Error estimates for mixed finite element approximations of nonlinear problems

inl J Engng Sci. Vol. 24, No. 8, pp. 12834293, Printed in Great Britain

1986

0020-7225/86 Pemon

ERROR ESTIMATES APPROXIMATIONS

$3.00 + .CUl Journals Ltd

FOR MIXED FINITE ELEMENT OF NONLINEAR PROBLEMS

MUHAMMAD ASLAM NOOR Mathematics Department, Ring Saud University, P.O. Box 2455, Riyadh, Saudi Arabia Abstract-The fmite element methods have proved a very effective tool for the numerical solutions of nonlinear problems arising in elasticity and other related en~nee~ng sciences. Relative to linear elliptic theory, little is known about the accuracy and convergence properties of mixed finite element approximation of nonlinear elliptic boundary value problems. The nonlinear problems are much more complicated, since each problem has to be treated individually. This is one of the reasons that there is no unified and general theory for the nonlinear problems. In this paper, the application of the mixed finite element method to a highly nonlinear Dirichlet problem, which arises in the field of oceanography and elasticity is studied and new results involving the error estimates are derived. In fact, some of the results and methods to be described in this paper may be extended to more complicated problems or problems with other boundary conditions. As a special case, we obtain the well known error estimates for the corresponding linear and mildly nonlinear elliptic boundary value problems. 1. INTRODUCTION

finite element methods for solving numerically elliptic problems are based on the variational principle of minimum energy. The mathematics theory of such methods is firmly estabhshed and is a part of the classical numerical analysis. In recent years, considerable efforts have been made to give a complete mathematical analysis of other finite element methods including mixed methods. In a mixed method, one introduces an auxiliary variable, usually representing another physically important quantity, in order to write the differential equation as a lower order system, thereby obtaining direct approximation to both the original and auxiliary variables. The analysis of the mixed methods makes use of the full strength of the theory so far has been developed. The analysis of the mixed methods represents excellent example of how the general approximation results and the interpolation theory, (see Oden and Carey [ 11)can be used to study many different types of finite element methods. Also these types of methods are very important in their own right. Mixed methods result naturally from finite element approximations of any variational boundary value problem with constraints. Thus, mixed methods are useful in the study of approximate methods for treating incompressible fluids and solids, contact problems in elasticity, fluid dynamics and other problems. All other works are related to linear elliptic problems except that of Noor [2], which provides general ideas to obtain error estimates for the mixed finite element approximation of mildly nonlinear elliptic problems. It should be pointed out that the nonlinear problems are much more complicated, since each problem has almost to be treated individu~ly. This is one of the reasons that there is no unified and general theory to obtain the error estimates for the nonlinear problems. The main aim of this report is to extend these methods for a class of nonlinear boundary value problems. It is well known that the mixed formulation is more complicated than the standard one because there are more independent variables. However, the recent investigations show that the mixed finite element methods are actually more efficient; see Oden and Carey [I]. The mathematical model we consider in this paper is highly nonlinear and has been subject of recent investigations by many research workers including Glowniski and Marroco [ 31, Pelissier [4], Noor [5], Babuska [6] and Bermudez and Moreno [7]. STANDARD

2. FORMULATION For simplicity, we consider the problem of finding the velocity u of the glacier, which is required to satisfy the nonlinear problem of the type: -V(lVulp-*Vu)

= f(u)

u=o 1283

in

s2

on

I ’ asz

(1)

M. A. NOOR

1284

where Q is the cross-section of the glacier and f( U) is the Coulomb friction. The presence of f(u) may be interpreted as a body heating term. It arises from the resistivity and is known as Joule heating effect. Similarly many other problems arising in elasticity equilib~um, fluid dynamics can be formulated in terms of eqn (1); see Oden and Carey [l] and Pelissier [4]. The problem (1) is a generalization of the nonlinear problem of finding u such that TU =.f; u = 0,

in on

Q dQ,

where Tu = -V(~VZ$-~VU), for which the error estimates have been derived by by using the mixed finite element approximation, The presence of the nonlinear needs a different approach for obtaining the error estimates. In order to obtain estimates, we use the techniques of Noor [9] and Falk and Osborn [lo]. The usual variational form of(l), see Noor [5] consists in finding u E V such

Noor [8] termf(u) the error that

where

is the energy functional associated with (I). Standard finite methods have been extensively studied and convergence results are now classical; see Noor [S]. On the other hand, it has been observed that one could weaken the requirement of interelement continuity for the function of the space and still obtain a convergent finite element method. This type of methods have been first introduced and studied by the engineers, (see Pian and Tong [ 111) and can be viewed as a generalization of nonconforming methods. In this report, we use a more general approach, which is based on extended variational principle, in which the constraint of interelement continuity has been removed at the expense of introducing a Lagrange multiplier. By this method, the known u and some of its derivatives are simultaneously approximated. This type of methods has been introduced and analyzed by Brezzi [ 121, Raviart and Thomas [ 131, Falk and Osbom [lo] and For-tin [ 141 for linear elliptic problems. An outline of the paper is as follows: Section 3 will be devoted for the abstract construction of the approximate method, which can be derived from a mixed variational principIe. In Section 4, we give the abstract error estimates for u - uh and $ - &. We also would like to remark that this approach can be applied to a wider class of nonlinear elliptic problems having more complicated boundary conditions and nonlinearities. 3. VARIATIONAL

PRINCIPLE

We begin by intr~ucing a general va~ational principle. Let I’, Wand Hbe three Sobolev spaces with their duals V’, IV’ and H’ and norms II* II“, II - II w and II- IIH respectively. Let T be a monotone operator on Vand ZJ(- , - ) be a continuous bilinear form on W X W. We denote by (a, . ), the pairing between V and I/’ or Wand IV’. We now consider the functional Itv, 41 =

(TV, v) + 2&u, 4) - 2F(v) - 2(g, 4,>,

c-3

where F is a continuous function from V into itself and g E IV’. If T is a symmetric and FE V’, then the problem of finding the saddle point of Z[V, 41, defined by (2) on V X W, see Brezzi [ 121 is equivalent to finding (u, 4) E I’ X W such that (Tu, v) f b(v, 9) = (F, c),

for all

vE V

b(u, cp) = ($5 6),

for all

f# E w.

(3)

1285

Mixed finite element approximations of nonlinear problems

For a Frechet differentiable nonlinear functional F on V, using the technique of Noor 19, 151, we can show that the saddle point of Z[u, Cp]defined by (2) can be characterized by the following variational eqns

(Tu, v) + NV, #I = (F’(u), v), b(ri, 4) = (g, (b),

for all

vE v,

(4)

for all

#E

(5)

Ix

Remarks 1. Since the operator T is symmetric on I’ X I’, one can easily check that the solution (u, $) of (4) and (5) may be characterized as the unique saddle point of the quadratic functional Z]v, #] defined by (2) over the product space I’ X W; i.e. (u, $) satisfies

for all

VE v

and

d, E w.

Thus it follows that u is the unique solution of the constrained minimization

problem

Z[u] = v E 2,inf Z[v], where Z[U] = (TV, v) - 2F(v), 2, = [v f

Y; for all p E W, b(v, p) = (g, p)],

while $ is the Lagrange multiplier associated with the constraint Z,. 2. In many applications, we are led to the problems (4) and (5) by the following procedure. Let I’, and V be Sobolev spaces, with I$, a closed subspace of V, and let T be a nonlinear operator on I’. We want to solve the problem: Find u E Va such that (TzI, v) = (F’(u), u).

for all

vEV,/’

I

where F’(u) E t/’ is the Frechet differential of the nonlinear functional F(U) = JO j:f(q) X do dQ at U. For this, we consider the space W = b’$(the polar space of VQ), which is a closed subspace of I”. Thus in this setting problem (6) is equivalent to: Find (21,$J)E I’ X W such that

(Tu, v) + (v, Ir/) = (F’(u), v),

for all

(% 6) = 0,

for all

(7)

Taking b(u, 4) = (#, v), for all v E V, 4 & W c H’, it is obvious that (7) is of the form (4) and (5). Following the duality approach we can associate with ( I), the Lagrangian Z[u, $1, defined by

From the above discussion, it follows that (u, $) E V X W is the saddle point of the function Z(v, #,I, defined by (7), if and only if (u, I/J) E V X W satisfies the va~ational equations

s

]VU]~-~VUVV dR +

52

s0

rc/div v d!C! =

fa

uf(u) dQ

for all

VEV

(9)

1286

M. A. NOOR

s R

4 div u d!J = 0,

for all

$ E w.

(10)

It is clear that eqns (9) and (10) are equivalent to eqns (4) and (5) with g = 0, (F’(U), 0) dQ, see [8, 161. = sii vfu) dQ, h(rd, I(/) = sn rl/d iv u dfl, (Tu, u) = s, ]VZ#‘-~VUVV Furthermore, it has been shown by Glowinski and Marroco [3] that the operator T have the following properties. Lemma 1

For all u, 2,E I’, we have (Tu - TV, u - v) z (~11~- v11’, (TeTv,u-v)

a @flu -

pa 2

412(1141f llwY

I/Tu - Tvll 6 @l/u - VI/~-‘, I(Tu - Tvll c pRPP211u- 1111,

(11)

I
1
(13)

p a 2

(14)

with R = max( l/u/], ]]v]])and cy > 0, p > 0 are constants independent 4.

ABSTRACT

ERROR

(12)

of u and v.

ESTIMATES

In this section, we first derive the general error estimates for u - u,, and It/ - jL*. To be more specific, we consider the following problem, which we refer to as problem (P). Find (u, $) E I’ X W such that (Tu, v) + NV, I/) = (F(u), v), bfri* 6) = (8, (b),

for all

vE Y

(15)

for all

cf,E lv.

(16)

We consider this problem for a subclass of data, that is, for (F’(u), g) E D, where D is a subclass of v’ X W’. We assume that: HI. For (F’(u), g) E D, (P) has a unique solution. In the analysis of(P), we also consider the adjoint problem: Given d E G’, where G is a Sobolev space such that WC G with a continuous embedding, find (y, A) = (ydr A,) E V X W satisfying

(TV, Y>+ &u, A) = 0, &Y, 4) = (d, #),

for all

VE v

(17)

for all

QiE w.

(18)

We shall assume that: H2. Problems ( 17) and (18) have a unique solution for d E G’. Let Vhand Wh be given finite dimensional subspaces of V and W respectively. We now consider the following approximate problem (Ph): Find (Q, &,) E V, X W, such that

(Tut,, V/z)+ &V/n$h) = (F’h), VII), b(%

6h)

=

(3%

&I),

for all

vh E

vh

(19)

for all

d’hE

w,.

(20)

We are interested in obtaining the error estimates for u - uh and $J - $h. We now state several assumptions which are required in the proof of our main results. It should be pointed out that these assumptions can be verified easily. H3. There exists a constant 01> 0 such that (Tu - TV, u - v) z= a/Iu - VII&

for all

vh

E

where & = {v E vh: b(vh, &) = 0, for all $h E w!,,).

&,

I287

Mixed finite element approximations of nonlinear problems

H4. There exists a number S(h) such that

H5. There is an operator Z,: Y -

b(Y-

ZhY, $h)

V satisfying for all

= 0,

and

YEY

d’h E

w,

where Y = span( {yd)GGt, u), (u, rc/)is the solution of (Z’)and (Y,, A,) is the solution of (17) and ( 18) respectively. H6. F’(u) is required to satisfy the Lipschitz condition, that is there exists a constant y > 0 such that

IIF’

- F’(V)llM =sYIIU - 4IH,

H7. F’(u) is antimonotone

for all

u, v E H.

on V, that is

(F’(u) - F’(v), u - v) G 0,

for

U, VE v.

We would like to remark that hypotheses (H 1-H5) are due to Falk and Osborn [lo] and (H6-H7) are due to Noor [ 151. From ( 14) we observe that

(Tlhu, Vh)

+

@Vh,

rc/) =

(F’(U),

V/I) +

(TzhU

-

for all

U, V/I),

v E v,

(21)

Now subtracting (19) from (21), we get for all uh E vh, (TZ,,u - U, vh) + b(+ - 1c//,,Vh)

=

(F’(U)

-

V)

F’(Uh),

+

(TZhu -

U, V/,).

(22)

This shows that in the nonlinear case, we have the term (F’(u) - F’(uh), vh), which is not present in the linear case considered by Falk and Osborn [lo]. We, therefore, introduce the concept of generalized pseudo-projection to obtain the inequalities bounding the errors u - uh and # - 1c/h.This idea is mainly due to Noor and Whiteman [ 161 and Noor [ 151 for deriving the error bounds for u - uh in the standard finite element methods for the mildly nonlinear elliptic boundary value problems. Define zi,,E vh to be pseudo-projection of u E V by the conditions:

( TIhU-

ih,

Vh)

+

b(Vh,

‘k -

$h)

=

(TIhU

-

for all

b(Zhu - i&r &) = 0,

for all

U, V/I) 4h

E

VhE

vh

(23) (24)

wh.

These conditions play an important part in deriving the error estimates. It should be noted that in the final analysis i&,E V, does not appear. We now state and prove the main results of this paper, which are the abstract error bounds for U - Uh and $ - +h. Theorem 1 Suppose that the hypotheses (Hl)-(H7) hold. If (u, #) and (uh, &) are solutions of problems (P) and (P,J respectively, then, for p 2 2, we have: l]u - Uhl,H< { 1 + (Ilz,,u - ~(~~)~2-p~‘~p~1~}~~I;I~ - &, + {1+

R[(P-2)l(P-1)1*(IIzhU X

IiU-UhIIVG

(ItzhU

-

Uk)

+R I/k1)

UllH)(2-PMP-w+

{R(p-2)‘(p-1)(

lb-zhUliv+s(h){l x

-

+

{ 1 +

11zhU -

(I[$

z&,)“(~-‘)

-

&

+

(II+

(P 2)/(P--1) + R(P-2)/(P-~)2(~~~hU(II+

_

4h 11w)[2-(P-1)21/(P-1)*}(

11w)(2-PY(P-1)*) -

&

IIw)“(‘-‘)}

(25)

UIIH)[2-_(P--1)21/(P--I)z} II+ _

&

11w)l/(P--l).

(26)

1288

M. A. NOOR

If in addition, Z, C Z = {VE I’, h(v, 4) = 0, for all CpE IV), then

IItf - uhI,f,< (I,lhu - ulIff)“(-‘+p)(l + (I&u - t&f+

( 2+PMP- I’(

R(P-W(P-~)*(III;~~

R(P--ZMP- 9

1 +

uIIu)[2-(p-1)211(~-l)2)

_

(27)

and

From b(u, 4) = (g, $), for all 4 E Wand (H5), we see that

b(h&,d'h) = b(u, cbh) = (R,d'h)>

for all

d)h E

wh.

(29)

Subtracting (20) from (29), we obtain for all Now taking uh = Ihu - r&,in (23) we get (Tlhll

-

Tlih,

I/,U

-

iih)

= (7&U

-

(TI,JA - Tu, I,,td - ii,,) + b&U - ti,,, \C/,, - $)

=

i%,

IhU

-

ii$’

+ b(lhU

-

Z&, $h -

#)

+ b(&l4

-

iih, $,, -

$,,).

(31)

Thus from (24) and (31), we have

from which, using (H3), (H5), (11) and (14) it follows that alllhu

-

~hll$f~

IITlhU

-

G pRP-211zhU

TUll~llZ,zU -

-

ChllH

UllHll~hU

-

+ P,llIhU

ChllH

-

&II”iI1c/

+ i=(h)illhU

-

-

@hIIF%’

ur,iWIt$

-

d’hIIW.

Hence, we have iizhU

-

&liH

G (~I~~-*&U <

-

&f

+ ~~(b)~~~

-

#/&i’)“p-’

C,RfP-21'(-'+P)(I]I~zt - Z&)“‘+) +

c&+!’

-

#h I/ p+r)*‘(P-l).

(32)

Consider now (Tti/, - TU/,, & - ldf,) < (i?i/, - Tu h, = (Tti/,

-

(F’(Gh)

-

F’(Uh),

TI,,u, ii/, - t&,) + (TIhU

-

Uh, i&, -

-

ch -

(F’(u/,)

vh)

-

- F’(U),

ii/, -

Uh)

-

Gh -

(F’(U)

vh),

by

(I-I@

Uh) -

F’(Uh),

i& -

I$,).

Taking t,$,= r&,- uh in (23) we have (Tiif, - Tlhu, iij, - El)= b(& - &, 11,- &) - (T&t& - Tu, i&,- &). Thus from (33) (34) and (H7), it follows that (Tiij, -

i%h,

ii/, -

vh)

from which it follows that

G (F’(U)

-

F’(%),

uh -

Uh)

s

YllU

-

llhIiffiuh

-

UhIiH,

(34)

Mixed finite element approximations of nonlinear problems

1289

or

(35) Now, by using (32) and (35) we obtain

llu - u,~IIHGllu - ~,4, + II&u - &llH + lluh- w,IIH G Il2.i- Z~UllH+ llZhU- z&llH+ c*(Ilzhu- &p-l) + C~(llZ~U - z&llffp-‘) < l/u- ZhUIIH{1 +

(IlZhU - z&#-p)‘(p-‘)}

+ (1 + (IlZhU - ~~(Iu)(2-p)‘(p~‘)}IIz~u - zih(IH

G c{ 1 + (IlZhU - 2.&)(2-p)‘(p-‘) } IIU - IhUllH + { 1 + +

(I/$

-

~WMP-lq

~hIIW)(2-P)l(P-l)Z}{R(P-2)l(P-l)(IlzhU

-

[I~/+

-

UIIH)ll(P-l)

+

UIIHyPMP-l)2 ($

-

$hW)llW)},

which is the required result (25) where c is generic constant. In order to prove (26) we first note that

Ilu-wzllv~ ll~-bI4lv+ IIhJ--Uhllv G IIU- bJ~ll v + moIlh - ~hIlH, by

(H4). (36)

By triangle

inequality,

we have

IlhZl - WlllHG IlbI~- GIL + lluh- %llH < Ill/p - ii/,llH + (IIZhu - u,&,)"~~') + (IlZp - i&,)"@-I), G

(IlbP - %Af)I/W1)

+

+

~WV(P~l)(II~hU

R(Pd)/(PPl)7

-

11Ih

_

UIIH)l/(P-l)

+

uII#(P-l~

+

([I$

(I/$

-

-

by

$hll

(35) w)l/(P-l)

(37)

c#qJW)1’(P-l)2.

Hence

(Iu - uhll v

<

IIu - Ihull

v

+

S(h){[l

R(pp2)'(p-') + R(p-2)'(p-1)2(IlZh~ - u~~~)[~-(~-')~~'(~-'~}

+

x (llZ/p - u)lfp~')+ {1 + (iI+ - &II w)~~-~~-‘~2~‘~~-‘~2}( I/$ - C#QJ&“@‘J which is the required (26). To prove (27) we observe that (30) and (24) together

&zhu -

Uh,

6)

’

for

0,

with Z, E 2 implies

that

$EW

(38)

and

(TZ,,u - Tiij,, Z,,u - ti,,) = (TZ,,u - TM,Z,,u - ii,,). Hence

(39)

from (H3) and (39) we get

~llzhu - i&II” <

11 TIhU- WI

’ !zhU

-

uhll

or 114~ - &&, < R(p-2)l(p-1) (Ill/& - &)“(p-‘). Now from (40) (35) and the triangle

IIU-

UhliH

inequality,

G

IIU

-

IhUiH

+

IlzhU

-

&llff

+

lluh

<

IIU

-

IhUllf,

+

IlzhU

-

tihl,f,

+

(IjlhU

<

IIU

-

z,Ulj,

+

(IlzhU

-

U[~J,)“(~~‘)

we get

-

U/k -

+

(40)

UIIff)l’(p-l)

+

R(p-2)‘(p-1)(IIzhU

(IlzhU

+

-

iihll)l’(p-l)

U~~~)l’(p-‘)

R(P-Z)/(P-l)2(

llZhU

-

uIIH)l/(Ppl)’

M. A. NOOR

I290

IlfhU- 4lH(1 + (llhlu - ad

=

(2 P)/tP- “(

-

1 +

RtP-2)/&‘-

0)

+ RtP-2)/tP--1)2(IIZhu _

uIIH)[‘-tP-‘)211tP-I)zj

which is (27) the required result. To prove (28), we note that

But

IIZN - 4lllH G Ilhlu- f&llH+ Ilu;,- WIIIH < [1 + Jzp2MP-l) + ~w/wq~~~hU

-

uIIH)Il-(P-I))lI(P-I)](llJhu

-

UII)wu<

Hence

which is (28).

Special Cases i. For p = 2, Theorem reduces to the following result of Noor [9]. Theorem 2 Suppose that the hypotheses (H 1)-(H7) hold. If (u, $) and (t6h,&) are solutions of problems (P) and (Pn) respectively, then

Ilu - w&f =scllu - h&f

+ CM - d%IlW,

for all

@h E

wh

IIU- wlllv~ IIU - hJ4lv+ CIIM- 4lH f cll+- &llw. If in addition Z,, c Z = {2)E V, &u, 4) = 0, for all d, E W}, then

where c is a generic constant independent of u and $. ii. If the function fis independent of U, that isf(u) = fl (say), then theorem 1 is exactly the same as proved by Noor in [8].

Theorem 3 Suppose that the hypotheses (Hl)-(H5) hold. If (u, $) and (uh, $+) are solutions of problems (P) and (PJ respectively then for p z 2, we have

ForZ,,EZ=

{vEKb(v,~#~)=O,forallQ,E

W},then

]]u - U/&G { 1 + R(p-2)‘(p-‘)(llu - ~~U~~~)~2-p~‘~p-‘~}~~~ - @&, and

liu - u/zIlv~Ilu - &4lv+

I-2fP - m-1)(

11u

-

ZhUIIH)W’f*

In the next theorem, we derive the error bound for # - $+.

Mixed finite element appro~mations

1291

of non&near problems

Theorem 4 Assume that (Hl)-(H3) and (H5) hold and if (u, J/) and (u,,, \Lh)are solutions of problems (P) and (Ph), respectively, then

Now using the techniques and idea of Falk and Osborn [lo] and Noor [9], we derive the error bound for II/ - Ii/h. Subtracting ( 19) from ( 15) and (20) from (14) we have

(Tu - TUh,vh)

+

&Vh,

II/ -

q/z)

=

(F’(U)

-

F’(tih),

?h)

=

0,

for all

for all

Vh),

vh

E

vh

(42)

and b(@-

uht

(43)

Combining (17), (H5), (42) and (43) we obtain ‘%yd,

\c -

=

bWd

= =

\ch) -

=

b(Yd -

Itid,

+

-

\th)

+

b(IhYd, ‘/ -

+h)

IhYd,

‘k -

6h)

+

(Tut,

-

Tu,

bd)

+ (F’(u)

b(Yd -

IhYd>

9

-

+h)

+

(Tub

-

Tu,

b.Yd)

+

&Yd

Itid,

$‘ -

&t)

+

(Tub

-

Tu,

ltid

-

+

bfu

-

b(u

-

F’(Q),

Ihyd)

Uhr Ad) +

(F’(u)

-

F’(zlh),

fhyd)

Yd) -

Uhr

Ad

-

?)

+

(F’(u)

-

Stahl,

Ihyd)*

(44)

Since

(44a) Thus (4 1) follows from (44) and (44a). Note that forfiu) = .&ay), the term (F’(u) - F’uh), IhYd) drops out, we have the same estimate for 1c,- t,&,as proved in Noor [SJ. Furthermore fory = 2 andf(u) =f(say), Theorem 4 is exactly the same as proved in [lo]. It is obvious that Theorems 1 and 4 are more general and include as special cases a11the previous known results of Noor [S, 91 and Falk and Osborn [lo] for nonlinear and linear elliptic problems. In particular, these results allow us to consider the highly nonlinear elliptic boundary value problems, which is the main motivation and inspiration for this paper. All the results proved in this paper appear to be new. 6. APPLICATIONS

We now consider our mode1 problem of finding u such that -V(]VZ&J-~CIU)= f(u), u = 0,

in

B

in

an

(45)

where the given function f(u) is Lipschitz continuous and antimonotone. Throughout, we shall use the classical Sobolev spaces, see Ciarlet [ 171, for notation and definitions. Following Raviart and Thomas [ 131, let

1292

M. A. NOOR

In this setting, the energy functional

associated

with (45) can be written

in the form

(46)

It follows that (u, rc/) E H(div, 0) X &,(a)is the saddle point of the function and only if (u, $) satisfies the following relations

s a

u-v dQ +

sR

rl/ div v da =

sn

f(u) - v da,

for all

s

4 div v dn = 0, 51

u E H(div,

I[v, $1, if

fi)

(47)

for all

One can easily see that (47) and (48) is an example of problem (P) with I/ = H(div, G), M’ = L,(Q), H = (&WY’, (7.~7 u) = j-s1 US u dfi, b(u, $) = sn J/ div v do, g = 0,

(F’(u), u) = ~JWu d% [161. The subclass D of data for which (HI) is satisfied is given by D = 0 X W’. Since the operator T is symmetric, it can be easily verified that (H2) and (H3) are satisfied with cy= 1. Let { Th)h,O be a regular family, see Ciarlet [ 171, of triangulations of fi and define V, = {vh E H(div v, 0),

for

TE

Th, Vh/T E QT},

where Qr = {v E H(div,

T): 8 E Q},

wh

for

see [ 141

and =

(6

E

&I@),

all

TE

Th,

bh/TfPk},

where Pk is a polynomial of degree k. To apply the results of section 5, we must verify that the appropriate hypotheses are satisfied. The hypotheses (H4) and (H5) can be shown to hold by using the technique of Falk and Osborn [lo]. Furthermore, it has been shown that for v E {H’-‘(R)JP, r > p, the following results hold.

112, - IhdO

G

hPllvll&‘,

1 < E < min(r

- p + 1, k + 1)

(49)

and lldiv(v - zhv)IIO G h”]ldiv VII,,

0 < m G min(r

- p, k + 1).

(50)

From the definitions of vh and wh, it follows that for al1 uh E vh, div VJ,/TEP~. Thus uh E Zh implies that div vh = 0 and so uh E Z. Thus Z, C Z and we are in special cases of Theorem 1. In order to apply our results, it remains to show that (H6) and (H7) are satisfied. Actually these have been proved in [ 161. Thus from the above discussion, we conclude that all the hypotheses (Hl)-(H7) are satisfied for our model problem (45), and so we can apply the results of Theorem 1. For illustration, we only consider the case 11 u - uhIIo. Assuming $ E H’(O), r 3 p, from (27) and (49), we obtain

11~~ - &I,,, < { 1 + (II l(

-

z,,t&)(p-2)‘(p-‘)( 1 + R(p-2)‘(p-‘)) +

<

{ 1 +

/?fl(P-2MP-l)l(

~(P-2MP-lP((~U

IIuIIp2Y(P-I)(

-

-

~hu)~o)~l(P-l)

1 + R(P-w(P-19

+ R(P-Z)l(P~1)2heI2-(P-L)3ll(P--I)Z

whereR = max(llulh,IlzhullO).

Jhu~~o)[2-(P-I)311(P-1)2)(~~u

(II UIIe)12~(P-1)311(P~l)2hal(P-l)(

IIullp)‘I(P-o

Mixed finite element approximations of nonlinear problems

I293

It is clear that the estimate obtained above is highly nonlinear in character, that is there is no counterpart in the linear elliptic theory. However, we note for the special case p = 2, andf(u) -f(say), these estimates are exactly the same as proved by Falk and Osborn [lo]. CONCLUSION

In this project, mixed finite element methods based on an extended variational principle are considered and studied. These methods enable us to approximate the unknown u and some of its derivatives simultaneously. As a special case, we get many previously known results. In fact, in this study, we have tried to present a more general and unified approach to derive the error estimates for nonlinear elliptic problems. Much work still remains to be done to give a complete mathematical theory for nonlinear problems and on streamlining the computational process of this method. REFERENCES

[I] J. ODEN and G. CAREY, Finite Elements Vol l-V01 V, Prentice-Ha11 (1983). (21 M. ASLAM NOOR, Error bounds for the mixed finite element methods. C.R. Math. Rep. Acud. SC. Canada 2,227-230 (1980). [3] R. GLOWINSKI and A. MARROCO, Sur I’approximation par elements finis d’ordre un et la resolution par pktlisation dualitt dune classe de probltmes de Dirichlet non-linhires, RAIRO-R-2, 4 1-76 ( 1975). [4] M. C. PELLISSIER, Sur quelques problemes non-linearies en gluciologie Universite Paris Xl. U.E.R. Math Orsay, France, T/R 110-75-24 (1975). [5] M. ASLAM NOOR, Finite element approximation theory for strongly nonlinear problems. Comm. Math. Univ. St. Paul. 31, 1-7 (1982). [6] I. BABUSKA, Singularity problems in the finite element method in Formulation and Computational Algorithms. In Finite Element Analysis (ed. by K. Bathe, et ul.), M.I.T. Press, U.S.A., 748-792 (1976). [7] A. BERMUDEZ and C. MORENO, Duality methods for solving variational inequalities. Comp. & Math. Appl. 7,43-58 (1981). [8] M. ASLAM NOOR, Mixed finite element analysis of nonlinear elliptic problems. Proc. Int. Conf: Fin. El. Math Australia, 11-14 (1982). [9] M. ASLAM NOOR, On mixed finite element techniques for elliptic problems. Int. .I. Math. Math. SC., 6, 755-766 (1983). [lo] R. FALK and J. OSBORN, Error estimates for mixed methods. RAIRO, Num. Analy. 14,249-277 (1980). [l 11 T. H. H. PIAN and P. TONG, Basis of finite element methods for solid continua, Int. J. Num. Meth. Eng. 3-28 (1969). [ 121 F. BREZZI, On the Existence, uniqueness and approximation of Saddle point problems arising from Lagrangian multipliers. RAIRO, 8-R,, l29- I5 1 ( 1974). [ 131 R. RAVIART and J. THOMAS, A mixedfinite element methodfor second order elliptic problem (to appear). [14] M. FORTIN, Analysis of the convergence of mixed finite element methods. RAIRO 11, 341-354 (1977). [ 15) M. ASLAM NOOR, On variational inequalities Ph.D. Thesis, Brunei University, U.K. (1975). [16] M. ASLAM NOOR and J. R. WHITEMAN, Error bounds for finite element solutions of mildly nonlinear elliptic boundary value problems. Num. Math. 26, 107- 116 ( 1976). [ 171 P. CIARLET, Thefinite element method for elliptic problems North Holland (1978).

(Received 26 October 1985)