An improved inf—sup condition for the spectral discretization of the Stokes problem in a cylinder

Computer methods in applied mechanics and engineering ELSEVIER

Comput. Methods Appl. Mech. Engrg. 175 (1999) 267-280 www.elsevier.com/locate/cma

An improved inf-sup condition for the spectral discretization of the Stokes problem in a cylinder Christine Bernardi* Analyse Num~rique, C.N.R.S. & Universit~ Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France

Received 23 March 1997; revised 22 May 1998

Abstract

We propose a spectral discretizationof the Stokes problem in a cylinder,that relies on the approximationby Fourier series with respect to the angular variable and high degree polynomialswith respect to the radial and axial variables. The aim of this paper is to prove some optimal Babugka-Brezzi type conditionsbetween appropriate discrete spaces of velocity and pressure, in order to derive the best possible error estimates on the pressure. © 1999 Elsevier Science S.A. All rights reserved. R6sum6

Nous proposons une discr&isationspectrale du probl~mede Stokes dans un cylindre,basre sur l'approximationpar srries de Fourier pour la variable angulaire et par des polynrmes de haut degr6 pour les variables radiale et axiale. Le but de cet article est de prouver une condition inf-sup de type Babugka-Brezzi optimale entre des espaces discrets de vitesse et de pression approprirs, de faqon "~obtenir la meilleure estimation possible de l'erreur sur la pression. © 1999 Elsevier Science S.A. All rights reserved.

1. Introduction The Stokes equations govern the flow of an incompressible viscous flow in the case of small velocities. Like most problems issued from physics or mechanics, when set in a three-dimensional axisymmetric geometry, they reduce without any approximation to one problem in the meridian domain in the case of axisymmetric data and to an infinite family of two-dimensional problems for general data: this relies on their formulation in cylindrical coordinates and a Fourier development with respect to the angular variable. Taking advantage of this property seems of high interest since most computations in three-dimensional domains are very expensive. Then, the key idea for the discretization is firstly to solve only a finite number of problems among the infinite family, which means using Fourier truncature, and secondly to apply spectral or spectral element discretization on the two-dimensional domain. Indeed, spectral methods that rely on the approximation by high degree polynomials, are fully appropriate for being coupled with Fourier series, since they have the same infinite degree of accuracy: the order of the discretization only depends of the regularity of the exact solution. We refer to Aza'/ex [1] for a complete analysis of the dimension reduction and of the spectral element discretization of the reduced problem. For the Stokes problem, the analysis of the continuous problem and of its discretization mainly relies on the existence of an i n f - s u p condition of Babu~ka [2] and Brezzi [6] type, which ensures the compatibility of the velocity and pressure spaces. In the spectral method, it is well known that using polynomials of the same degree for each component of the velocity and pressure leads to the existence of spurious modes on the pressure, so that no i n f - s u p condition holds in this case. So several other choices of the space of discrete pressures have been

* E-mail: [email protected] 0045-7825/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S0045-7 825 (98)003 57-0

C. Bernardi / Comput. Methods Appl. Mech. Engrg. 175 (1999) 267-280

268

proposed in the Cartesian case and the behaviour of the best inf-sup constant has been fully identified. In most cases, it depends on the discretization parameter [4, Sections 24-26], however a new choice has been recently introduced in [5], leading to uniform inf-sup condition. Its proof relies on Fortin's trick [7, Section II.2.3, 8], combined with an appropriate projection operator. Most of these results have been extended to the axisymmetric case in [1, Chap. X] but not the uniform condition of Bernardi and Maday [5]. The aim of this paper is to prove an analogue of this inf-sup condition in a model axisymmetric geometry, namely a cylinder. Several projection operators are needed in this case, since the variational formulation of two-dimensional reduced problems involves weighted Sobolev spaces of several types and, even if studying their stability properties is rather technical, they allow for proving two optimal inf-sup conditions: one in the axisymmetric case and one for general Fourier coefficients. The main consequence of this result is of course a better error estimate on the pressure, which is now of the same order as the error estimate on the velocity. An outline of the paper is as follows. In Section 2, we describe the Stokes problem in a cylinder and its reduction to two-dimensional systems, together with the corresponding spectral discretizations. Section 3 is devoted to the stability analysis of several projection operators. In Section 4, we apply the previous results to the proof of the inf-sup conditions and we derive the error estimates on the pressure.

2. T h e Stokes p r o b l e m

Let /2 denote the rectangle f2={(r,z); 0 < r < l

and - l < z < l } .

(2.1)

Its boundary 0/2 is made of two parts: F0 = {0} X] - 1, 1[ and F = 0 / 2 \ F0. The three-dimensional cylinder ~ is built by rotating 12 U Fo around the axis r = 0: /')={(r, 0, z); 0 ~ < r < l , - n - ~ < 0 < z r and - l < z < l } ,

(2.2)

and its boundary a / ) is obtained by rotating F around the axis r = 0. Note that, for a generic point in ~3, we use both Cartesian coordinates (x, y, z) and cylindrical coordinates (r, 0, z) in ~+ x ] - ~-, n-] x R, with r=~x2+y 2

and

0

[-arcc°sX =], x arccos r

if y < 0 , if

0 y ~> "

(2.3)

The Stokes problem in Cartesian coordinates writes -vA~+grad/5=f

in ~ ,

div t~=0

in ~ ,

~=0

on a J ) ,

I

(2.4)

where the data are a positive constant viscosity v and a density of body forces f while the unknowns are the velocity ~ and the pressure/5. Only for simplicity, we assume that the velocity satisfies zero boundary conditions and we refer to [1, Section IX.l] for a more general analysis. In this formulation, Cartesian components of the velocity and the data are used. The new unknowns for the reduced problems are the radial component ~r, the angular component ~0 and the axial component t7 of the velocity, plus the pressure fi, and we consider their Fourier development as a function of 0: -

1

k iko

1

Eu~ei, o

~=1

2,i,

u.e

o

,

1

/ 5 - ,,2/~w ~

ik0

p*e

And it is readily checked that the Stokes problem (2.4) is equivalent to the infinite family of uncoupled problems: for all k in Z',

C. Bernardi / Comput. Methods Appl. Mech. Engrg. 175 (1999)267-280

{

-a~uk~-la,.u~-a]u~+~-Ukr+2i~k2uko+O~pk=f~ r

~

r

2 k 1 k 2 k -a,Uo-rO,.Uo-O:uo+

~-o~u

I

1 +k r2

in/2

r

2

k 2ik k ik uo-~-ur+TP

1 2 ---ou -o_u_ ~ u _ : r r z " " r- "

|a~u~ ~+lu~ l r

269

k

k =fo

in/2, (2.5)

o_p =j_

+ i k u~ + a u ! = O r " "

in /2, in/2

(u~, u~o,u~)=(o, o, o)

onV, k

k

k

where the Fourier coefficients f r , f o and.f: are defined as above. In the case of axisymmetric data, only the Fourier coefficients of order zero of fr, fo and f.= do not vanish, and it is readily checked that the same axisymmetry property holds for fir, fie, fi~ and ft. Note also that problem (2.5) for k = 0 is different from the O other ones: indeed, it results in a Laplace equation for the component u o that we do not consider here (we refer to [1, Chap. VII] for its analysis and its discretization) and an uncoupled problem for the radial and axial components of the velocity and the pressure (we omit the exponents o for simplicity): -

02 1 U ,-.,aU ., r-a-U+ .,~Ur+Orp=rf ,

, ~

1 r

in ,(2

1

__OrU " ---r OrU __{~ U _~_

in /2,

1 OrUr+--U~+&u_=O

in /2

r

-

"

,.(ua~, u~)=(O, O)

(2.6)

on F .

2.1. Variational formulations

In order to write the variational formulations of problems (2.6) and (2.5), we need weighted Sobolev spaces, since using cylindrical coordinates transforms the Lebesgue measure dx dy dz into the weighted one r dr dO dz. So, we firstly introduce two spaces of square-integrable functions on /2"2: L+~(/2)=

v :/2-->C measurable;

Iv(r,z)[gr ~1 d r d z < + zc .

(2.7)

1

Next, we build from L2j(,Q) the full scale of Sobolev spaces HI(/2): when s is an integer, H'1(/2 ) is the space of functions in L2~(/2) such that all their partial derivatives of order ~
(2.8) 1

I

and the two subspaces H l ~ ( / 2 ) , respectively V ~ ( / 2 ) , of all functions in Hll(/2), respectively V j(/2), which vanish on F. All these spaces are provided with the natural associated norms. We refer to [1, Section II.2] for their main properties. The equivalent variational formulation of problem (2.6) is now obvious, it reads: find (u r, u:, p) in 2 Vf~(/2) × H{O(/2) X L~.(/2) such that: V

(Ur,

v : ) ~ V 1,~(/2) x H 1j~(/2) ,

a(u~, Vr)+ V (

J~

Vq~L2~.(/2),

U~V~r J dr dz. + a(u_, v : ) + b ( v r, v.; p ) = ( f , v,.) + (f.:, v:), b(%,u:; q ) = O ,

(2.9)

C. Bernardi / Comput. Methods Appl. Mech. Engrg. 175 (1999) 267-280

270

where L~.(S2) stands for the space of all functions in L21(~2)which have a null integral for the measure r dr dz, while the bilinear forms a ( ' , ") and b(. ;-) are defined by

a(u,v)=v fagradu'grad6rdrdz

(v~,vjq)=-f~ q(a~Or+IO~+a~O<)rdrdz.

and

(2.10)

The continuity properties of these forms is obvious, and the ellipticity of a(', .) on HIO(~2) and the inf-sup condition for b(., .) can easily be derived from the properties of their analogues on J2. So, for any data (fr, ~ ) in the dual space of V~O(J2) X HIO(J2), problem (2.9) has a unique solution. As far as problem (2.5) for k ~ 0 is concerned, we introduce the spaces I X H l "J2" ({(Ur'V°'Uz)EH~(J2) ~k,t ) = l V l ( a ) x V , ( a ) x v l ( ~ )

l H'(O)XV~I(J'2);

Vr+iku°ELZ-'(S2)}

if ]k]-- 1, iflk[1>2,

and their subspaces H i k , O ( O ) = H i k ) ( ~ ) fq Hllo($'2) 3. These spaces are provided with the norm, for 2

+lklllvll ,. > ) 2

,

V=(Ur,UO,Uz),

with

Ikl/> 2, (2.11)

with obvious extension to Ikl = 1. Indeed it is proven in [1, Chap. II, Theorem II.3.6] that the mapping which k k k associates with a function 0 on ~0 its Fourier coefficients (u~, u o, V~)~EZ,k~0 is an isomorphism from H l ( j ' ) ) 3 1 J .(2 ). onto the product IIkE~.k~o H~k~($2) and also from HIo($')) 3 onto 17kEZ.k~0 H~k)O( Thus, problem (2.5) admits the following equivalent variational formulation: find (u k, pk) in Hik~O(O)X L2~(S2), such that

V v@HI~,O(O) , ,~l,(u~,v)+ ~,(v, p ~ ) = < f ~ , v ) , V q~L2,(J2),

(2.12)

~(u~,q)=0,

where the bilinear forms ~¢~(., .) and ~ ( ' ,

~(u,w)=a(u,w)+p

f(l~- - ( u fef : C~k r

.) are defined by

2ik

/,:~

)

+uo~o)+---~(uo~-u~V~o)+---Tu~ ~ rdrdz, r r

~k(w,q)=-jaq(OrVvr+l(wr+ikwo)+Oz~z)rdrdz,

with

U.=(blr, blo, blz),

W=(~&r, Wo, Wz).

Thanks to the previous isomorphism, the continuity, ellipticity and inf-sup properties of these forms are deduced from their three-dimensional analogues. So, for any data f~ in the dual space of H ~ o ( S 2 ) , problem (2.12) has a unique solution.

2.2. The discrete problems We now intend to discretize problem (2.9) and (2.12). For each nonnegative integer n, we define the space P,(J'2) of polynomials on .(2 with degree ~(S2)= Pn(S2) N H~jo(J2) (made of all polynomials in 0z,,(J2) vanishing on F ) and P,°,(J'2)= P,,(J2) N V J~o(~) (made of all polynomials in P n ( ~ ) vanishing on 0J2). This definition is extended to non integral values of n in a trivial way: for any nonnegative real number/z, Iz (J2) coincides with ~Zm(/2), where m stands for the integral part o f / z . Next, for a fixed integer N t> 2, we denote by ~ci and ~, 0 ~
(.,

N+I

N

i=l

j=o

=

E ,(r,,

(2.13)

C. Bernardi / Comput. Methods Appl. Mech. Engrg. 175 (1999) 267-280

271

The discrete space of velocity associated with problem (2.9) is P°u(~2) X PN~(~). In analogy to [5, Section 3], for a real number A such that 0 < A < 1, we choose the discrete space of pressure as

mu*(O) = Pmin{AU,N 2}(f~)OL21*(f2) '

(2.14)

Thus, the discrete problem reads: find (URN, U:N, PN) in P°($2) X PN~(,Q) X M * ( O ) such that V (l)rN , VzN) ~ [~l)/~/(J~) X []])N~(J~) , aN(UrN , OrN) ~- .(URN , O~Nr 2)N d-aN(UzN , UzN) d-bN(U rN, lflzN; pN)=(fr, DrN)N'~-(fz, lflzN)N ,

(2.15)

V qNEMN(~2) * , bN(blrN,UzN; q N ) = 0 , where the bilinear forms aN(', ") and bN(', ") are defined by

aN(I.IN, ON)=l)(gradbIN, gradON)N

and

bN(OrN,UzN; qN)=--(qN, Or UrN-]-UrNr l ~-OzOzU)N.

(2.16)

Note however that, since M* is included in ~DN_Z(f~) and due to the exactness properties of the quadrature formulas, this form bN(" , ") can be replaced by b(., .) in the discrete problem. The spurious modes on the pressure, i.e. the polynomials qN ¢ 0 in PN(~) such that V (UrN , O z U ) ~ O ( f ~ ) X

~N(~'~),

bN(UrN , UzN ; qN) = 0 ,

have been fully identified in [1, Prop. X.1.2], none of them belongs to MN(O ). So, the constant flu defined by

fiN =

inf

b(OrN, l')zN; qN)

sup (UrN,VzN)CP N( )

,

(2.17)

~/Nfl(.(2)

is positive. Consequently, by the same arguments as for the continuous problem, we prove that problem (2.15) has a unique solution. Moreover, the discrete velocity is the solution of a separate elliptic problem set in the discrete kernel of b(-, "), and the distance of the exact velocity to this kernel is evaluated in [1, Prop. X.1.4]. So the following estimate holds: if the solution (u r, u:, p) of problem (2.9) belongs to H~([2) 2 x HI 1(,(2) for a real number s > 1 and if the data ( f , ~ ) belong to H°-(,.Q)2 for a real number o - > 3 :

Hu~-ur~Hvha~+~[u -uz~H"~a~~c~N~-`'~Hur~;¢m+Hu:~;~m)+N-~(~f[~"1"`m+~`n~)~

(2.18)

Things are very similar for the discretization of problem (2.12). We now work with the space XN(f~ ) ~-PN(O)3 N H~k~(O) for discrete velocities, and the discrete space of pressure is even simpler: m N ( f ~ ) = ~[~min{)tN,N_2}(f~) .

(2.19)

The discrete problem now reads: find (Uku,p~) in XNU2) X MN(~2), such that V lfl ~XN(~'~),

V quEmN(f2),

~kN(l~lkN , ON) -~ ~kN(ON, pkN) = (fk, [fiN)N'

(2.20)

~ku(UkN, qu)----0,

where the bilinear forms '~ku(", ") and ~ku(", ") are constructed by replacing, in the definition of ~¢k(", ") and ~k(", "), each integral with respect to the measure r dr dz by the discrete product introduced in (2.13). And as previously, due to the choice of MN(J2), the form ~kU(" ' ") can be replaced by ~ ( ' , -) in the discrete problem. The spurious modes on the pressure in this case have been identified in [1, Prop. X.l.15]. None of them belong to MN({2), SO that the constant t~k) =

inf

sup

~k(ON ' qN)

,

(2.21)

is positive and, for any continuous d a t a f k on ~0 problem (2.20) has a unique solution. We refer to [1, Theorem X.1.21] for the corresponding error on the velocity.

C. Bernardi / Comput. Methods Appl. Mech. Engrg. 175 (1999) 267-280

272

~(k) The end of the paper is mainly devoted to the evaluation of the constants/3 u and/.-.~ N introduced in (2.17) and (2.21), respectively.

3. Some projection operators In this section, we work in the reference interval A = ] - 1 , l[. We observe that the change of variables r = (1 + ( ) / 2 maps A onto ]0, 1[, so that the spaces HI(A) and VJ~(A) are defined exactly as in Section 2, relying on the spaces L2,(A) =

Iv(()12(1+()+-'

v : A--->C measurable;

d(<+~

.

(3.1)

1

Here, we also need the space, for 1 < p < +0%

{

L~;(A) =

v :A---~C measurable;

f' Iv(()lt'(l+()d(<+~ )

.

(3.2)

-I

We introduce the subspaces H~I+(A) and V tt~>(A) of functions in H JI(A) and V~I(A), respectively, which vanish at 1. As in Section 2, for n ~> 0, we define the space P,,(A) of polynomials with degree ~-c,a 1. In this section, we intend to construct several operators with values in P~j+u~M(A), which leave invariant polynomials in PM(A) and are stable in appropriate norms. We firstly recall the result of [5, Section 3]. Let L,,, n 1> 0, denote the Legendre polynomials on ] - 1, 1[: each L,, has degree n, satisfies L,,(1)= 1 and is orthogonal to the other ones in L2(A), moreover it satisfies the differential equation

(( 1 - ( 2 ) L ' ) ' + n(n + I)L,, = 0 .

(3.3)

Then, writing each function ~ in HJo(A) as +~

g)(()= Z

,'~n(L,,+l--L ,)(•),

(3.4)

it = I

we define the operator 7r~4 by (7"rM~p)(() = ~

A,,X

tt=l

(5) (L,,+,-L,,_,)(().

(3.5) I

The following stability results are proven in [5, Lemmas 3.2 and 3.3]: for any function q~ in Ho(A), 1

t

p

1

II(~M~)IIL2.,.~
(3.6)

for a constant c independent of M. By an interpolation argument, this yields the more general result: for 0 ~< s ~< 1 and any function q~ in H~(A),

I1~-~,~11.,, ~, <~cll~ll.,,~,.

(3.7)

We now intend to exhibit analogous operators in the weighted spaces.

3.1. Projection operator in H II(A) We firstly define the polynomials M,,(()=

L,,(() + L,,+, ( ( ) 1+( ,

n~0.

(3.8)

C. Bernardi / Comput. Methods Appl. Mech. Engrg. 175 (1999) 267-280

273

These polynomials, which are introduced in [1, Section III.l.b], are orthogonal to each other one in LZ~(A). We just recall [1, L e m m a III.1.2] that they satisfy the differential equation (( 1 + ()2(1

- ()M'~)'+

n(n + 2)(1 + ( ) M . = 0 .

(3.9)

We now consider the family of polynomials ~o., n/> 0, defined by

M.(() dE.

~p.(() = -

Each function ~p in Ha(A ) admits the expansion +2

~(()= ~

A.~p.((),

(3.10)

n=0

and we define, for an integer M i> 2, the operator 7r~<> by +~

(°) @.(().

1~~---- Z .~,, ~ 77"M

(3.11)

n=O

The first result is nearly obvious: indeed, since the first derivative of each q~,, is M,,, (3.10) and (3.11) provide expansions of both ~p' and (7"rM ~) in the same orthogonal basis. Moreover, the x(n/M) are positive and smaller than 1.

PROPOSITION 3.1. The following stability property holds for any function ~p in H~>(A):

IILwA,--II IIL,,A)

(3.12)

To study the stability in L~(A), we need the next lemma.

LEMMA 3.2. Each polynomial ~o,, n/> 1, satisfies n+2

2

~-(n+l)(2n+3)M.+,

(2n+l)(2n+3)M.

n (n+l)(2n+l)M_,.

(3.13)

PROOF. Thanks to the definition of the M., the polynomial ~p. is orthogonal in L21(A) to all M m, m > n + 1. On the other hand, by integrating by parts and using Eq. (3.9), we have

1 f

fl -1

~p.(f)Mm(f)(l+()d(-m(m+2~

1

M,,(()(1-(

hence it is orthogonal in LZj(A) to all M m, m < n -

2 ,

1

)Mm(()( + ( ) d ( ,

1. So, it can be written

~. = cr.M.+l +fl.M.+T.M._j . In order to identify a., /3,, and y., we firstly look at the coefficient of the leading term ('+~ in q~.(() and we derive that kn+l o~- (n+ 1)k.+ 2 '

where k. denotes the coefficient of ( " in L.((). From the formula k. = (2n)?/(2"(n!)Z), we deduce the value of a.. Since q~. vanishes in 1, we also have

a,,+ fl.+ ~,.=O. Finally, we compute q~'.(1)= 1 = a.

( n + 1)(n + 3 ) n(n + 2 ) ( n - l ) ( n + 1) 2 +/3. 2 +Y" 2 '

and we obtain the desired result by combining these two last equations.

274

C. Bernard, / Comput. Methods Appl. Mech. Engrg. 175 (1999) 267-280

PROPOSITION 3.3. There exists a constant c independent of M such that the following stabili~ property holds for any ~ > 0 and for any function g~ in H~(A): I~

IlrrM ~llc~,a,~
1/2

I1~1[~,~(~,+~ 1/2 I1~11.~(.,,).

(3.14)

PROOF. From Lemma 3.2, we write a different expansion for q~:

q ~ = ~ Iz,,M,, ,,=o

with

] n+l [/Z~=n(~n~l ) A._,

2 ( 2 n + l ) ( Z n + 3 ) A.

n+l (n+2)(2n+3) A,,+,,

n~>l,

and, similarly, for ~'M ~: ,77-M l ~ @---- E t-£nin , n=O

with 2

1

/2o = - ~ ao - ~

a,

n+l /n-l'~ ~z,, n ( ~ X ~ - - ) A , ,

2 (n) (2n+l)(2n+3)X ~

,

a,,

n+l /n+l\ (n+2-)(~n+3)xLY)A"+"

n>~l"

Next, we have M--I 17=0

(I +/~)M

I/x,,I I1 ,,IIL~,A)+

2

I/X,,I IIM,,IIL~¢A),

n=M

(3.15)

and the first term in the right-hand side is obviously bounded by 1]~,l[2L~(a).To estimate the second one, we write n n n-1 I~,=X(~)tz,_(X(~)_X(__~_))

( ( " ] ' "]] n +( nl

n+l

n(2n~

l ) /~n 1 - ~ . x - ) ( ~ . x y / /

n+l

--/~/k M - ] / / (n-l- Y ) ~ ? / - t - 3 ) / ~ n + l

•

Since the derivative of X is bounded by c/x ', this yields 2 1~2,,12<~21/x,,12+c#-2M 2n 2(/A, ,12+1 /~,+11)

.

(3.16)

Now we must compute the A, as a function of the /x,,. When setting ~"

z

an n+l

and

Pn-

Un--E,- l 2n+l '

the recurrence formula can be written P,,+l-

/z,, n+l

P"'

whence the formula A,=(n+l)

1+

(-1)~(2/+ 1) Ao+ =

(2/+1)~ l=l

k=O

( - 1 ) t k /zk ~

"

When inverting the two last summations, this yields

Ia.l<~c(n+l) (n+l)laol+

k=0

~k~i

'

2 is equal to 2/(k + 1)), whence, by a Cauchy-Schwarz inequality (note that IIM ~ll~(a,

C. Bernardi / Comput. Methods Appl. Mech. Engrg. 175 ( 1 9 9 9 ) 2 6 7 - 2 8 0

la°l=c(n+l) (n+l)laol+

~

k+l

:

275

/

<-c(n + I)2(Iaoi ÷ (log(n 4- I))"211~II~,~,~,). To bound Itol, we observe from (3.10) that, since M 0 is equal to 1,

L (f I,~o1< %=

¢ ' ( ()( 1 + ( ) d ( = -

1

¢( O d ( . --1

So, we use a HSlder's inequality: for ~ > 0 ,

I¢(()l~"+~kl+O d

~)'/'="+~')(f]

(14

l

,,)

So combining all together and using once more the formula

~,

n=M

-

1/(1 + 2 e ) ) ( I

d(

+ 2~)/(2(I +e))

l

2

2

~

2

÷

-2

l~,,lllMoll~,A,--cll~ll~i,a, I ~ M

-2

~

IIMolkw~,= 2/(n + 1) yield 2

(n+l)log(n+l)

n=M

( I +I~)M

÷ c'~-'ll~ll~,~,

=

rt=M

When combined with the imbedding of H~(A) into LZl(j +~)(A) [3, Theorem 1.d.2] (note that its norm is bounded independently of e), this gives the desired result.

3.2. Projection operator in V I(A) Here, we work with the polynomials ~,, n i> 1, defined by 1 - ( t / L ', ( ( ) L ,'+ l ( ( ) ) n+lk n + n+2

~0(¢)=

The idea is firstly that these polynomials vanish in -+ 1, secondly that they satisfy, due to (3.3), (1+()

' ( ( l + ( ) ~ b , ) ='

m

,

,

and that the operator in the right-hand side is involved in the divergence operator. Writing any function q~ in VJ~(A) as

¢ ( O = ~] & 6 . ( O , we introduce the operator ~

'° = rrM¢

defined by

(") ~,.(0 •

+~

~] ,kx ~

n=l

We observe that, for any function ~o in V{~(A) (hence vanishing in _+ 1), jr_

p 2

t 2

So the ~9 are orthogonal to each other One in

_

V J~(A) and

2

.

(3.17)

the first result is obvious.

PROPOSITION 3.4. The following stability property holds for any function q~ in Vll~(A): 1o

So, it remains to prove the stability in L~(A).

(3.18)

C. Bernardi I Comput. Methods Appl. Mech. Engrg. 175 (1999) 267-280

276

L E M M A 3.5. Any polynomial ¢t , n >! 1, is written 1

2

1

0n-2n+3M~+~+(2n+l)(2n+3)Mn

2n+lM~-~"

(3.19)

PROOF. As in the proof of L e m m a 2.2, from orthogonality arguments, we firstly derive that ~0 is a linear combination of M,+~, M and M,_~. Secondly, we compute the coefficient of M,+~ by comparing the coefficients of ("+~, and the two other ones by noting that ~b vanishes in -+1. PROPOSITION 3.6. There exists a constant c independent o f M such that the following stability property holds f o r any function ~o in V ~ ( A ) : 1o

liabilitY,a, ~cll~ll~,=~ • PROOF. It is very similar to the proof of Proposition 3.3, so we present it in an abridged way. We write

f q~=~

/*,M,,,

with

l

n=O

1 /*o = - ~ al 2 1 /'1 = ~ - al - ~ a2 2 1 /*~ = 2n + 1 A,,_I + ( 2 n + l ) ( 2 n + 3 )

An

1 2 n + 3 A"+I'

n~>2'

and, similarly, for 7r M ~: l~ ~= 77"M

E /2M,, n=O

1

with

1 /2n

(n-l)

2n+l

X Y

2 An- ' + ( 2 n + l ) ( 2 n + 3 )

(n) X ~-

1 A , - - -2 n + 3

{n+l'~

n~>2.

Formula (3.15) is still valid in our case, and we now write n

n

n-1

{

{n+l'~

n

So inequality (3.16) is still valid here. The formula for the A~ now reads

an=-~](-1)"

'(2l+l)

/=1

(Yo)

/*k ,

so it is readily checked that n--I

IA°I

nE

_<

2

k=O

Combined with (3.16), this gives the desired result. REMARK. By a closer look at the proofs of Propositions 3.3 and 3.6, it is readily checked that the constant c in I/2 (3.14) and (3.20) can be bounded by Co# , where now c o depends on neither M nor/*, and this also holds for the constant c in (3.6) [5, Remark 3.4]. So we can also use these results when the constant/* is replaced by a function of M.

C. B e r n a r d i

/ Comput. Methods Appl. Mech. Engrg.

175 (1999) 267-280

277

4. Back to the Stokes problem Estimating the quantities J~N and t"N[~(k) is now easy: the idea consists in applying Fortin's trick [8] with the previous projection operators. Thanks to the change of variables r = (1 + ( ) / 2 , we now define these operators ~-]4<> and ~-~ on ]0, 1[, we keep the same notation for simplicity. We begin with the constant flu defined in (2.17).

PROPOSITION 4.1. Let A be a real number, 0 < A < 1. There exists a positive constant c o such that flN>~c( A)(log N) 1/2,

(4.1)

with c(A) = Co(1 - A)l/Zllog(1 -- A)1-1/2. PROOF. We consider two cases. (1) When N - 2 is ~
flN>~CN I/2(logN) I. So estimate (4.1) holds with c = c'(1 - A)~/2llog(1 - A)1-1/2 (2) When N - 2 is > AN, with any qN in M*, we associate the pair (v r, v~) in V11©(S2)× H ~ ( S 2 ) which realizes the inf-sup condition of the continuous problem [1, Prop. IX.I.1]:

b(Vr, Vz; qN) = IIqNIl~,~.~>

_<1

and

Ilvrllv,,,~, +[Ivzll..,.,., ~

IIqNIIL,~,~,.

where fl is a positive constant. Next, we set: M = (1 + A)/2N and/~ = (1 - A)/(1 + A), so that N coincides with (1 + / x ) M, and we take UrN :

lo(r) 077" M I(Z) V r 77"M

where the superscript belong to pON(S2) and the M,(2r - 1)/Lm(Z), divergence of (v r, v ) .

and

VzN :

l~(r)

77"M

l(z)

077" M

(4.2)

Vz ,

(r) or (z) represents the variable the operator is applied to. Indeed, these polynomials PN<>(S2), respectively. Moreover, writing the divergence of (Vrg, VzN) in the basis made of we observe that its coefficients of order (m, n), 0 ~< m, n ~2/(1 - A ) , M - 1 is i> AN, and this yields 2

b(v r, v ; qN)= b(V,N, V~N; qN)= liqNll~(.)

-

So, it remains to estimate the norm of (VrN, Vm). Firstly, with obvious notation, we have <

lo+

1(.

i1

I;+o

,,z,

and applying (3.5), (3.18) and (3.20) yields

2cllqNll~,a,.

II~NII~I,-, ~
liV~NIIH',,,~,~IIOr( ~ M

l(z)

°~'M

l~(r)

V=)II~,:(0.I:L=(~, +11~

I(z)

.

o~M vrll~(O,l;.,(A))

and we derive from (3.6), (3.12) and (3.14) that

IlvzNll.:(,~,<~c(llo~vzll~,(o.I:L:(A))+(IogN)I/21IVzlIL:,(O.

--

I:H'(A,, ~-~

-1/211

l(z)

IlrrM Ur H~(O.I;H'(A,,) '

The dependency of the previous constants c with respect to A is made precise in the final remark of Section 3. Only the last term requires more attention. We use the inverse inequality [3, Cor. 3.h.3]

V +%ePu(A),

I1+

II.'<-cN:~II+~II.'-:, e

where the constant c is independent of ~. Since HI(S2) is imbedded in H~ (0, 1; H

l--e

(A)), using now (3.7) yields

278

C. Bernardi / Comput. Methods Appl. Mech. Engrg. 175 (1999) 267-280 H[}zNIIHII(12) ~ C

(1 +(log N ) ' / z + e'/eU2%lv~Nnl~a ) .

Taking ~ equal to 1/(log N) gives the desired result. An immediate consequence is the estimate on the pressure. C O R O L L A R Y 4.2. If the solution (u r, u~, p) of problem (2.9) belongs to HSl (~2) 2 X H~-J(~2) f o r a real number s > 1 and if the data ( f , ~ ) belong to H~(,(2) 2 f o r a real number o" > 7, 3 the following error estimate holds between the pressure p o f problem (2.9) and the discrete pressure PN o f problem (2.15): lip -PuHL 2,(a)<~c( /~)- ' (log

u) l/2((aN)~-~(llurllH;,~) + IluJl.w, + Ilpll,,~ ,,,~))

+

+

(4.3)

f o r the constant c( A) of Proposition 4.1. So, up to log N, the error on the pressure is of optimal order, i.e. of the same order as the error on the velocity estimated in (2.18). We now turn to the case k # 0. The proof is very similar to the previous one. P R O P O S I T I O N 4.3. Let A be a real number, 0 < A < 1. There exists a positive constant c o such that fl'k'>~c(,~) inf{(logN) -j/2 ~ Ikl-'} N

(4.4)

,

with c(A) = Co(1 - A)l/211og (1 - A)I -I/2 PROOF. In view of the definition of HI,)(,(2), we must handle separately the case k = _+1. We omit the proof when N - 2 is ~
~k(v, qN)=]lqullZC~
IlvllH?,,~ IlqNll~,:
(note that this constant /3 does not depend on k). (1) For = +_1, in view of the definition of HI+I)(S2), we define v N = (VrN, VO,, V~N) by

Ikl

VrN+ ikVoN = l~(r)

Uz N = 7"fM

° 1(Z)t"' + i 7"fM(r)°TTM ~vr kv O) ,

VrN--ikVoN = 77"lM~(r)o77"lM(z)(Vr--ikVo) ,

(4.5)

l(z)

o "t7"M Vz ,

By the same arguments as for Proposition 4.1, we derive /3~u-+1~~>C(1Og N)-1/2

(4.6)

(2) For Ikl i> 2, we choose v N in a simpler way: Io(r) l(z) V N 7--- 77" M o ,77"M V .

Indeed, it is readily checked that ~3k(v, qN) =

~k(VN ,

2

and, from (3.6), (3.18) and (3.20), that

However, we must evaluate the norm of v N in HIk)(S2), as defined in (2.11):

Ikl Ilvll

< clkl IIqNII v,, , .

279

C. Bernardi / Comput. Methods Appl. Mech. Engrg. 175 (1999) 2 6 7 - 2 8 0

which yields (4.7)

.

REMARK. The fact that the inf-sup constant depends on k seems unavoidable, as explained in [1, Remark X. 1.30]. So the final error estimate on the pressure depends on the largest Ikl for which problem (2.20) is solved. k Instead of stating the resulting error estimate between pk and PN, we have rather go back to the three-dimensional domain and conclude the analysis of the Spectral-Fourier discretization of problem (2.4). So, we fix a positive integer K and we set

1 UrKN -- X ~

~ k ikO [kl~ K UrN e ,

~ U°KN =

l

Z k ikO , Ikl~K H ON e

~

~ IgzKN --

1

~

k

ikO

~ blzN e Ikl~g

,

(4.8)

P~N-- ~

1

~ p~ el/,0 Pkl~

where o o • for k = 0, UON is an appropriate approximation of u o obtained by the spectral discretization of the o o o corresponding Laplace equation and (U~N, U~N, PN) is the solution of problem (2.15), k k k • for Ikl/> 1, (UkrN, UON , UzN, PN is the solution of problem (2.20). We denote by i~KN the function with cylindrical components ~rKU, ~tOKUand / i r x. Combining Propositions 4.1 and 4.3 with [1, Chap. X] leads to the following result.

COROLLARY 4.4. If the solution (~, ~) of problem (2.4) belongs t o H'~($'))3 )< H ' 1 ( ~ ) f o r a real number s > 1 and if the data f belong to H~(f)) 3 for a real number o-> 3 the following error estimate holds between the solution (ti, fi) of problem (2.4) and the discrete solution (aKN, PXN) defined in (4.8):

Ila-a NIl,,< ,, <~c(g'-~(lla

+c(a)a

' inf{(log N) ,/2, K + IIPII,,

,,

+ (g

`~ +

(4.9)

for the constant c(A) of Proposition 4.3. REMARK. In practical situations, the Fourier coefficients of the data f are not computed exactly but via a quadrature formula with respect to 0, see [1, Section VI.4]. Estimate (4.9) is still valid in this case, but with K - ~ - ~ replaced by K -~. In the previous estimate, the o- in K - ' - ~ only depends on the regularity of the data. So, for smooth functions f, we can work with very low values of K. In this case the estimate on the pressure is nearly optimal.

REMARK. The regularity of (tL/~) is explicitly known as a function of the smoothness of the data [1, Section IX.l.b]. This leads to the final estimate, when f belongs to H 4 ( ~ ) 3 for instance,

IIti-tiKNll,,,~)~ + c( A)A 4 inf{(log N)-~/e, K - ~}[lp- PKullL2,~)<-c(f)(N 4(log N)3/Z + K - 5) . Up to l o g N and log (1 - A), the error on the pressure behaves like (1 - A) '/2(AN) 4 and it is better the space of pressures is chosen equal to ~N_2(~), whenever N is larger ,than A 8 (for instance, if must be />30, which is always the case). It seems that taking the discrete space of pressure equal to M* or M N for 0.7 ~< A ~< 0.9, leads approximation of the pressure in all practical situations. Numerical experiments are presently under tion.

References [1] M. Azai'ez, C. Bernardi, M. Dauge and Y. Maday, Spectral Methods for Axisymmetric Domains, to appear. [2] I. Babugka, The finite element method with Lagrangian multipliers, Numer. Math. :20 (1973) 179-192.

(4.10) than when A = 0.8, N to a better considera-

280

C. Bernardi / Comput. Methods Appl. Mech. Engrg. 175 (1999) 267-280

[3] C. Bernardi, M. Dauge and Y. Maday, Polynomials in weighted Sobolev Spaces: Basics and trace liftings, Internal Report 92039, Universit6 Pierre et Marie Curie, Paris, 1992. [4] C. Bernardi and Y. Maday, Spectral methods, in: EG. Ciarlet and J.-L. Lions, eds., Handbook of Numerical Analysis, Vol. V (North-Holland, 1997), 209-485. [5] C. Bernardi and Y. Maday, Uniform inf-sup conditions for the spectral discretization of the Stokes problem, Math. Models Methods Appl. Sci., to appear. [6] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, R.A.I.R.O. Anal. Num6r. 8 R2 (1974) 129-151. [7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods (Springer-Verlag, 1991). [8] M. Fortin, An analysis of the convergence of mixed finite element methods, R.A.I.R.O. Anal. Num6r. 11 (1977) 341-354.