh;phÞj2d

X

!) : ðA:3Þ

By using the inequality(A.1)with

c

= b in the following identity

b ð1 þ bÞ Z Xe

ae

ðheÞ2

l

@ph @xi 2 ¼ b ð1 þ bÞ Z Xe

ae

ðheÞ2

l

jRiðuh;phÞ þ

l

D

uhi

l

‘ 2

DD

uh ij 2 d

X

; ðA:4Þ we have b ð1 þ bÞ Z Xe

ae

ðheÞ 2

l

@ph @xi 2 6 b ð1 þ bÞ Z Xe

ae

ðheÞ 2

l

ð1 þ bÞjRiðuh;phÞj2d

X

þ Z Xe 2

ae

ðheÞ 2

l

1 þ 1 b ðj

l

D

uh ij 2 þ j

l

‘2

DD

uh ij 2 Þ d

X

! : ðA:5Þ

It follows from(46), (48), (50), (51) and (A.5)that

b ð1 þ bÞjp h_j2 ae;he;l;H1 ðXeÞ6b ju h_j2 l;H1 ðXeÞnþ ju h_j2 l;‘;Lapþ X e0_>e Z Cee0

su

@ue @ne þ@ue0 @ne0 2 d

C

þ Z Xe

ae

ðheÞ 2

l

jRiðuh;phÞj2 ! : ðA:6Þ

Combining(45) and (A.3)with(A.6), we ﬁnd

Ahðuh_;_ph_;_uh_;_ph_{Þ P kjðu}h_;_ph_Þkj2 b;h;X Xne e¼1 ph e

l

1=2 2 L2 ðXeÞ ! : ðA:7Þ (2) k > 1 Shp L 2 ðXÞ and Vhp L 2 ðXÞ

From the result presented in Fortin[31](cf. also[32]) and Lemma (4.19) given in reference[10], we have that for all k > 1 there exists a positive real constant C⁄,Fortin_{> 0 such that, for all p}h_{2 V}h

p, there is

v

h2 V h;k u satisfying Xne e¼1 ð

r

v

h e; p h eÞL2_ðX eÞ¼ Xne e¼1 ph e

l

1=2 2 L2_ðX eÞ ; ðA:8Þ k

v

h_k2 H1 ðXeÞn6C ;Fortin X ne e¼1 ph e

l

1=2 2 L2_ðX eÞ !1 2 : ðA:9Þ

Furthermore, using the inverse inequalities(38) and (39)together with the regularity condition MC1, and by inequality(A.1)

with an appropriate value of

c

and the Fortin’s result(A.9), after some algebraic manipulation, we have

Xne e¼1 Z Xe

l

‘2ð

D

v

Þ2d

X

þ Z Xe

l

j

r

v

j2d

X

þX e0_>e Z Cee0

su

@

v

e @ne þ@

v

e0 @ne0 2 d

C

! 6_CX ne e¼1 k

v

ek2H1 ðXeÞnþ X e0_>e k

v

e0k2_H1 ðXeÞn ! 6_C_ðN_face_{þ 1Þ} X ne e¼1 k

v

ek2H1 ðXeÞn ! 6_C_ðN_face_{þ 1ÞC};Fortin X ne e¼1 pe

l

1=2 2 L2_ðX eÞ ! ¼ eC2 X ne e¼1 pe

l

1=2 2 L2_ðX eÞ ! ðA:10Þ

where Nfaceis the number of faces of the elementXe.

A straightforward calculation yields

Ahðuh_;_ph_;

_v

h_;_{0Þ ¼ A}0 ðuh_;

_v

h_{Þ þ}X ne e¼1

r

v

h e; p h eþ ^p h e L2_ðX eÞþ Xn i¼1 Z Xe

ae

ðheÞ2

l

Riðuh;phÞ

l

D

v

hi d

X

e ! ; ðA:11Þ

(18)

A0ðuh_;

_v

h_{Þ ¼}X ne e¼1 Xn 1¼1 Z Xe

l

r

uh e;i

r

v

h e;id

X

eþ Z Xe

l

‘2

D

uh e;i

D

v

h e;i d

X

e ( þX e0_>e Z Cee0 1 2

l

‘ 2

D

uh eþ

l

‘ 2

D

uh e0 @

v

e @ne þ@

v

e0 @ne0 d

C

þ Z Cee0 1 2 @ue @neþ @ue0 @ne0

l

‘2

D

v

h eþ

l

‘ 2

D

v

h e0 d

C

þ Z Cee0

su

@ue @neþ @ue0 @ne0 @

v

e @neþ @

v

e0 @ne0 d

C

!) : ðA:12Þ Since Xne e¼1 k

r

v

h ek 2 L2 ðXeÞ6n Xne e¼1 k

v

k2H1 ðXeÞn; ðA:13Þ by Cauchy–Schwartz inequality in L2₍

Xe),(44), (49), (A.8) and (A.9)and by Cauchy-Schwartz inequality in Rne, we have

Xne e¼1

r

v

h e; p h eþ ^p h e L2_ðX eÞP Xne e¼1 ph e

l

1=2 2 L2 ðXeÞ ! C2 Xne e¼1 ph e

l

1=2 2 L2 ðXeÞ !1 2 _Xne e¼1 b 1 þ bjp h_j2 ae;he;l;H1_ðX eÞ !1 2 ; ðA:14Þ C2¼ C;Fortin sup n

l

ð1 þ bÞ b

ae

1 2 CePoinc;e ¼ 1; . . . ; ne ( ) : ðA:15Þ

Similarly, by Cauchy–Schwartz inequality in L2₍_X

e),(50), (A.9)and by Cauchy–Schwartz inequality in Rne, we have

Xne e¼1 Xn i¼1 Z Xe

ae

ðheÞ2

l

Riðuh;phÞ

l

D

v

hi P

l

b 1 b 1 2 C;Fortin X ne e¼1 ph e

l

1=2 2 L2_ðX eÞ !1 2 X ne e¼1 ð1 bÞX n i¼1 Z Xe

ae

ðheÞ2

l

jRiðuh;phÞj2 !1 2 : ðA:16Þ

Next, by using the Cauchy–Schwartz inequality in L2(Xe), the inequalities relations deﬁned in (MC1),(38)and by Cauchy–

Schwartz inequality in Rne_{, we can ﬁnd a constant C}c

1>0 such that A0 ðuh;

v

hÞ P Cc1 Xne e¼1 Xn 1¼1 Z Xe ð

l

ð

r

uh e;iÞ 2 d

X

eþ

l

‘2

D

uhe;i 2 d

X

eþ X e0>e Z Cee0

su

C

! ( )1 2 X ne e¼1 Xn 1¼1 Z Xe

l

r

v

h e;i 2 þ

l

‘2

D

v

h e;i 2 d

X

eþ X e0_>e Z Cee0

su

@

v

e @neþ @

v

e0 @ne0 2 d

C

! ( )1 2 ðA:17Þ

It follows from the result(A.10)that

A0 ðuh_;

_v

h_{Þ P C}c 1eC ð1 bÞ Xne e¼1 Xn 1¼1 Z Xe ð

l

ð

r

uh e;iÞ 2 þ

l

‘2ð

D

uh e;iÞ 2 Þ d

X

eþ X e0>e Z Cee0

su

C

!!1 2 1 ð1 bÞ Xne e¼1 ph e

l

1=2 2 L2_ðX eÞ !1 2 : ðA:18Þ

From(A.11), (A.14)–(A.18)and by Cauchy–Schwartz inequality in R3_{, we infer that}

Ahðuh_;_ph_;

_v

h_;_{0Þ P} X ne e¼1 ph e

l

1=2 2 L2_ðX eÞ ! C3 Xne e¼1 ph e

l

1=2 2 L2_ðX eÞ !1 2 X ne e¼1 ð1 bÞ juh_j2 l;H1 ðXeÞþ ju h_j2 l;‘;Lapþ Xn i¼1 Z Xe

ae

ðheÞ2

l

jRiðuh;phÞj2d

X

þX e0>e Z Cee0

su

C

! þ b 1 þ b jphj2ae;he;l;H1_ðX eÞ !1 2 ; ðA:19Þ C3¼ ð3Þ 1 2_sup

l

b 1 b 1 2 C;Fortin;Cc1eC 1 1 b 1 2 ;C2 ( ) : ðA:20Þ

(19)

Combining the classic inequality(A.1)with

c

= 1 and(A.19)we ﬁnd Ah ðuh_;_ph_;

_v

h_;_{0Þ P}1 2 Xne e¼1 ph e

l

1=2 2 L2 ðXeÞ ! C4 Xne e¼1 ð1 bÞ juh_j2 l;H1_ðX eÞþ ju h_j2 l;‘;Lapþ Xn i¼1 Z Xe

ae

ðheÞ2

l

jRiðuh;phÞj2d

X

þX e0>e Z Cee0

su

C

! þ b 1 þ b jphjae;he;

l

;H 1 ð

X

eÞ 2 ! ;: ðA:21Þ where C4¼ ðC3Þ 1

2_{=2. It follows from the norm}₍₄₅₎_that

Ahðuh_;_ph_;

_v

h_;_{0Þ P}1 2 Xne e¼1 ph e

l

1=2 2 L2_ðX eÞ ! C4 kjðuh;phÞjk2b;h;X Xne e¼1 ph e

l

1=2 2 L2_ðX eÞ !! : ðA:22Þ (3) Choosing h¼ 1 1 þ C4þ1₄ ; ðA:23Þ and setting

wh_{¼ ð1 hÞu}h_{þ h}

_v

h _{and q}h_{¼ ð1 hÞp}h_; _ðA:24Þ

and by combining(A.7)with(A.22)(k > 1), we ﬁnd

Ahðuh_;_ph_;_wh_;_qh_{Þ P þðð1 hÞ hC} 4Þ kjðuh;phÞjk2b;h;X Xne e¼1 ph e

l

1=2 2 L2_ðX eÞ !! þh 2 Xne e¼1 ph e

l

1=2 2 L2_ðX eÞ ! ; ðA:25Þ

and therefore it follows that

Ahðuh;ph_;_wh_;_qh Þ Ph 4kjðu h_;_ph Þjk2b;h;X: ðA:26Þ Since ðwh_;_ph_{Þ ¼ ð1 hÞðu}h_;_ph_{Þ þ hð}

_v

h_;_0Þ; _ðA:27Þ

we have by triangular inequality,(45), (50) and (A.9)that

kjðwh;qh Þjkb;h;X6ð1 hÞkjðu h_;_ph Þjkb;h;Xþ hkjð

v

h_;_0Þjk b;h;X6ðð1 b 2 Þ

l

Þ12_C 5kjðuh;phÞjkb;h;X; ðA:28Þ C5¼ C;Fortinð1 þ kð‘; MhÞÞ: ðA:29Þ Setting CInfSup¼ h 4ðð1 b2Þ

l

Þ14_C 5 ; ðA:30Þ ﬁnally yields Ahðuh_;_ph_;_wh_;_qh_{Þ P C} InfSupkjðuh;phÞjkb;h;Xkjðw h_;_qh_Þjk b;h;X; ðA:31Þ

and the result follows immediately, completing the proof. h References

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