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On the convergence and stability of the standard least squares finite element method for first-order elliptic systems

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APPLIED

MATHeMATiCS AND

C ~ ~UTAT[©N ELSEVIER Applied Mathematics and Computation 93 (1998) 51-62

On the convergence and stability of the

standard least squares finite element method

for first-order elliptic systems

S u h - Y u h Y a n g 1

Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan

Abstract

A general framework of the theoretical analysis for the convergence and stability of the standard least squares finite element approximations to boundary value problems of first-order linear elliptic systems is established in a natural norm. With a suitable density assumption, the standard least squares method is proved to be convergent without re- quiring extra smoothness of the exact solutions. The method is also shown to be stable with respect to the natural norm. Some representative problems such as the grad-div type problems and the Stokes problem are demonstrated. © 1998 Published by Else- vier Science Inc. All rights reserved.

AMS classifications." 65N12; 65N30

Keywords. Least squares; Finite elements; Convergence; Stability

1. Introduction

T h e p u r p o s e o f this p a p e r is to e s t a b l i s h a g e n e r a l f r a m e w o r k o f the a n a l y s i s for the c o n v e r g e n c e a n d s t a b i l i t y o f the s t a n d a r d least s q u a r e s finite e l e m e n t m e t h o d w h i c h is a p p l i e d to b o u n d a r y v a l u e p r o b l e m s o f f i r s t - o r d e r l i n e a r ellip- tic systems. S o m e e x a m p l e s such as the g r a d - d i v t y p e p r o b l e m s a n d the S t o k e s p r o b l e m a r e o f p a r t i c u l a r interest in this f r a m e w o r k .

1 E-mail: syyang@math.nctu.edu.tw.

0096-30031981519.00 © 1998 Published by Elsevier Science Inc. All rights reserved. PII: $0096-3003(97) 10050-9

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52 S.-Y. Yang / Appl. Math. Comput. 93 (1998) 51~52

In the last ten years, the applications of the use o f least squares principles in connection with finite element techniques have been extensively studied for the approximations in many different fields such as fluid dynamics, elasticity, elec- tromagnetism, and semiconductor device physics. The approach offers certain advantages, especially for large-scale computations. F o r example, it leads to minimization problems rather than saddle point problems by the mixed finite element approach, thus it is not subject to the restriction of the Babu~ka-Brezzi condition; a single continuous piecewise polynomial space can be used for the approximation of all the unknowns; the resulting algebraic system is symmetric and positive definite; accurate approximations of all the unknowns can be ob- tained simultaneously.

The least squares finite element approach represents a fairly general method- ology that can produce a variety o f algorithms. Roughly speaking, according to the boundary treatment, these methods can be classified into the following two categories (see Ref. [1] and references therein for more details): the stan- dard least squares finite element method [2-7] and the weighted least squares finite element method [8,9,1]. Here, standard means that the associated least squares functional is defined to be the sum of the squared L2-norms of the re- siduals o f the differential equations.

In the error analysis o f the least squares methods mentioned above for first- order elliptic systems, there is a problem that the error estimates require rela- tively smooth exact solutions. The error estimates do not guarantee any con- vergence when the methods are applied to problems with low regularity solutions. Accordingly, in Ref. [10], a new least squares finite element method based on a discrete minus one inner product for first-order systems is proposed. The least squares method developed therein is shown to be convergent and sta- ble in some Sobolev's norm as long as the solution belongs to the space Hl+~(f2), for any e > 0. However, this method seems rather tricky to implement in practice.

In the present paper, we shall establish a general framework o f the analysis for the convergence and stability of the standard least squares finite element approximations to first-order elliptic systems. By using the standard density ar- gument [11], we prove that, without requiring extra smoothness o f the exact so- lutions, the standard least squares method is convergent in a natural norm associated with the least squares bilinear form. We also show that the method is stable with respect to the natural norm. Furthermore, for many examples as we shall present, the natural norm is equivalent to some appropriate Sobolev's norm. Therefore, for these model problems at least, we have established the convergence and stability in some Sobolev's norm without any extra require- ment on the regularity o f the exact solutions.

The remainder of the paper is organized as follows. In Section 2, we intro- duce the standard least squares finite element method for first-order elliptic sys- tems. In Section 3, we establish the main results for the convergence and

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S.-E Yang I Appl. Math. Comput. 93 (1998) 51-62

53 stability. In Section 4, some representative examples are given. Finally, in Sec- tion 5, some concluding remarks are drawn.

2. The least squares finite element method

Throughout this paper, the classical Sobolev space

HS(f2),s >~ 0

integer, with its associated inner product (., ')s,a and norm [[. [Is,o, are employed [11- 13]. As usual, L2(f2) = H°(g2). For the product space [H~(f2)] m, the correspond- ing inner product and norm are also denoted by (.,

"),,0

and [[ " [[s,a, respective- ly, when there is no chance for confusion.

As usual, L~(~2) will denote the subspace of square integrable functions with zero mean, i.e.,

fov d x = O

for all v E L02(f2). By L~(f2) and L~(Sf2) we denote the usual Banach spaces of measurable and essentially bounded real-valued functions defined on f2 and 5f2 with the norms [[. 1[~,o and [[ " [[~,ao, respectively. Let @(f2) denote the linear space of infinitely differen- tiable functions with compact support in f2, and let @(9) denote the restric- tions of the functions in ~(Nd) to 9. It is well-known that ~ ( ~ ) is dense in Hi(Q).

We shall consider the standard least squares finite element approximations to the boundary value problems of first-order linear elliptic systems in the gen- eral form:

f~tA ~ U

' ~ x i + A ° U = F

inf2, (2.1)

BU = G

on 5f2, (2.2)

where f2 C Rd, d ~> 2, is an open bounded connected domain with a smooth boundary 5Q, and U = ( U l , . . . , U m ) T , F = ( J i , . . . , f m ) T , G = ( g l , . . . , g , ) a'. In this paper, we shall always assume that the entries of m × m matrices

AiE[L~(f2)]m×m,o<<,i<<,d,

and the entries of

n x m

boundary matrix B E [L~(512)] "×m are regular enough on D and 5Q, respectively, such that prob- lem (2.1) and (2.2) has a unique strong solution U C [H 1 (f2)] m with the given functions

F ~ [L2(Fa)]m,G C

[LZ(~f2)] ". For simplicity, we also assume that G = 0 on the boundary 5f2.

We now introduce the standard least squares finite element method for problem (2.1) and (2.2). Define a function space ~ for our problem by

= { V ~_ [Hl(~'~)]m;BV = 0 on ~f2}, (2.3)

and then define a standard least squares energy functional J : ~ ---+ ~ by d 5V - F 0 2,a"

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54 S.-Y.

Yang / Appl. Math. Comput. 93 (1998) 5142

Obviously, the exact solution U E ~/r of problem (2.1) and (2.2) is the unique zero minimizer of the functional J on ~U, that is,

J ( U ) = 0 = m i n { J ( V ) ; V E ~ } . (2.5)

Applying the variational techniques, we can find that (2.5) is equivalent to

~ ( U , V) = ~ ( V ) VV E ~ , (2.6)

where the bilinear form ~(., .) and the linear form ~ ( . ) are defined, respective- ly, by

:f(di~lAO

Vi_~_AO

V ~ x i )(~__~Ai~..~.._~_AoW~dx,~=]d

~Woxi

J (2.7) ~(V, W)

~ ( V ) = F I ~ , A i - - + A o V dx (2.8)

for all V, W E ~U. Therefore, the standard least squares finite element method for problem (2.1) and (2.2) is to determine Uh ~ ~ h such that

~(Uh, Vh) = ~(Vh) VVh E Uh, (2.9)

where the finite element space ~ h C ~ is assumed to satisfy the following ap- proximation property. For any V c V

N [Hp+l(~)]m,p

~ 0 integer, there exists Vh ~ "Us such that

[I v - Vhl[1,~ ~< ChPllvllp+l,~, (2.10)

where the positive constant C is independent of V and the mesh parameter h. Approximation property (2.10) is satisfied for usual finite element spaces pro- vided the associated family of triangulations {Y'h} of S2 is regular [11].

Throughout this paper, in any estimate or inequality the quantity C will de- note a generic positive constant always independent of h and need not neces- sarily be the same constant in different places.

3. Convergence and stability

It is clear that ~(-, -) defines an inner product on ~ × ~ since the positive-def- initeness is ensured from the fact that problem (2. l) and (2.2) possesses the un- ique solution U = 0 for F = 0 and G -- 0. Denote the associated natural norm by

IlVllb = { ~ ( v , v)} 1/2 Vv ~ ~ . (3.1)

Although we do not know whether the I[.

lib-norm

is equivalent to the ][. Ill,a- norm or not, evidently there exists a positive constant C such that

IIVllb~< CIIVlll, ~ VV ~ ~ (3.2)

d

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S.-Y. Yang / Appl. Math. Comput. 93 (1998) 5 1 4 2 55 We first state some fundamental properties of the standard least squares fi- nite element scheme (2.9).

Theorem 3.1. Let U E [H 1 ((2)] m be the exact solution of(2.1) and (2.2) with the given functions F C [/,2(0)] m and G = O.

(i) Problem (2.9) has a unique solution Uh E ~Uh which satisfies the following stability estimate."

IIU,,L <<. IIFII0,~.

(3.3)

(ii) The matrix of the linear system associated with problem (2.9) is symmet- ric and positive definite.

(iii) The following orthogonality relation holds:

~ ( U - Uh, Vh) = 0 VVh E ~t/~h. (3.4)

(iv) The approximate solution Uh is a best approximation of U in the II • lib-

norm, that is,

IIU- u~llb

-- inf

I I g - ~11~.

(3.5)

Vh E'~h

Proof. T o prove the unique solvability, it suffices to prove the uniqueness of solution since "¢h is a finite-dimensional space. Let

Uh

be a solution of (2.9), then we have

Iluhll2b = ~ ( u h , uh) = ~ ( u ~ )

IIFll0,•

ai~lAi~-Uh

+ AoUh

~xi

II0,~

~< IIFll0~llU~llb,

which implies (3.3). Consequently, the solution Uh of (2.9) is unique.

Assertion (ii) follows from the fact that the bilinear form ~(.,-) is symmetric and positive definite. (iii) is obtained by subtracting Eq. (2.9) from Eq. (2.6). Using (3.4) and the Cauchy-Schwarz inequality,

IIU - U~[[2~ = ~ ( U - U~, U - U~)

= ~ ( u - uh, u - vh) vvh ~ ~ h <<. I I u - uhll~llu -

~11~ v ~ c ~ ,

we prove (iv). []

Estimate (3.3) indicates that the standard least squares method is stable with respect to the ]]. [[b-norm, that is, when we change the given data function F slightly in the L2-norm, the least squares solution Uh changes only slightly in

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56 S.-Y. Yang / Appl. Math. Comput. 93 (1998) 51-62

the [I " lib -n°rm. Moreover, by using the standard density argument [11], we can obtain the following results for the convergence.

Theorem 3.2. Assume that there exists a subspace 5 ¢ C ~ A [1t q+l (•)]m, for some integer q >1 1, which is dense in the space ~U with respect to the [[ " Ill,Q- norm. Then the standard least squares finite element method (2.9) is convergent with respeet to the [[ • lib-norm without requiring any extra regularity assumption on the exact solution U, i.e.,

d ~Uh 0,4

lim ~ - " A i - - + A o U h - F = 0 . (3.6) Moreover, if the exact solution U E ~/~ A [H p+l ([2)] m, then we have the following error estimate."

IIg -

ghtlb <-

Ch~llgllp+l,~,

(3.7)

where C is a positive constant independent o f h.

Proof. Since the subspace ~ c ~ ~ Egq+l(~)] rn, is dense in ~ with respect to the II • lit,Q-norm, for any e > 0, there exists U E 5 ~ independent o f h such that

£

IIg

- 0[II,~ < ~---~,

where C is the same constant as in (3.2), which implies

(3.8)

I I g - Oils < ~.

F o r this fixed smooth function 0 E 5e C [gq+l(~)] m, q • 1, by the approxima- tion property (2.10), we can find Oh E Uh so that,

II 8 - 0~ II ~,~ ~<

ChUll

Ollq+l,~ which implies, for sufficiently small h,

118-

Ohllb---< ctlg-

l[-~]h[[1,

~ < ~.

(3.9)

Combining inequalities (3.8) and (3.9) with (3.5), we immediately obtain

0~< I I u - u~ll~ ~

< I I U - O~llb ~ I I U - OIIb + I I 0 -

Ohllb < e

which implies (3.6). We now assume that U E "f/- ~ [H p+I (f~)]m. By (3.5), (3.2) and the approximation property (2.10) of the finite element space ~/Fh, we ob- tain (3.7). This completes the proof. []

4. Examples

Unless otherwise specified, we assume in this section that the dimension d is two or three. We begin with two preliminary lemmas.

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S.-Y. Yang / Appl. Math. Comput. 93 (1998) 5 1 ~ 2 57 Lemma 4.1 (The Poincar6-Wirtinger inequality [13]). L e t f2 be an open bounded connected subset o f ~d, d ~ 2, with a C 1 boundary ~f2. Then there exists a constant C = C(d, f2) > 0 such that f o r every v E H 1 (f2), we have

IIv - vollo,~ ~ c I I V v l l o ~ (4.1)

where

1 /

va . - meas((2) vdx a is the average o f v over f2. Defining the function space

-~(0) = .~(~) n Lo2(~),

we obtain, from Lemma 4.1, the following result.

(4.2)

(4.3)

Lemma 4.2. The space ~(-~) is dense #t Ht (f2) ~ Lo2(f2).

Proof. Since ~(O) is dense in H 1 (O), for every v E H 1

((2) N

Lg(f2), there exists a sequence {vn} in ~(f2) such that

IlVn -- V 2 112

IIv.-vtl2o,~+llV(v.-v)tl~o,~o

a s n - - + o e . Define the following sequence {c,} of real numbers,

1 / c, . - meas(f2) v, dx, f2 then

1 /

c. - meas(f2) (v. - v)dx Q

since f o v dx = 0. Applying the Poincar4-Wirtinger inequality to v. - v for all n, we have

o 4 II(v. - c . ) - vll~,~ = II(v. - c . ) - vll20+~ + I l V ( v . - c . ) - Vvl{~,~

= II(v. - en) - vl120,~ + IIV(v. - v)lP~,~

~< C l l V ( v ~ - v)llo~ ~ 0 a s n ~ o c .

This completes the proof. []

Let Ho 1 (f2) be the closure of ~(f2) in H 1

(Q),

then

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58 S.-Y. Yang / Appl. Math. Comput. 93 (1998) 5142 We n o w introduce o u r first example.

Example 4.1 (The velocity-vorticity-pressure Stokes equations). Let f2 C ~a,d = 2, and let f = ( f l , f 2 ) T E [L2(f2)] 2 be a given function representing the b o d y force. T h e Stokes equations with h o m o g e n e o u s velocity b o u n d a r y conditions can be posed as:

- Au + grad(p) = f in f2, (4.5)

div(u) = 0 in O, (4.6)

u = 0 o n ~f2, ( 4 . 7 )

(p, 1)o,n = O, (4.8)

where u = (ul, u2) v is the velocity and p is the pressure, all o f which are as- sumed to be nondimensionalized. I n t r o d u c i n g the auxiliary variable 09 = curl(u) := ~u2/~x - ~Uy/~y, which is k n o w n as the vorticity, we can trans- f o r m (4.5)-(4.8) into a first-order system [14] as follows:

curl(09) + grad(p) = f in f2, (4.9)

curl(u) - 09 = 0 in [2, (4.10)

div(u) = 0 in f2, (4.11)

u = 0 o n ~I2, ( 4 . 1 2 )

(p, 1)0,a = 0, (4.13)

where curl(09) := (09y,-09x) T is a n o t h e r curl operator.

Applying the least squares finite element scheme (2.9) to the first-order sys- tem (4.9)-(4.13) with

t / ' = {(v,<p,q) T E [H~(Q)] 2

×HI(Q)

xHI(~-2);qEL~([2)}

and taking

__ ~ __

= ~ ( ~ ) × 9 ( 0 ) × ~ ( ~ ) × ~ ( ~ ) ,

which is contained in V fq [H2(g2)] 4, we have the following convergence results that follow f r o m (4.4), L e m m a 4.2, and T h e o r e m 3.2,

[[curl(09 - 09h) + g r a d ( p - Ph)II0,~ + Ilcurl(u - Uh) -- (09 -- 09h)110,~

+ I l d i v ( u - Uh)ll0,o ~ 0 as h ~ 0, (4.14)

provided the exact solution U = (u, 09,p)X E • ; and

I l c u r l ( 0 9 - 09h) + g r a d ( p - ph)ll0,o + I l c u r l ( u - uh) - ( 0 9 - 09h)110,~

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S.-Y. Yang / Appl. Math. Comput. 93 (1998) 51~62 59 if U = (u, o J , p ) T E "//" fq [Hp+I((2)] 4, where Uh = (Uh, Ogh,ph) T E 3V'h is the least squares finite element solution.

For the case of sufficiently smooth exact solutions, the error estimates can be found in, for example, Refs. [15-17]. More specifically, in Ref. [17], we have proved that there exists a positive constant C such that

[Ivll,,~ + [[~°110,~ + [Iqll0,o <~ C(llcurl(q~) + grad(q)l]0,~ + ][curl(v) - q~]]0,~.

+lldiv(v)ll0~)

VV

=

(v, qLq) T E 3U.

(4.16)

Combining (4.16) with (4.14) and (4.15), we have, respectively,

Ilu - uhll,,~ + IIo~ - cohllo,~ + lip --Phll0,~ --' 0 as h ~ 0, (4.17)

Ilu

-

uhlll,~ + IIco - ~ohlt0,~ + lip - phll0,~ ~<

ChPllUIIp+l,~.

(4.18)

In this example, we focus our attention on the two-dimensional velocity- vorticity-pressure Stokes equations with velocity boundary conditions. All of the results developed here can be extended to the three-dimensional case direct- ly [15,17].

So far, we have been mainly interested in the case of ~ c [H' (I2)]m; but all of the results developed in Section 3 can be easily applied to the function space V with less regularity. Bearing this in mind, we introduce the following func- tion spaces:

H(div; f2) := {V E [L2(f2)]d;div V C L 2 ( ( 2 ) } , (4.19) H0(div; O) := the closure of [~(f2)] a in H(div; O), (4.20) where the space H(div; f2) is equipped with the following inner product and n o r m ,

(V, W)n(aiv;~ ) := (V, W)0,o + (div V, div W)0,a VV, W E H(div; (2), (4.21)

11V[lH(div;f2)

: =

(ll Vll~,,~

+ Ildiv Vll20,,~)'/2 v v

~ H(div; f2), (4.22)

which make it a Hilbert space [12].

The following lemmas will be required later and their proofs can be found in [12].

Lemma 4.3. The space [~(~)]d is dense in H(div; f2).

Lemma 4.4. Let n be the outward unit normal vector to ~£2, then we have Ho(div; f2) = {V E H(div; f2); V. n = 0 on Of 2}. (4.23) E x a m p l e 4.2 (Grad-div type problems). Let the smooth boundary 8 0 of domain f2 c ~ d , d = 2 or 3, be partitioned into two disjoint open parts, FD and FN,

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60 S.-Y. Yang / Appl. Math. Comput. 93 (1998) 5142

such that ~f2 = FD U FN. We consider the following b o u n d a r y value p r o b l e m with mixed type b o u n d a r y conditions:

- div(A grad u) = f in g2, (4.24)

u = 0 on FD, (4.25)

( A g r a d u ) • n = 0 on IN, (4.26)

where A = (aij(x))a×a, aij C L~(-O), and we assume that A is symmetric and uni- formly positive definite on f2. It is u n d e r s t o o d that if F u = ~(2, we further re- quire the compatibility condition, f o f d x = 0, and impose an additional constraint on u such as fo u dx = 0, for the well-posedness.

I n t r o d u c i n g the auxiliary variables, p = - A grad u on O, we can reformulate p r o b l e m (4.24)-(4.26) to the following equivalent first-order form:

p + A g r a d u = 0 in f2, (4.27)

div p = f in O, (4.28)

u = 0 on FD, (4.29)

p - n = 0 OnFN. (4.30)

Following Section 3 with m i n o r modifications, we can apply the s t a n d a r d least squares m e t h o d (2.9) over the extended first-order system (4.27)-(4.30) with

• : { ( v , q ) a ' E H l ( ~ 2 ) x H ( d i v ; O ) ; v = 0 o n F D a n d q - n : 0 o n F N } ,

provided the associated finite element space ~ h c ~ also possesses the follow- ing a p p r o x i m a t i o n p r o p e r t y [2,7]. F o r any V = (v,q) a" E ~ N [HP+t(O)] a+l, p 7> 0 integer, there exists Vh = (vh,qh) a- E ~ h such that

2 , (4.31)

IIv -

vhll~,,~ +

IIq -- qhlln(~iv;~/~< Chp { llvll2+l,~ +

llqllp+l,~} 1/2

where the positive constant C is independent o f V and the mesh p a r a m e t e r h. Similar to T h e o r e m 3.2, we can prove that, w i t h o u t any extra regularity as- sumption on the exact solution U = (u, p)X,

II

(p

- ph) + A grad (u - uh)[[0,~ + [Idiv(p - ph)II0,~

= IlPh +Agraduhll0,o + Ildivph - f t l 0 , o --* 0 as h ~ 0, (4.32) w h e n F D = SO o r FN = ~ ' 2 , where Uh = (uh, p h ) T E 3~h is the least squares fi- nite element solution, and we choose

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S.-Y. Yang / Appl. Math. Comput. 93 (1998) 51-62 61 a n d

5 ~ = ~ ( ~ ) x [@(Q)]a i f F u = 012

b o t h o f which are c o n t a i n e d in ~ fn [H2(O)] d a n d dense in U with respect to the H I (f2) x H(div; f2) n o r m (cf. (4.4), L e m m a 4.3, a n d L e m m a 4.2; (4.20), L e m m a 4.4, respectively).

F o r sufficiently s m o o t h exact solutions, the error estimates can be f o u n d in, for example Refs. [2,3,7].

5. Concluding remarks

In this paper, we establish a general f r a m e w o r k o f the theoretical analysis for the convergence a n d stability o f the s t a n d a r d least squares finite element m e t h o d for first-order problems. W i t h a suitable density assumption, the meth- od is p r o v e d to be c o n v e r g e n t in a natural n o r m w i t h o u t requiring extra regu- larity o f the exact solutions, a n d with respect to the same n o r m , the m e t h o d is also stable.

By the example shown in the previous section, we observe that c h o o s i n g a suitable dense subset 5e o f ~U plays a crucial role in the analysis o f the conver- gence. A useful tool for c h o o s i n g 5 e is based on the following p r o p e r t y o f Ban- ach spaces [12], p. 26:

A subspace 5P o f a Banach space ~ is dense in ~U i f and only i f every element o f U" that vanishes on 5 ~ also vanishes on ~ ,

where V ' denotes the dual space o f ~ . Actually, L e m m a 4.3 a n d L e m m a 4.4 are two applications o f the p r o p e r t y , whose p r o o f s c a n be f o u n d in Ref. [12].

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[15] P.B. Bochev, M.D. Gunzburger, Analysis of least squares finite element methods for the Stokes equations, Math. Comp. 63 (1994) 479 506.

[16] C.L. Chang, A mixed finite element method for the Stokes problem: An acceleration-pressure formulation, Appl. Math. Comput. 36 (1990) 135-146.

[17] C.L. Chang, S.-Y. Yang, Piecewise linear approximations to the Stokes problem with velocity boundary conditions, preprint, SIAM J. Numer. Anal. (submitted).

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