a least squares petrov-galerkin finite element …...volume 60, number 202 april 1993, pages 531-543...

mathematics of computationvolume 60, number 202april 1993, pages 531-543

A LEAST SQUARES PETROV-GALERKIN FINITE ELEMENT METHODFOR THE STATIONARY NAVIER-STOKES EQUATIONS

TIAN-XIAO ZHOU AND MIN-FU FENG

Abstract. In this paper, a Galerkin/least squares-type finite element method is

proposed and analyzed for the stationary Navier-Stokes equations. The method

is consistent and stable for any combination of discrete velocity and pressure

spaces (without requiring a Babuska-Brezzi stability condition). The existence,

uniqueness and convergence (at optimal rate) of the discrete solution is proved

in the case of sufficient viscosity (or small data).

1. Introduction

For mixed finite element methods solving the stationary Navier-Stokes equa-

tions, it is an important convergence stability condition that the Babuska-Brezzi

inequality holds for the combination of finite element subspaces (see [1, 13]).

Recently, in an attempt to circumvent this constraint, the so-called CBB [6] or

stabilized finite element methods [2-5] have been developed, motivated by SD

(or SUPG) methods [7, 8]. In addition to works [3-6] on the Stokes problems,

the paper [9] proposed and analyzed a stabilized SD method for time-dependent

N-S equations, and a stabilized, piecewise discontinuous vorticity-stream func-

tion formulation of mesh-dependent type for stationary N-S equations has beendiscussed in the paper [ 10] based on so-called homology families of generalized

variational principles.The present paper considers the stationary N-S equations in primitive vari-

ables. In this direction, L. Tobiska and G. Lube [12] proposed a penalty finiteelement method of streamline diffusion type. It is a stabilized method in whichthe finite element spaces of velocity and pressure are not required to satisfy the

discrete B-B condition. But it is not consistent with the exact solution, owing

to the addition of the penalty term a(Vp, Vq), and the optimal estimates of

convergence rate cannot be achieved. In this paper, another stabilized finite

element method is studied, which is different from the method in [11, 12]. It

is an application of the Galerkin/least squares method [14] and its alternative

[16] to nonlinear equations. Least squares forms of residuals are added to the

Galerkin method for enhancing its stability without degrading accuracy.

For the following presentation we introduce the following notation: X =

VxQ, V = Hx(ü)", Q = L2(Çl) = {qe L2(Çi)\Jaqdx = 0}, (o, o)G the

Received by the editor January 9, 1991 and, in revised form, August 21, 1991 and March 31,

1992.1991 Mathematics Subject Classification. Primary 65N30.The research was supported by the Chinese Aeronautical Science Foundation.

©1993 American Mathematical Society0025-5718/93 $1.00+ $.25 per page

531

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

532 TIAN-XIAO ZHOU AND MIN-FU FENG

inner product in L2(G) and L2(G)n, respectively, C7CÍ2. Let \\w\\k p G and

\w\k,p,G be the usual norm and seminorm on the Sobolev space Wk>p(G),

respectively. For vector-valued functions u = (ux, ... ,un) £ Wk'p(G)n and

v = (vx, ... , v„) £ L°°(G)n we use the following norms and seminorms, respec-

tively: \\u\\PkpG = E"=lHMC,G> H,p,G = 2ZU\Ui\Pk>p>G, Ho.oo.G =max, ||v,||o,oo,g • In the case of G = £2 and p = 2 we omit the indices G and

P-Throughout the paper, C indicates a positive constant, possibly different at

different occurrences, which is independent of the mesh parameter h , but may

depend on Q, on the Reynolds number and other parameters introduced in this

paper. Notations not especially explained are used with their usual meanings.

An outline of the paper follows. In §2 we present the new finite element

variational formulation for the N-S equations. The existence and uniqueness

of the finite element solution is studied in §3. Its error analysis is performed in

§4, and concluding remarks are made in §5.

2. Finite element formulation

Let Q be a convex domain with boundary r in Rn (n = 2, 3). We consider

the following stationary Navier-Stokes equations with boundary conditions:

-pAu + (uV)u + Vp =f in £2,

(2.1) divw=0 inQ,

u = 0 on T,

where u = (ux, ... , u„) is the velocity vector, p the pressure, /' = (f , ... , fn)

the body force, and p the constant inverse Reynolds number. Problem (2.1 ) is

equivalent to the following variational problem:Find (u, p) £ X such that for all (v , q) e X

(2.2) pa(u, v) + b(u; u, v) - (p, divv) + (q, divu) = (/, v),

where

a(u, v) = VuVv dx,Jn

b(u;v,w)= 52 / Uidvj/dXiWjdx Vu, v , w e V,, Jn',7=1

b(u; v , w) = \{b(u; v , w) - b(u; w, v)} Vu,v,weV.

We define

N= sup f^y;^, u/u. = sup iL^l.u,v,w€V \u\l\V\x\W\x V£V Ml

Theorem 2.1 [1]. If f £ H~x(Çl)n , then the problem (2.2) has a solution which,

in addition, is unique provided that p~2N\\f\\+ < 1.

Let {S?~h} be a family of triangulations of Q into affinely equivalent finite

elements K with Q = {jKe^- K, which is assumed to be regular in the usual

sense, and let /?* = diam/Y. We also assume that h/hx < C, VX e ^,

h = maxtf hfc, so that we can use inverse inequalities.


A PETROV-GALERKIN FINITE ELEMENT METHOD 533

We introduce the following finite element spaces of velocity and pressure:

Vh(Q) = {v£ Hx(Ü)n:v\K £ P,(x), Vie^},

Qh(n) = {q£QnHx(n):q\K£Pk(x), Vtfe^}.

Here, P¡(x), Pk(x) denote piecewise polynomials of degree / and k , respec-

tively. We let Xh = VhxQh.Paper [12] proposed the following penalty finite element method of stream-

line diffusion type with the penalty term a(Vp, Vq) for (2.2):

Find ûn = (Uf¡, Ph) £ Xn such that for all v £ Xh

pa(uh , v) + b(uh ; uh,v)- (ph , dxsv) + (q, divuh)

+ Yl ÖK(-P&uh + (uhV)uh + Vph , (uhV)v + Vq)K

(2.3) *+ a(Vph, Vq)

= (f,v) + YlWAuhV)v + Vq)K.K

In order to establish existence, uniqueness, and convergence of the solution of

(2.3), the parameter Sk is required to satisfy the condition 0 < 6k < Cxp~xh2 ,

where Ci is a certain constant.

In this paper, we present the following Petrov-Galerkin finite element formu-

lation for problem (2.1): Find «/, = («/,, pn) £ Xh such that for all v £ Xh

pa(uh, v) + b(uh; un,v)-(ph, divv) + (q, divuh)

+ YôK(-pAuh + (uhV)un + Vph , -pAv + (uhV)v + Vq)K(2.4) r

= (/, v) + 5]^(/, -pAv + (uhV)v + Vq)K,K

where Sk = ahj,, and a > 0 is arbitrary.

For u £ V , uh £ Vh , v = (v , q), w = (w , r) e V x (Q n Hx (Q)), we define

Bô(u,un;v,w) = pa(v , w) + b(u; v , w) - (q, divu;) + (r, divu)

+ 52 ôK(-pAv + uVv + Vq, -pAw + uhVw + Vr)K ,K

Ls(uh ; w) = (f, w) + ^5K(f, -pAw + uhVw + Vr)K ,K

where ô is the piecewise constant function defined by ô\k = ôK ■ Then (2.4)

can be rewritten in the following form: Find û/, = (un, Ph) £ X^ such that

(2.5) Bs(uh,uh;ûh,w) = Ls(uh;w) Vu) £ Xh.

Remark X. Assume f belongs to L2(Q)n and the solution (u,p) of (2.1)

belongs to (V n H2(Q)n) x(Qn //'(«)) ; i.e., there holds

-pAu + uVu + Vp = f in L2(Q)".

Then û = (u, p) satisfies

(2.6) Bs(u,uh;û,w) = Ls(uh;w) Vw/, £ Vh, w £ Xh ,

i.e., (2.5) is consistent with the exact solution of problem (2.1) or (2.2).


534 tian-xiao zhou and min-fu feng

3. Existence and uniqueness of the finite element solution

For the discrete problem (2.5) we will establish existence and uniqueness of

an approximate solution without requiring the B-B condition.

Theorem 3.1. If f £ L2(Q)", then (2.5) has at least one solution uh = (un , ph)

Proof. We use Brouwer's fixed point theorem to prove our theorem. The proof

proceeds in two steps.(I) For a given vh £ Xn, the following linearized problem of (2.5) has a

unique solution:

Find un = (un,pn) £ Xn such that

(3.1) Bs(vn,vh;ûh,w) = Ls(vh;w) Vw £ Xh.

In fact, we have

Bs{vh, vh ; ûn, ûn) = p\uh\2 + \\ôx/2(-pAuh + vhVuh + Vpn)\\20Jl,

where || o ||0 h = Ç£/K || ° ||q a:)1^2 • By virtue of the coercivity of the bilinear

form B¿(vh , vn ; u, w) there exists a unique solution of (3.1), and since

Ls(vh ; ûh) = (f, uh) + 52 Wf, -p&uh + vnVuh + Vph)KK

<(p-x\\f\\l + \\ôxl2f\\2o)xl2

x (p\un\\ + \\6xl2(-pAuh + vhVuh + Vph)\\lh)x'2,

we get

(p\uh\\ + \\ôx'2(-pAuh + vhVuh + Vph)\\lh)x'2

<(p-l\\f\\l + \\Sl/2f\\2o)l/2

or

(|M„|2 + p-x\\ôxl2(-pAuh + vhVuh + Vph)\\lh)x'2

(3'2) <U\\f\\l + H\\SV2f\\2o)1/2.P

Therefore, for arbitrarily given vn £ Xh, the solution of (3.1 ) determines a map

F:vh^ûh = F(vn).For convenience, let 1 = (||/||2 + p\\Sx/2f\\2)x'2, R=X/p.

(II) For the set BR = {vn £ Xh: \vh\x < R], F is a continuous map from

Br to Br . In fact, by virtue of the estimate (3.2), it is easy to show that

F : BR -* BR . Thus, we only need to prove F is continuous.

Let for arbitrarily given v'h £ Br the approximations u\ = F(v'h) (i = X, 2)be defined by (3.1); then we have

(3.3) Bs(vh,vih;ûih,w) = Ls(v,h;w) Vw £ Xh

and

(3.4) (lu'^ + p-'llo^-pAu^ + vi-Vu^ + Vp^lhy/'^R (i = l,2).



By (3.3), we have

Bs(v\ ,v\,ûxh,w)-B¿(v¡, v¡;ù\,w)

(3"5) = '£öK(f,(vx-v2)Vw)K VW£Xh.

K

Now we choose w = û\ - û2h , i.e., w = (w , r) = (uxh - u\, p\ - p2h). We have

(3.6) Bô(vxh ,vxh;w,w) = p\w\\ + \\ôx'2(-pAw + vxhVw + Vr)\\20h

and

Bs(vxh ,vxh;w,w) = Bô(vxh ,vxh;û\,w) - Bô(vxh , v\ ; ù\ , w)

= Bô(v2h , v¡ ; ù\ , w) - Bô(vxh ,vxh;û2h,w)

+ Y,¿K(fAvlh-v2)Vw)K (by 3.5)K

= b(v¡ -vxh;u2h,w)

(3.7) + 52<MK2 - vxh)Vu2h , -pAw + vxhVw + Vr)KK

+ EôK(-pAu2h + v¡Vu2h + Vp¡, (v¡ - vxh)Vw)KK

+ YiôK(f,(vxh-v2)Vw)KK

=: Sx + S2 + S3 + S4.

For S4, it is easy to get

(3-8) |54|<lK-^||o,ocMli||¿/||o.

By means of Sobolev's embedding theorem and an inverse inequality, we can

prove that

(3.9) Ho,«. < CoA-'Mi Vv£Vh

with x > 0 arbitrary in the case of n = 2 and x = 1/2 in the case of n = 3 .Thus, (3.8) yields

(3.10) |£,| < C0h-*\\Sf\\o\vJ,-vl\x\w\i < C0px/2o¡l2h-XR\vx-v2\x\w\x,

where Ôm = maxxeíi¿ = ah2. Similarly, by (3.4), (3.9) and the Cauchy-Schwarz inequality, we get

(3.11) \Sx\<NR\v2-vx\x\w\x,

(3.12) |52| < CoRh-*ôli2\v2 - vx\x\\ôx'2(-pAw + vxVw + Vr)||0>A ,

(3.13) |S3| < CoPx/2oj¿2h-XR\v2-vx\x\w\x.

Combining (3.10), (3.11), and (3.13), we have

\Sx\ + \S3\ + \S4\ < (NR + 2CoPX/2ax¿2h-*R)\vx - v2\x\w\{,

i.e.,

(3.14) |S,| + |S3| + |S4| < M(R)\vxh - wA2|iHi.



By (3.12), (3.14) and Cauchy's inequality, we obtain

4

(3.15) J] |5/| < L(R)\vxh - v2\x(ßM\ + \\Sl/2(-pAw + vxhVw + Vr)\\2ih)x'2.;=1

Then, by (3.6), (3.7), (3.15), we finally get

(3.16) (p\w\2 + \\Sx'2(-pAw + vxhVw + Vr)\\lh)x/2 < L(R)\vxh - v2\x,

where M(R), L(R) are constants independent of v'h and ulh. Noting that

w = ûxh - u\ and (3.16), we conclude that F is a continuous map from Br

to Br . By Brouwer's fixed point theory, this implies that F has at least one

fixed point un = F(uh), i.e., the problem (2.5) has at least one solution ûn =

(uh,pn)£Xh. D

Theorem 3.2. Assume that f £ L2(Q)" , p~2N\\f\\t < X. Then there is a con-stant ho > 0 such that for all h < ho the problem (2.5) is uniquely solvable,

and the solution ùn = (Uh , pn) satisfies the estimate

iji\Uk\\ + \\Sxl2(-pAun + uhVuh + Vph)\\lh)1/2

<(ß-l\\f\\l + \\Sl/2f\\2o)l/2.

Proof. From Theorem 3.1 we know that the problem (2.5) has at least one

solution ûn = (un , Ph) £ Xh such that

(3.18) Bs(uh,uh;ûh,w) = Lô(uh;w) Vw £ Xh.

Setting w = ûh in (3.18), we have

(3.19) Bô(uh, uh;ûh, ûh) = Ls(uh; ûh),

(3.20) Bô(uh , uh ; ûh, ûh) = p\uh\2 + \\Sx/2(-pAuh + uhVuh + VpÄ)||^A ,

Ls(uh;ûh)<(p-x\\f\\l + \\ôx/2f\\2)x'2

x (p\uh\2 + \\Sx'2(-pAuh + uhVuh + Vph)\\lh)"2-

By (3.19), (3.20), and (3.21), we have

(p\uh\2x + \\ôx'2(-pAuh + uhVuh + Vph)\\lh)l/2

<(ß-]\\f\\l + \\sl'2f&/2.

If we let R = /¿-'(ll/ïl^ll«*172/!!2,)172, then (3.22) can be rewritten as follows:

(3.23) (\uh\2x + p-x\\ôx'2(-pAuh + uhVuh + V/ja)||2iA)'/2 < R.

In order to prove «/, e Xh to be the unique solution of (2.5), we suppose û'h

(i = 1,2) are two solutions of (2.5); by the above results we easily conclude

that

(3.24) (K|2 + //-1 \\ôxl2(-pAu'h + u'hVu'h + Vp'h)\\lh)l/2 <R 0' = 1, 2)

and

Bg(uxh ,uxh;ûxh,w)- Bs(u2h ,u\;û2h,w)

(3-25) =Y/SK(f,(uXh-u2)Vw)K Vw£Xh.K



Now let w = uxh - xjt\ , i.e., w = (w , r) = (u\ - u\, pxh - pi). We have

(3.26) Bs(uxh , u\ ; w , w) = p\w\\ + \\Sxl2(-pAw + uxhVw + Vr)\\2h

and

(3.27) Bs(uxh ,uxh;w,w) = Bs(uxh, u\ ; u\ , w) - Bà(u\ ,uxh;u\, w).

By using the same arguments as in the proof of Theorem 3.1.(11), based on

(3.24)-(3.27), we can get an estimate similar to (3.16):

p\w\\ + \\\ôxl2(-pAw + u\Vw + Vr)\\lh

< (NR + 2CoPxl2ox¿2h-*R + x2CiôMh-2xR2)\w\2.

Since

R = p-x(\\f\\l + p\\ôx'2f\\2o)x'2 < p-x\\f\U + p-WôjfWfWo,

(3.28) becomes

plw^X-p-'NWfW.-p-^NWfWo(3.29) - 2c0p-x/2ol¿2h-XR - \C2p-xôMh-2^R2)

+ \\\6xl2(-pAw + u\Vw + Vr)Ho,Ä < 0.

As p~2N\\f\\* < X, there exists a constant w\ £ (0, 1) such that p~2N\\f\\* <

oj\ < X . Since Sm = ah2, ôj^2h~x -> 0 as h -» 0, there is a constant h0 > 0

such that for all h < ho

(3.30) p-y2oü2N\\f\\o + 2Cop-x'2oÜ2h-XR+2-C2p-xoMh-2XR2 < ±(l-w,).

By (3.29) and (3.30), we obtain

(3.31) i(l - cox)[p\w\2 + \\ôx'2(-pAw + uxhVw + Vr)\\2h] < 0.

This means that \w\2 = \\px/2(-pAw + uxhVw + Vr)||2 h = 0, that is, u[ =

ü\. D

4. Convergence of the method

This section is devoted to establishing convergence results on the

Galerkin/least squares finite element approximation for any choice of discrete

velocity and pressure spaces.

Theorem 4.1. Assume that f £ L2(ÇÏ)n , and let {«/,} be a sequence of solutions

of (2.5) as h tends to zero. Then there is a subsequence {uh} which converges

strongly to a solution û of (2.2) in the sense of

(4.1) lim(|M - uh\x + ax'2h\\Vp - Vph\\o) = 0.h—>0

Proof. From Theorem 3.2 we see that {ûh} is uniformly bounded with respect

to h , i.e., there exists a constant C independent of h and ûh such that

(4.2) IplUhÏÏ + ^ÔKW-pAuh + UhVUh + VphW2, 2 A <C.


538 TUN-XIAO ZHOU AND MIN-FU FENG

Using h/hf[ < C, Sk = ahK , (4.2) and an inverse inequality, we obtain

(4.3) |wA|i+a1/2/*||VpA||o<C.

By (4.3) we have

(4.4) \\Ph\\o<C(a),

where C(a) denotes a constant dependent on a and independent of h and

ûh . Therefore, we get

(4.5) |M*li + l|P*llo<C(a).

Consequently, by (4.5) there is a weakly converging subsequence in V x Q,

which for simplicity we denote again by {«/,}. We will show that the weak

limit w is a solution of (2.2). For this, let Ih = (Ixh , I2h): (V n H2(Çï)n) x(Qn

Hx(£l)) -* VnxQh be the usual Lagrangian interpolation operator [15]. Setting

w = Ihv = (I\v, i\q) in (2.5), we conclude that, as h tends to zero,

(4.6) pa(u, v) + b(u;u,v)-(p, divv) + (q, divu) = (f, v) + XimFhx,h—>0

where

Fh = Y,ôK(f + pAuh - uhVuh - Vph , -pAIlhv + uhVl\v + VI¡q)K.K

By (4.2) and Cauchy's inequality and an inverse inequality, we have

1^1 < £ Wllo,*ll - M/> + uhVIxhv + VI2hq\\o,K

+ 1^2sk\\- pAuh + UhVuh + Vph\\oyK)

1/2

(4.7)52*5*11 - pAIxhv + uhVIxhv + VI2hq\\ltK

1/2

< {ÔmUWo + Cá]//2)(|| - pAIxv\\o,h + \\UhVI[hv + VI2q\\0)

< (SmUWo + Cô]l2)(\\ - pA(v - Ixv)\\o,h + II - Mwllo

+ C0h-x\Ixhv\x + \q - l\q\x + \q\x)

< (C«5]f + <Wllo)(C|M|2 + CA-^llüll, + C\\q\\x)<C(f,v)ô\l2h-*<C(f,v)hx-*,

where C(f, v) denotes a constant dependent on / and v , but independent

of h. This gives lim^n ^a = 0, since 1 - x > 0; i.e., for v = (v, q) £

(VnH2(Q.)n) x(QnHx(Q)), (4.6) becomes

(4.8) pa(u, v) + b(u; u,v)-(p, divu) + (q, divu) = (/, v).

Since (V n H2(£l)n) x(Qn HX(Q.)) is dense in V x Q, we obtain that

pa(u, v) + b(u; u, v) - (p, divv) + (q, divu)

{ ' ' ={f,v) Vv = (v , q) £ V x Q;

i.e., (u, p) is a weak solution.



Now we prove that limA_0(lM _ maIi + ctxl2h\p - ph\x) = 0. In fact, for

/ £ L2(Çl)n , the solution û = (u,p) of (2.2) belongs to \h¿{Q) n H2(ü))n x

(HX(Q)/R) and there holds pa(u, u) = (f, u). Therefore,

p\uh-Ixu\2 + \\ôx/2V(Ph-I2hP)\\lh

= pa(uh -Ixhu,Uh-Ixhu) + Y^MV(Ph - I2hP), V(pA - I¡p))KK

(4-10) = pa(uh , uh) + Y,SK(Vph , Vph) - pa(2uh -Ixhu,Ixhu)K

-Y,SK(V(2Ph-I2hP),VI2hP)K.K

Recalling that ûh = («/,, pn) satisfies (2.5), we have

(4.11) pa(uh , uh) = (f, uh) + Fh2,

where

Fh = 52 <M/ + M"A - uhVuh - Vph, -pAuh + uhVuh + Vph)KK

and

Fh2 = ~YjÔ^VPh ' VPh) + £<W/ + M«A - UhVuh , VPh)K

(4.12) K K+ Y^0Kif + ß^h - uhVuh - Vph , -pAuh + uhVuh)K.

K

By (4.10), (4.11), and (4.12), we get

/i|MA-/^|2 + ||<î1/2V(Jp,-/2p)||2jA

= (f, uh) -pa(2uh -Ilhu,Ixu) - 52<MV(2p„ - I2p), VI2p)KK

(4.13) +^2¿K(f + pAuh-UhVuh, VPh)KK

+ 51 <M/ + M"a - tihVuh - VPh , -pAuh + uhVuh)KK

=:(/» uh) - pa(2uh - Ixhu, l\u) + Fh3.

By using the same techniques used in deriving the estimates of Fhx , we obtain

(4.14) \F¿\ < CSl¿2h-*.

Thus, we conclude from (4.13) and (4.14) that as h —> 0

(4.15) Xim(p\uh - Ixhu\\ + \\ôxl2V(ph - I2hP)\\l) = (f, u) - pa(u, u) = 0.

By virtue of the definition of ô and the assumption h/hn < C, (4.15) yields

Xim(\uh-IXhU\x+ax'2h\ph-I2hp\x) = 0.h—»0

Finally, limA^0(l" - "aIi + axl2h\p -ph\x) = 0. ü



Remark 2. In the proof of (4.14), we used (4.2) and ||maIIo oo < Coh~x\uh\\

and (EK\\^h\\lK)1/2<Ch-x\uh\i.

Theorem 4.2. Assume that p~2N\\f\\t < X and that the exact solution û = (u, p)

of (2.2) belongs to the space (W0x'°°(Çl) n Wl+x(Q.))n x Wk+x(Ü), I, k £ N.

Then there is a positive constant h* such that the following error estimate holds

for the solution ûh = (uh, Ph) of (2.5) for h < h* :

(p\u - uh\\ + \\8xl2[-pA(u - uh) + uhV(u - uh) + V(p - Ph)]\\2o,h)l/2

<C(hl + hk+x),

where the constant C depends on the seminorms |«|i,oo. M/+i. \p\k+i of the

exact solution of (2.2).

Proof. According to Theorem 2.1 and Theorem 3.2, both problems (2.2) and

(2.5) have uniquely determined solutions. Let it)A = Ihû - ûh , i.e., wh =

(Wh , rh) = (lin - Uh, I\p - Ph) ■ It is easy to see that

(4.16) Sx =: Bs(uh ,uh;wh, wh) = ß\wh\\^\6xl2{-pAwh+uhVwh+Vrh)\^ih ,

and from (2.5), (2.6) we derive that

(4 17) Si "5^"A' uh;hû,Wh)-Bs(uh, uh\ ûh,wh)

= Bs(uh, uh;Ihu,wh)-Bs(u, uh;ù,wh);

i.e.,

S\ = S2 + Si + S$,

where

52 = pa(Il\u -u,wh)- {l\p-p, divwA),

53 = 52 sK(-lià(Ilhu -u) + uhVl\u-uVu + V{I¡p - p), -pAwh + uhVwh + Vrh)K ,K

54 = b(uh;Ilhu,wh)-b(u; u, wh) + (rh , div{Ilhu - «)).

By using well-known interpolation error estimates [15], it is easy to get

(4.18) \S2\<C(h' + hk+x)\wh\x,

m < j E^ll - Mw* + uhVwh + Vrh\\lK + CÔM(h21-2 + h2k)K

(4.19) + CôM\\UhV(Ixhu- u) + (uh - u)Vu\\l

< -\\ôx'2(-pAWh + uhVwh + Vrh)\\lh

+ CSM\wh\2x + CÔM(h21-2 + h2k + h2'-2* + h2l+2).

It should be mentioned that the constants C in (4.18) and (4.19) depend on

the seminorms of the exact solution.For 54 , by using Green's formula, we have

(rh , div(Ixhu- u)) = (uhVwh ,l\u-u)- (uhVwh + Vrh ,Ixhu-u),



and then

S4 = b(uh ;Ixhu-u,wh) + b(Ixhu -u;u,wh)- b(wh ;u,wh)

+ (uhVwh ,Ixhu-u)- (uhVwh + Vrh,Ixhu- u).

Recalling that wA is bounded and that for the exact solution û we have |m|i <

», and using an inverse estimate, we get

|S4|<CA/|u>*li+/*"1^ll/IUI«'*lî

+ 52(-M^A + uhVwh + Vrh,Ixhu- u)K

(4.20)52(-mw/m ixhu-u)KK

< Chl\wh\x +p-iN\\f\U\Wh\2x + CS-xDh2l+2

+ \\\ôxl2(-pAwh + uhVwh + Vrh)\\lh .

where ômin = minxe£î(î = infKSK. Combining (4.16) with (4.17)-(4.20), wehave

(4.21)p\w\2(X - p~2N\\f\U - CÔM) + \\\ôxl2(-pAwh + uhVwh + VrA)||g>A

for all h < h

(4.22)

< C(h' + hk+x)\wh\x + CSM(h2'-2 + h2k + h21'2*) + COZ¡nh2,+2.

Taking into account öm = ah2 and p~2N\\f\\* < 1, we may conclude thatthere exists a sufficiently small h* > 0 such that

(p\wh\\ + \\ôxl2(-pAwh + uhVwh + Vrh)\\lh)i/2

<C(hl + hk+x+ô-^2h'+x)

Since Sk = ahK , h/hK < C, we then obtain

(p\wh\\ + \\ô{l2(-pAwh + uhVwh + Vrh)\\lh)l/2

<C(hl + hk+x).

Noting that Wh = hû - ûh , and using the triangle inequality, we finally get

(p\u -uh\2x + \\Sxl2[-pA(u - uh) + uhV(u - uh) + V(p -Ph)]\\o,h)1'2

<C(h! + hk+x). D

Remark 3. By using Nitsche's duality technique, we can also get L2-error esti-

mates for velocity and pressure.

Remark 4. If the finite element pressure subspace Q belongs only to Lq(Q) ,

we need to add the boundary integral term Y,k ßnK^dAa^r^s t0

Bs(u,uh;v,w) (where ß > 0, [q] = q+ - q-) in order to obtain corre-

sponding convergence results.

5. Conclusion

A finite element method of Galerkin/least squares-type for approximating the

stationary N-S equations in primitive variables is presented with the following

characteristics:



(i) The method exhibits stable and convergent approximation with optimal

rate for any choice of the discrete velocity and pressure spaces, in contrast with

the Galerkin mixed methods, in which the discrete B-B condition is required.

For the 3-dimensional analysis, this point has important significance because of

the implementational simplicity of lower and equal-order interpolations.

(ii) The method is variationally consistent, and the parameter a > 0 can

be arbitrarily chosen, yielding practical convenience and improved convergence

error estimates compared to the associated penalty-type method [12].

Acknowledgment

The authors would like to thank the referee for helpful comments and sug-

gestions.

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Computing Technology Research Institute, Chinese Aeronautical Establishment,

Xi'an, China 710068

Department of Mathematics, Xi'an Jiaotong University, Xi'an, China


a least squares petrov-galerkin finite element …...volume 60, number 202 april 1993, pages 531-543...

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