A conforming mixed finite element method for the pure traction problem of linear elasticity

APPLIED MATHEb{ATDC5 CON PUTATR©N ELSEVIER

Applied Mathematics and Computation 93 (1998) 11-29

A conforming mixed finite element method for the pure traction problem of linear elasticity 1 Chang-Ock Lee 2 Department of Mathematics, Inha University, Yonghyun-dong, Nam-gu, Inchon, 402-751, South Korea

Abstract

A conforming mixed finite element method using the method of reduced integration is developed for the two-dimensional pure traction boundary value problem of linear elasticity. The convergence is uniform as the material becomes nearly incompressible. © 1998 Elsevier Science Inc. All rights reserved. Keywords: Pure traction problem; Linear elasticity; Mixed formulation; Conforming finite

elements; Lam6 constants

I. Introduction

Let f2 be a b o u n d e d convex polygonal d o m a i n in N2. The pure traction b o u n d a r y value p r o b l e m for planar linear elasticity is given in the f o r m -

2tr(e(u))a

(2#~(u) + 2 t r ( ~ ( u ) ) ~ ) v = g

.

in

on 0~2,

(2)

where u denotes the displacement, f the b o d y force, g the b o u n d a r y traction, /* > 0, 2 > 0 the Lam6 constants, a~d v is the unit ot~ter normal. In addition, the Lam6 constants (/*, 2) belong to the range [/*l,/*2] x [20, oo), where/.1, #2, ~.0

1 This research was partially supported by BSRI-96-1436. 2 E-mail: [email protected]. 0096-3003/98/$19.00 © 1998 Elsevier Science Inc. All rights reserved. PII: S0096-3003(97) 10073-X

12

C-O. Lee I Appl. Math. Comput. 93 (1998) 1 1 ~ 9

are fixed positive constants. The explanation for the notation used in Eqs. (1) and (2) is given in Section 2. It is well known that the finite element method using conforming piecewise linear (P-l) finite element converges for moderate fixed 2, but as )~ ~ ~ , i.e., the elastic material becomes incompressible, the conforming finite element method seems not to converge at all [1,2]. In order to overcome the so-called locking phenomenon, Falk [3] introduced a nonconforming P-1 finite element space for the mixed formulation of the pure traction boundary value problem by using the method of reduced integration, i.e., by using an LZ-orthogonal projection in the discretized weak formulation. The finite element method employed by Falk is robust in 2, i.e., it gives the uniform convergence rate as 2 ~ ee. In [3], Falk also suggested the conforming mixed P-1 finite element space using the same local projection and a slightly different weak formulation. Brenner and Sung [4] proved that the conforming finite element method for the pure traction problem converges and is robust in 2 for the displacement formulation (i.e., not mixed formulation) with the same projection employed by Falk. In [5], Brenner proved the convergence of the P-1 nonconforming finite element methods for the mixed formulation and robustness in 2 using a modification of the space used by Falk. In addition, Brenner adopted the W-cycle full multigrid method as a numerical solver for the resulting linear system and obtained the convergence of the multigrid method, which is robust in 2. In this paper we present a P-1 conforming finite element method for the mixed formulation of the pure traction boundary value problem by using the same local projection used by Falk [3] and Brenner and Sung [4]. We show that the convergence is uniform with respect to 2. In seeking effective methods which avoid this phenomenon, one naturally looks to finite element methods for the limiting problem, that is the stationary Stokes equations. For the Stokes problem, the method presented here was first analyzed by Bercovier and Pironneau [6] when displacement boundary conditions were assumed. However the extension to the case of traction boundary conditions does not seem to be straightforward. Also, even though Brenner and Sung [4] obtained the robustness of the conforming finite element method for the displacement formulation, it does not guarantee the robustness of the mixed finite element method because the convergence of the pressure ph depends on 2. The key result in this paper is the Babugka-Brezzi stability condition giving the convergence of the finite element method in the saddle point problem resulted from the mixed formulation. To establish the stability condition [3,5], use Korn's second inequality and a classical lemma which associates the existence of the inverse of the divergence operator from the P- 1 finite element spaces to the P-0 (piecewise constant) finite element spaces. We follow the argument of [5] and use many of the results in [5] to prove the analogue of the classical lemma for the existence of an inverse of the divergence operator, which is essential for the convergence of our conforming mixed finite element method.

C.-O. Lee/Appl. Math. Comput. 93 (1998) 11-29

13

In [7] we developed the W-cycle multigrid method as a numerical solver for the resulting linear system and proved the convergence and robustness in 2 as Brenner did in [5]. This paper is organized as follows. In Section 2 we explain notations and the finite element space we employ. The Babugka-Brezzi stability condition is discussed in Section 3. In Section 4 we establish the fundamental estimates in the theory of finite elements.

2. The finite element method

Throughout this paper, an undertilde is used to denote vector-valued functions and operators, their associated spaces, and double undertildes are used for matrix-valued objects. The letter C denotes a positive constant independent of the Lam4 constants and the mesh parameter k, which may vary from occurrence to occurrence even in the proof of the same theorem. We define various standard differential operators as follows: OUl

OU2 Oy '

divu=~-~

Old1 Obl2 Oy + Ox '

rotu-

(orl,lOx +&l:lOy) - ~ &~l& + o~2~lOy

div r =

,

grad u =

~ -

(O~,lOx O~,lOy~ , \ Ou21Ox o~21Oy/

and ~(u) = l[gradu + (g~du)t]. We also define 8=(10

01)

tr(~) = r: 6,

where 2

2

i=1 j = l

Since the domain (~ is a polygon which has corners, the boundary conditions (2) must be handled carefully. We shall denote by Si, 1 ~ i ~ n, the vertices of 0 ~ (note that S,+I = S~); by Fi, 1 ~
at Si+l

14

C.-O. Lee I Appl. Math.

Comput. 93 (1998) II-29

if

J14(s)-PC-S)I2$ <

W?

0

where s is the oriented arc length measured from S,+i and t is a positive number less than min{ ]r;l: 1 < i < n}. We are able to write Eqs. (1) and (2) more precisely as

(4) where f E L2(Q), and gi E g”‘(ri) gi

. vi+l

z

gi+l

’ ,Vi

at S+l

satisfy for 1 < i < n.(5)

In order for a solution of Eqs. (3) and (4) to exist, f and gi must satisfy the compatibility condition

where R-M, the space of rigid motions, is defined by R_M := {zj: u = (a + by, c - bx), a, b, c E R}. When this compatibility condition holds, the pyre traction boundary problem (3) and (4) has a unique solution E E e (Q) where

value

Here, Hk(S2), k 2 0, denotes the usual L2-based Sobolev spaces of vector-valued functions (see [S]). Moreover, we have elliptic regularity (see [4,9]): there exists a positive constant Cc such that ‘.

Following Brenner [5], we adopt the space

(7)

15

C.-O. Lee/Appl. Math. Comput. 93 (1998) 11-29

which results in better design of multigrid methods for which convergence is proved in [7]. The space L 2 (f2) is interpreted as H ° (O). We assume that the origin of our coordinate system is chosen to be the centroid of f2 so that R M has the following orthonormal basis:

1 (10)

01

lal,/~

,

1 (01)

02

iol,/2

,

O~

1(7)

~o

'

where 09 = V/ jr~ (x ~ + 9 ) d x d y . For k >~ 1, define the operators

and r~:8~(a) -~ g~(a)

72:H~(f2) -~ ~k(f2) by

rotu dx dy I//3

T2u := u - 2 ~ f2

and

Then we have

T~(T~u) = u Vu E H~(O) and T~(T~u)= u Vu E H~k(f2), ~(~) E U

~

E(T2u)

Vu EH~(O)

div(u) = div(T~u)

and

Vu E H~(O).

(8) (9) (10)

Moreover, there exist positive constants CI and C 2 such that

c, IIT2 II ,

II ll k¢ C211T211,¢)

(11)

By Friedrichs' inequality (cf. [10]) for I" [H'(at (1 ~ l ~ k) we have

Note that [ulm(o ) becomes a norm on H~(Y2) for 1 ~< l ~
16

C-O. Lee/Appl. Math. Comput. 93 (1998) 11-29

{

.

)

i=1

Henceforth, by taking 7 = 2/2# and p = 7 div u, we consider the mixed weak formulation for Eqs. (3) and (4) as follows: Find (u,p) E HI(p) x L2(~2) such that S~(u): ~(v)dxdy+

Jp(divv)dxdy

f2

(2

2t~

-~lvd~

.vdxdy+

,

(14)

Fi

J

(div u)qdxdy - ~

Spqdxdy=O

f2

(15)

Q

for all (v,q) ~ H](O) x L2(O). Eqs. (14) and (15) are equivalent to the following formulation. Find (u,p) ~ H)(O) x L2(O) such that

f.vdxdy+_

N((u,p), (v, q ) ) = ~

,=1

-~g"vl~ds~,

(16)

Fi

for all (v,q) E H) (f~) x L2(f2), where " ( ( u , p ) , (v,q)):= i ( ~ ( u ) : ~ ( E ) + p ( d i v v ) + ( d i v u ) q - ! . q ) . . clxdy. O

Eq. (16) is uniquely solvable since the Babugka-Brezzi stability condition holds in HI (f2) x L2(•). The Babugka-Brezzi stability condition is based on two ingredients, Korn's second inequality and the existence of an inverse of the divergence operator whose properties are described in Lemma 4. (See [5] for more details.) Let {y--h}, k >/0, be a family of triangulations of f2, where j-k+l is obtained by connecting the midpoints of the edges of the triangles in )--~. Let hk := maxreyk diamT, then hk = 2hk+l. The conforming finite element spaces are defined as follows:

:_- {w:

l.islinear orall

f2

w isconti.uouson

C.-O. Lee/Appl. Math. Comput. 93 (1998) 11-29

17

To describe the mixed finite element method, we define Qk := {q: q c L2(O) and q Iv is a constant for all T E j k } . 1 Observe that Wk c Wk+l c HI(O) and W~ c W~-+Ic H ~±(O). For each k,define a bilinear form ~ on H~(~) x L2(O) by

dive) 0

÷(P~_I d i v u ) q - ~ p q ) dxdy, where Pk-1 is the LZ-orthogonal projection onto Q~-I. Note that the bilinear form ~k is symmetric and indefinite on H1([2) x L2(~). Cauchy-Schwarz inequality implies that there exists a positive constant C, independent of (u,p) and (v, q), such that

for all (u,p), (v, q) E W; x Q,-I. We have the following conforming discretization of Eq. (16), which is a modification of the one proposed by Falk in [3]: Find (uk,pk) E W~- x Qk-1 such that ~k ((u,,pk), (v, q)) =~-fi~

f.vdxdy+

g,.vlr, ds

(18)

Fi

for all (v, q) E W~- x Q~-I. 3. The Babu~ka-Brezzi stability condition

In this section, we show that Eq. (18) is uniquely solvable by proving the Babugka-Brezzi stability condition for the mixed finite element space W~: x Qk-]. The following Lemma is the well-known Korn's second inequality in [3,11], which plays a key role in the proof of the stability condition. Lemma 1 (Korn's Second Inequality). There exists a positive constant C, independent of u, such that

Corollary 1. There exists a positive constant C, independent of u, such that

18

C-O. Lee I Appl. Math. Comput. 93 (1998) 11-29

Proof. Using Eq. (9), Lemma 1 and Eq. (12), we have

>

cl~2ul~,.,) []

Let a/(-,-) be an inner product on H i defined by ag(u,v) := f ( ~ ( u ) :

~(£) + (divu)(divv)) dxdy.

t2

Then it induces a norm I1~11.~,:= ~¢(u, u) w2. As is in [4], define Jk: H2(f2) --+ ~k as follows: let m be a vertex of some triangle in 3-k. If m is also a vertex of a triangle in J'k-l, then (Jk~)(m) = ~(m).

Otherwise, there are two vertices ml and m2 of a triangle in 3-k_1 such that m is the midpoint of the edge e connecting ml and m2. Then 2 /~ds (Jk~) (m) = ]-~

~b(m,) + ~b(m2) _~ 2 ~

e

Also define an interpolation operator FI,: ~2(f2) --+ ~k by Flk~ := Jk~ -

JkCdxdy • O

From [4] we have

I1~ - n*~ll~(~)

+ h~l~ -

n~lw¢o) ~
(19)

Lemma 2. There ex&ts a positive constant C, independent of u, such that

Proof. Since T~u C H~2(f2), using Eq. (19), we have

IT2u - n.(~2~) I.,~.~ ~< Ch, l~2ul,2~,).

(20)

C-O. Lee/Appl. Math. Comput. 93 (1998) 11-29

19

Since IIk(T_~u) E ~ C 1:11 ~ (a), Eqs. (8) and (20) imply that

It follows from Eq. (12) that

Also, by applying the standard interpolation error estimate in [8, p. 122], we obtain an estimate that (21) where C is independent of qk Now, let us define an operator Divk : H ~ l ~ ---, Q~ by DiVku := Pk div u.

(22)

Lemma 3. Given u E H I ( a ) , let uk be the ,d(., .)-projection to W{. Then we have

I1~- u~lle<~>~ ch~lu- ~,1 ~'<~, where C is independent o f u.

Proof. For any f~ E Lz(a), there is a unique solution ~~bf E H~2 ±(a) of -~v{~(~f)+tr(~(~f))6}

=f

(~(~f) +tr(~(~f))f)Vir~ = O,

ina, 1 <<.i<~,,

because 0 c, 0) satisfies condition (5) and (6). By elliptic regularity (13),

I1~11~,~ +

IIdiv~ll,,,~,~-< clldle(,~.

Since

d(pj,~) = fs ~dxdy Q

we have

N(ps, ~ - , ~ ) = f f . (.~ - ~)dxdy 0

and

~(u

_ 9

o Vvew~.

(23)

20

C.-O. Lee/Appl. Math. Comput. 93 (1998) ll 29

Therefore, E

w~.

0

Since

ff.(u

- u , ) dxdy =

I~(u- ~.,~ - ~)1

-< cir. -,..u, [#l(r~) I ~x - £1~'(,~),

(24)

it follows from Eq. (24), Eq. (23) and Lemma 2 that ~<

clu-~.l.,~)inf

I O f - v IH'(.)

(25) Using a duality argument and Eq. (25), we get

IL (u-u~)~dxdyl Ilu-~,ll~-~<~) =

sup

~<~)~o

Ilsll~

Chk[~ u, [~,(~/. -

[]

The following Lemma is an analogue of the classical results found in [3,5,12]. Lemma 4. Given q C Qk-1, there exists u E W~ such that

Divk l u = q

and

]u I~(~) <-

C[[q[[L2(~),

with C independent o f u and q.

Proof. From Lemma 2.1 in [5], there is v E H1, (O) such that divv = q

and

[Ivllg~(n)~
Let vk be the au¢(., .)-orthogonal projection of ~vfrom H1~± into W~-, i.e., ~(~-~k,~)=0

V~W~.

(26)

C.-O. L e e / A p p l .

Math. Comput. 93 (1998) 11-29

21

Define u E W~- by (i) u(m) = vk(m) for the vertex m of T E J-k-l, (ii) fe uds = f~ v d s for the edge e connecting ml and me which are vertices o f T E 3"-k-1 • R e m a r k 1. By (ii), U(me) is determined for the m i d p o i n t connecting ml and m2 which are vertices of T E 3--k-1.

Remark 2. u # vk because fe vk ds #

me

of the edge e

fe vds.

By definition of the Divk o p e r a t o r (22) and (ii), for any q E Qk-1

f

q Divk_, (u - v ) d x d y

=/qdiv

-

dxdy

12

: rc~_ q r / d i v ( u - v ) dxdy T

=

Z TE3-k I

qrZJ(u-v)nds ecOT e

=0. Since DiVk_l(U--V)E Qk-1, D i V k - l ( U - v ) = we have

0. Therefore, since divv E Qk-l,

Divk_l u = Divk-i v = divv = q. Let z := v - v k and zk := u - v k . Then z and zk satisfy that (iii)

zk(m) = u(m) - v~(m) = 0 for the vertex m o f T E J-k-l,

/iv/

/z ds-/lu ld e

e

=

e

= [zds de

(v)

zk = Z

Z

T~3-k l me6T

for the edge e connecting ml and m2 which are vertices of T E 3--k 1.

zk (me/7~me where me is the m i d p o i n t of the edge e E T E Y-k-i and 7~meiSthe nodal basis function defined on me.

22

c-o. Lee/Appl. Math. Compur 93 (1998) 11-29

By following the argument in the proof of Lemma 8 in [12], we estimate Iz~(me)[. First we introduce a nondegenerate reference triangle K of •z as in [12]. By using (iii), (iv), a change of variable and the formula (3.15) in [13], we get [zk(me)[ ~< C[ det(B) -1/2

2

2

1/2

2

where T is mapped from K by an affine invertible mapping F induced by 2 × 2 matrix B. (See Lemma 3 of [12] for more details.) Hence, we obtain

Iz~l.,(~> ~ c By Lemma 3,

~l~'i~/+ h-~ll~ll~/~>

Ilzll~=(~ ~< Ch~lzl~,(~>, Therefore

we have

Iz~I~,<~>~ clzl~,~,~.

(27)

Using Eq. (27), Corollary 1, the definition of ~¢(., .) and Eq. (26), we get

~
<~c(11~(£ - ~,) I1~,~>+ I1~) 11~2~,~) ~
[]

The proof of the Babu~ka-Brezzi Stability condition is the same as the proof of Proposition 2.2 in [5]. But we include the proof for completeness. Proposition 1. There exists a positive constant C, independent o f (u,p), such that for any (u,p) E W{ × Qk-l, sup ~k((u,P), (v,q)EW~xQk 1\(0,0)I~l.,(o)+

(v, q)) ~> c(lul.,<~ +

IlqllL2(~)

[IpJ]L2(a/).

(28)

C-O. Lee/Appl. Math. Comput. 93 (1998) 11-29

Proof. Given (u,p) E W~- ×

s=

23

Qk-1, let

~k((u,p), (v, q))

sup

(v,q)EW~×Qk_I\(O,0) IUIHI(~2)-'~

IlqllL=(~) '

First, we prove the proposition for the special case: p = 7Pk-i divu. By Corollary 1, M,((u,p), ( u , 0 ) ) : f ( ~ ( u ) : ~(u)

+ p(Pk_l divu) ) dxdy

I2

7 (Pk_l divu) 2) dxdy [2 2

>1 Clul~,(~t. Therefore we have

s >1Clulq.(~l.

(29)

Since p E Qk-1, by Lemma 4, there exists v E W~: such that Divk_l v = p

and

I~1~,(~) ~
(30)

with C independent of v and p. Using the definition (22) of the Divk operator, the definition of s, Eqs. (29) and (30), we have IIPlI~2(~) = Mk((u,p), ( v , 0 ) ) - f ~ ( u ) : ~(v)dxdy f2

~< ~k((u,p),(V,0)) + f~(U): ~(v) dxdy

<. Csllpllv(,~). Therefore we have I[pIIL2(o/~< Cs.

(31) Combining Eqs. (29) and (31), we obtain Eq. (28) in the special case where p = 7divu. Now, we prove the proposition for general case using the result for the special case: For any (u,p) C W~- × Qk-1, by Lemma 4, there exists w E W~- such that

Div~_lw=~p--DiVk_lU~

and

yPl _ Div~_lu L2(a)' (32)

24

C-O. Lee I Appl. Math. Comput. 93 (1998) 11~9

with C independent of u, w and p. Then 7Divk-1 (u + w) = p. From the special case, we know that sup

~k((u + w,P), C~,q))

(~'q)cWt×Qk 1\(0'0)

I~lu~>

+

Ilqllm,~/

~>c(lu + wl~,~ + IIplIL2(~))[. (33)

Therefore it follows from Eq. (33), the bilinearity of ~ and Eq. (17) that

lUl~l(o/ + IIPlIL2(Q/ ~< [u + wl~,(~) + Ilpllmo/+ Iwlu(.~ ~
sup M~ ((u +w'P)' (2' q)) (v,q)EW~-xQk 1\(0,0) [EIHI(~'~)Af_IlqllL2(ol

~
~ Iwl.,~.~ (34)

On the other hand, by the definition of Divk operator (22) ~P-1 DiVk_lU 2 = / { ( P k - l d i v u ) ( - ~p + Divk_, u) r2(~) f2

~p(-~p+Divk_lu)}dxdy

~
(35)

Therefore Eqs. (32) and (35) imply that [WIHI(~) ~ as.

(36)

Combining Eqs. (34) and (36), we have

s >/C(lul~,(~/+ llplIL~(~/).

[]

Corollary 2. For any linear functional F on W~ × Qk-l, there is a unique solution (u,p) c W ~k± × Qk-~ such that

~k((u,p),(v,q)) = F ( ( v~, q ) )

V(v,q) C W ~k±

×

Qk-1.

C-O. Lee/Appl. Math. Compur 93 (1998) 11-29

25

4. Diseretization error estimate

In this section we shall show the following fundamental estimate for the discretization error

<~Ch,2{ll fl] ~ ~2(f2) +

£11 g/l[ i=1 ~

}

H1/2(ri)

(37)

•

Theorem 1. If (u,p)EH~(Q)×L2(~2)

is the solution of Eq. (16) and (uk,pk) E W~- × Q/~-I is the solution of Eq. (18), then

(ll

i=1

)

where C is independent of (u,p), (uk,pk). Proof. For any (v, q) C W~

x

Qk-1, using Proposition 1, we have

Iu - Uk[~l(~) + lip - P~IIL2(~/ ~<]E- VIHL(a)+ liP-- qllt2(a) + IV- ukln,(a)+ IIq--pkIlL2(~) ~< ]U -- vl~,(~ ) ÷ lip + C

q[]L2(f~)

~k((V--Uk,q--p,),(w,r)) sup (w,r)cW~×Qk1\(0,0) ]~'~]HI(Q)÷ ]]rllz2((2)

In the above last inequality, by Eq. (17),

~k((V--uk,q--pk),(w,r))

+C

]~k ((u,P), (w, r)) -- 1/2#[ f~ f . w dx dy + 2i~, fr, ~g" w Ir, ds]]

+ IIrllL2(

26

C-O. Lee I Appl. Math. Comput. 93 (1998) 11-29

Therefore we have

tu - u~ I~-'lo) + lip- p~ll~.
I~((~,p),

sup (w,r)eWff×Qk-, (w,r)#(o,o)

c(l~- Vl.l(o> + liP- qllL,(a))

(w,r)) -

" frigi.wirds]l ~ 1/21x[faf .wdx dy-i- Ei=l

Since

1"2

Q

we have ~k((u,p), (w, r))-~-fi~1 [SQf . wdxdy +

DS

gi • wit, ds ]

Fi

= S @(u): %(w)+p(Pk_,divw)+

(Pk_,divu)r~pr)dxdy

Q

= f(Pk_ip-p)(divw)dxdy

IlPk-ip- pllL.
In the above inequality, we have v = 2 and Eq. (21), we have +

T~l-IkT~uand q = Pk-ip. Then by Lemma +

/38/

Since p = 7divu, by elliptic regularity Eqs. (13) and (38), the theorem holds.

[]

C-O. Lee I Appl. Math. Comput.93 (1998) 11-29

27

Theorem 2. If (u,p)CH~(f2)×L2(f2) is the solution of Eq. (16) and (uk,p~) E W{ × Qk-1 is the solution of Eq. (18), then

2(11

£11

)

i=1

Proof. In order to get the LZ-estimate of the approximation error, we use a duality argument:

Ilu- u~lle(~) =

I f~ (u - u~) • wdxdyl

sup

(39)

For any w E L2(f2), there is a unique solution ~ E H2(f2) of - ~v{2/~(() + 2tr(~(())6} = w (2pe(~)+2tr(~(g))6)Vilr , = 0 ,

in I2, l<~i<~n.

Take ¢ = (2/2#)div~ so that ((, ~) satisfies the equation ~((~,¢),(v,q)) = ~

1/

w.vdxdy

V(v,q) EH~((2) xL2(~c~).

f2

Let (~k, ~k) E W~- × Qk-l be the solution of the weak problem ~@k((~k, ~k), (v, q))

=~fw.zdxdy V(v,q)E W~- x Qk-,. f2

Then we have 1 f ( u - uk)'wdxdy = ]~@((~,¢), (u,p)) - Mk(((k, ~k), (uk,pk))l

~< I(I)l + I(II)l, where (I) := ~ ( ( ( - (k,~ - ~,), (u - uk,p--pk)) and (II) := ~@((~k,~k), (u,p)) + ~((~, ~), (u,,pk))

-

~((5, ~), (u~,p~)) ~((5, e~), (u~,p~)). -

(40)

28

C-O. Lee/Appl. Math. Comput. 93 (1998) 11-29

It follows from Eq. (17) and Theorem 1 that

I(I)l = [~((~ - ~k, ~ - ~), (u - u k , p - pk))l

2 <. Chkl[wllL2(~)

(ll fllL2(o) . . . + . £[I i=1

gillHl/Z(Fi) ) •

To estimate (II), we have ~((~k,~k), (u,p)) = ~

f.;~dxdy+

gi" ;kit, ds Fi

: ~((~,p~), (~k, ~)). w e((~' ~)' (~k'P~)) = V1 f-.~,dxdy = e,((~, ~), (u~,p~)).

12

Therefore, since Pk, ~k E Qk-1, (II) = Nk((~k, (k), (u~,pk)) -- M((:k, (k), (uk,Pk))

=/{(Pk_,

div~)p~ + ~k(Pk_, divuk)

f2 - (div~k)pk- {e(divuk)}dxdy =0. Combining Eqs. (39) and (40) and the estimates of (I) and (II), we complete the proof. [] Combining Theorem 1 and Theorem 2, we obtain the discretized error estimate Eq. (37).

Acknowledgements

The author thanks Professor S.V. Parter for his valuable advice and encouragement, and Dr. Sze-ping Wong for helpful discussions.

C-O. Lee/Appl. Math. Comput. 93 (1998) 11-29

29

References [1] I. Babu~ka, B. Szabo, On the rates of convergence of the finite element method, Internat. J. Numer. Meth. Eng. 18 (1982) 323 341. [2] M. Vogelius, An analysis of the p-version of the finite element method for nearly incompressible materials. Uniformly valid, optimal error estimates, Numer. Math. 41 (1983) 39-53. [3] R.S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp. 57 (1991) 529-550. [4] S.C. Brenner, L.Y. Sung, Linear finite element methods for planar linear elasticity, Math. Comp. 59 (1992) 321-338. [5] S.C. Brenner, A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity, Math. Comp. 63 (1994) 435-460, S1-$5. [6] M. Bercovier, O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables, Numer. Math. 33 (1979) 211-224. [7] C.-O. Lee, Multigrid methods for the pure traction problem of linear elasticity: Mixed formulation (Preprint). [8] P. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [9] P. Grisvard, Singularit6s en elasticit6, Arch. Rat. Mech. Anal. 107 (1989) 157 180. [10] J. Ne6as, Les M6thodes Direct en Th~orie des t~quations Elliptiques, Masson, Paris, 1967. [I1] J. Nitche, On Korn's second inequality, RAIRO 15 (1981) 237-248. [12] M. Crouzeix, P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, RAIRO Anal. Num6r. S6r. 7 (R-3) (1973) 33-75.