Qualitative Analysis and Finite Element

Approximation of a Class of Nonmonotone

Nonlinear Dirichlet Problems

J.T. Oden, C.T. Reddy, and N. Kikuchi

The Texas Institute for Computational Mechanics

The University of Texas at Austin

July, 1978

Qualitative Analysis and Finite

Element Approximations of a Class of

Nonmonotone Nonlinear Dirichlet Problems

J.T. Oden, C.T. Reddy, and N. Kikuchi

Summary: Existence and convergence of finite element approximations of aclass of nonlinear Dirichlet problems involving pseudomonotone operatorsare considered. A model problem, which is analyzed in some detail, ischaracterized by an operator which satisfies a generalized Garding inequal-ity and, thus, leads to nonlinear indefinite forms. Under assumptions onthe local regularity of solutions, error estimates in the Wl'P-norm areobtained.

Contents

1. Introduction

2. Variational Formulation

o3. An Existence Theorem for Garding-Type Operators

4. Some Inequalities in ]Rn

5. Some Properties of the Operator A

6. Existence Theorem

7. Finite Element - Galerkin Approximations

8. Linear Auxiliary Problem

9. Error Estimates

Acknowledgement

References

1

3

5

8

15

27

28

34

41

45

45

.

1. Introduction

In this paper, we consider a class of nonlinear Dirichlet problems

of the type

uf in n=p-2 V ) + a.

(1.1)

-V· (IVul u ~

0 on anu =

with 1 < P < co

where n is an open bounded domain in ]Rn with Lipschitzian boundary an

and a. is a finite constant. The operator appearing in (1.1) is not nec-

essarily monotone, and, consequently, multiple solutions may exist for

fixed data f.

Our aim is to investigate the qualitative behavior of solutions to

(1.1), particularly conditions for their existence, uniqueness, and bound-

edness, and to study finite element-Galerkin approximations of these solu-

tions. We also obtain a-priori error estimates for finite element approxi-

mations under assumptions on the regularity of solutions.

The special case a. = 0 of (1.1) was studied in some detail by

GLOWINSKI and MARROCO [6]. In this case, the operator appearing in (1.1)

is strongly monotone from wl,p(n) into its dual w-l,p' (n) and uniqueo

solutions exist for any f in W-l,p' (n) • When a. I 0 , we will show

that the operator is of the G~rding type studied by ODEN [9 ] and, there-

fore, is pseudomonotone in the sense of BREZIS [3 ] and LIONS [7]. Analy-

ses of finite element approximations of certain one-dimensional pseudo-

1

2

monotone problems were investigated by ODEN and NICOLAU del ROURE [10] and

similar techniques were used in the study of'certain nonlinear elliptic

systems encountered in plane elasticity problems by ODEN and REDDY [11,12].

Additional references on finite element approximations of nonlinear strongly-

monotone problems can be found in the work of GLOWINSKI and MARROCO [6]

and in the survey article of BABUSKA [2]; see also ODEN and REDDY [11,12].

Following this introduction, we describe a variational formulation

of problem (1.1) in Section 2 and we record a useful existence theorem for

operators A of the type in (1.1) in Section 3 which establishes sufficient

conditions for A to be pseudomonotone and surjective. Sections 4 and 5

are devoted to studies of properties of the specific operator A in our model

problem (1.1), with Section 4 containing some preliminary inequalities in

]Rn • Conditions for the existence of weak solutions of (1.1) are laid

down in Section 6 and finite element approximations of (1.1) are introduced

in Section 7. There we discuss conditions for weak and strong convergence

of sequences of approximations generated by regular refinements of a mesh

containingoC -elements.

The lack of monotonicity of A leads to indefinite forms which

resemble G~rding's inequality in linear elliptic theory; e.g.

(A(u) - A(v), u - v) >

for positive constants C and y

p p'cllu-vlll - yllu-vllo,p ,q

The presence of the negative term

presents some difficulties in the error analysis. One way of overcoming

these difficulties is to first determine some "local" estimates for an

appropriate linearized auxiliary problem. We analyze such a linearized,.

3

problem in Section 8 on the basis of assumptions on the global regularity

of solutions to (l.l). Finally, in Section 9 of the paper, we obtain a-

priori estimates of the error in the Wl,p-norm. In particular, we show

of our problem, thenu

that whenever a sequence of piecewise linear finite element approximations

wl,p(n) to a solutiono

O(hl/(p-l» as h + 0 .

uh converges strongly in

2. Variational Formulation

We will consider a variational formulation of problem (l.l) which

has meaning in the context of Sobolev spaces. Following standard notations,

we denote by ~,p(n) the Sobolev space of order (m,p) consisting of

When equipped with the norm

equivalence classes of functions with generalized derivatives of order

< m in LP(n), m > 0 , 1 2 p < 00

m > 1

(2.1)

= ess sup L ID~ u(x)1x E n I~L~_m -

the spaces ~,p(n) are Banach spaces. Here

x = (xl' x2' ..•, xn) En, and ~ is a multi-index, ~ = (0.1,0.2, ..•, an) ,

o.i~ 0 , a.i = integer, I~I = 0.1 + 0.2 + ...+ an. We are particularly

1 p 00

interested in the space w' (n) which is the closure of C (n) in theo 0

"·I/m,p-norm, and which is a reflexive Banach space equipped with the norm

The dual of wl,p(n) is the negative Sobolev spaceo

4

p' = p/(p-l) (2.3)

equipped with the norm 11·11 * where

v I 0 (2.4)

wherein <.,.) denotes the bilinear duality pairing on

Returning to (1.1), we introduce an operator

defined by

u,v

<A(u), v> J2 2 -1/2

n [IVulp- Vu· Vv + 0.(1+ IVul) uv] dx

E wl,p(n)o (2.5)

Likewise, if f 1 'is a given bounded linear functional in W- ,p (n) , we

write f(v) = (f,v) V v E wl,p(n) .o

We will consider the following variational boundary value problem:

Given f E w-l,p' (n) , find u E wl,p(n)o such that

."

<A(u), v) (f , v>

5

(2.6)

Since00

c (n)ois dense in wl,p(n) , we take

o and conclude that

A(u) in (2.6) is formally given by

A(u) up-2 ) + 0.-v . (I VU I Vu Ii+lVufL (2.7)

where A(u) E V'(n) ; i.e., (2.7) is to be interpreted in the sense of

distributions. With this interpretation, the equivalence of (1.1) and

(2.6) is apparent.

3.o

An Existence Theorem for Garding-Type Operators

The first question that arises in the study of problem (2.6) is

that of existence of solutions. We will record here an existence theorem

which will prove to be critical in the subsequent studies of approximations.

We first review some definitions and preliminary results.

Let

•

U and V be reflexive Banach spaces with the

property that U is compactly embedded in V,

cU ~ V

A: U + U' is an operator mapping U into its

strong topological dual U'

l(3.1)

6

The operator A is bounded from U into U' if it maps (strongly)

bounded sets in U into (strongly) bounded sets in U' • A is hemicon-

tinuous at a point u E U if the real-valued function

~(t) = (A(u + tv), w) t E [0,1] (3.2)

is a continuous function of t V v ,w E U. In (3.2), <. , .) denotes

duality pairing on U' xU. Moreover, A: U + U' is coercive on U if

lim~I/vilUoo Ilvllu

+co (3.3)

The following existence theorem was proved by aDEN [9]:

Theorem 3.1. Let conditions (3.1) hold and let A: U + U' be an

operator having the following properties:

(i) A is bounded

(ii) A is hemicontinuous

(iii) A is coercive

(iv) If B (0) is the open ball of radius µ in U , i.e.,µ

then,

B (0)µ

V u , v E B (0) ,µ

{v E U: Ilvllu < µ} (3.4)

(A(u) - A(v), u-v) ~ -H(µ, Ilu-vllv) (3.5)

+ + + +where H:]R x]R + JR (]R

tion with the property that

[O,co» is a continuous real-valued func-

lim t H(x, ey)

e +0+

a x , y E ]R+ (3.6)

7

Then A is surjective, i.e., V fEU' there exists at least one u E U

such that

A(u) f o

•

We remark that operators satisfying the conditions of Theorem 3.1

are monotone only in the special case in which H = O. Operators of this

type were referred to as G~rding operators by aDEN [9] owing to the formal

osimilarity of (3.5) to the Garding inequality for linear elliptic operators.

Note that H depends upon the norm II u - v II V in the space V in which U

is compactly embedded. This property is crucial in the study of the exist-

ence and approximation of solutions to abstract problems involving this

class of nonmonotone operators. It effectively establishes that the opera-

tor A differs from a monotone operator by a component which is completely

continuous from U into U' (whenever U is dense in V). It is not

difficult to show that when the conditions of Theorem 3.1 hold (in particular,

conditions (ii) and (iv», the operator A is pseudomonotone in the sense

of BREZIS [3] and LIONS [7]; i.e.,

If u E U converges weakly to u f. U as m+co and ifm

lim sup (A(u ), u -u) < o , thenm m -m+co (3.7)lim inf (A(u ), u - v) > (A(u), u - v) V v E Um m -m+CO

4. Some Inequalities in ]Rn

We will now establish several inequalities for vectors in ]Rn

which will prove to be important in studying properties of the operator A

of (2.5). We use the following notation.

8

and

(~ ' y)

(x , x) 1/2

nfor x, y E ]R (4.1)

(4.2)

Lemma 4.1. Let x, y be any two vectors in ]Rn .

(i) for p ~ 2 ,

Then,

(4.3)

(ii) for 1 < p ~ 2

2(p - 1) Ix - yl- - (4.4)

Proof:

(i) For p = 2 , the inequality is obvious. Also the equality occurs.'

for x = y So we consider p > 2 and x 1 y By direct expansion

(1~IP-2~_lyIP-2y, ~-y)

9

Noting that122 2

(x,y) ="2 (I~I + I~I - I~-~I ) , we get

The second term on the righ hand side is positive for p > 2. Therefore,

( Jrr r r a+bFor a, b, r > 0 , we have a + b > [max(a,b)] ~ -Z- . Hence,

(ii) For p = 2 or x = y , the above inequality obviously reduces to- -an equality. We therefore consider 1 < P < 2 . When I~I = Irl we

get

2-pwhere we note that 2 > P - 1 for 1 < P < 2. Therefore we need only

•consider the case I~I I Irl In this case, let

(I~I + Irl)2-p(I~IP-2~ - IrIP-2r, ~-r)2I~-rl

10

In the above expression we made use of the expansion of 2(x,y)

222I~I + Irl - I~-rl . Denoting r = 2-p so that 0 < r < 1,

1[IXI+IYI]r ~ r r (Ixl + Iyl) Ilxlr _IYlr~

> - - - Ixl + lyl - - - - -2 1~llrl - - I I~I - Irl I

Without loss in generality, let I~I > Irl and denote

Ixlt =--==-> l.

Irl "

Then

~2(~'~) ~ } (1 + ~)r ~ + tr _ (t+1) t:=lj1. (1 + 1) r t(t - tr)] r

= t -1> 1--2 t t-l - t-1

> 1 - r = p - 1

11

Here we used the fact thatr

t -1~ < r for t > 1 , o < r < 1 .

o

Lemma4.2. Let x,y be arbi.trary vectors in ]Rn. Then

(i) for p ~ 2 ,

(ii) for 1 < p ~ 2 ,

• I IP-2x x -- - < (4.7)

12

Proof:

(i) Cases p = 2 or ~ = ~ or I~I = Iyl are obvious. We will

therefore consider p > 2 and I~I 1 I~I .Let

1~IP-2~ _ 1~IP-2y II~ - r I ( I~ I + I~ I )p-

2

Then by direct expansion and using 2(x ,y) = I ~ 12

+ I ~ 12

- I ~ - ~ 12 , we

get

I I~IP -lrlPI1~_~12(1~1+1~I)p-2

I 1~IP-2_lrIP-21(I~I + Irl )p-2

< 1 + (1~1+lrl)2(1~1-lrl)2

I I~IP -lrlPI(I~I + I~I)P

I I ~ Ip-2 - Ir Ip-21( I~ I + Ir I )P- 2

Without loss in generality, we take I~I > Irl and denote

Iyls = -..::.-- < 1. Then,

I~I

p-2( J

2 1 sP 1 - s1+s -2 1 + p_~3(~'~) ~ l-s (l+s)P (l+s)

1 1 - sP 1 - sp-21 + 2 ( 2)

(l+s) P- l-s l-s

< 1 + 1 - sP 1 - sp-2

l- s 1-s

.•

for 0 < s < 1 ,

Therefore

rl-sl-s

when

when

r > 1

r < 1

13

{

1+P(P-2)~2 <3 -

1 + P

The result follows immediately.

2(p - l) p ~ 3

(ii) When P = 2 , the inequality is obvious. When I~Iobserve that

Irl we

We now consider 1 < P < 2 and I~I I Irlthat a < r < 1. Let

Let r = p -1 so

=1~lr-l~ _ Illr-lll

I~_ylr

• (x , y)Also we note that cos e = ~ , where e < n represents the

1~llll

angle between vectors x and y

l4

vll~12r + 1~12r _ 21~lr 1~lr cos e

(M + Irl2 - 21~1 Iri--~~~r

Without loss in generality, let

Then,

Ixl > lyl- -lyl

and let t = --=:- < 1 .

I~I

2r r_ 1 + t - 2t cos e- 2 r

(l+t - 2t cos e)

r 2 r 2(l- t) + 4t sin (e/2)

2 2 r[ (l - t) + 4t sin (6/2) ]

=

<

r 2(1- t ) +

r[(1-t)2 + 4t sin2(e/2)]

[ ]2 r 2

l- tr 4t sin (6/2)

l- t + 4rtr sin2r (e/2)

r 24t sin (6/2)

2 2 r[(l-t) +4t sin (e/2)]

< 1 + 41-r < 5

GLOWINSKI and MARROCO [6] proved the inequalities of Lemmas 4.1

2and 4.2 for vectors in JR • In addition to extending these results to

nJR , we have also obtained here sharper estimates on the constants appear-

ing in these inequalities.

As a final lemma, we establish a useful elementary inequality:

15

Lemma 4.3. Let a, b E ]R • Then,

(4.8)

Proof: We multiply and divide the left hand side by Jl+a2 + Jl+b2 .

Thus,

J(A+a2 - Jl+b2) (Jl+a2 +Jl+b2)LJl +a2 + Jl +b2

~

Jl+a2 + Jl+b2

= I~(Jl+aZ + Jl+b2)

la - bl

We obviously have la+bl/(Jl+a2 + Jl+b2) <

ity (4.8) follows immediately. []

1 and thus inequal-

•

5. Some Properties of the Operator A

We now return to the operator A defined in (2.5). In this section,

we establish a number of properties of A that are crucial in proving the

existence of solutions to (2.6) and to subsequent studies of finite element

approximations.

Theorem 5.l. Let the operator

16

defined in (2.5), and let

3np > n+2 (5.1)

where n is the dimension of n

(i) P > 2

Then, for u ,v ,w E Wl,p (n) , we haveo

where

I <A(u) - A(v), w> I < gl(u,v)llu-vlll,p Ilwl/l,p (5.2)

gl(u,v) (5.3)

where 83 is defined by (4.6) and C1 is a positive constant.

(ii) 1 < p < 2

I (A(u) - A(v), w)1 2 g2(u,v)lIu-vlli~~ Ilwl11,p (5.4)

15 + Cl 10.1 (1 + Ilvlll ) Ilu- vlll2-P,p ,p (5.5)

Proof: It is convenient to decompose A into two parts,

where, formally,

A A + Ap 0

(5.6)

l7

A (u) = - V • (I VU Ip-2 Vu) VuE Wl,p(n) (5.7)P

uVuE wl,p(n)A (u) = 0. (5.8)0 Jl+ Ivul

2

Then,

(A (u) -A (v), w) = J (IvulP-Zvu - IvvIP-2vv) • Vw dx (5.9)P P n

For p ~ 2 , we make use of the inequality (4.5) of Lemma 4.2 to obtain

where 83 is defined in (4.6). Now using Holder's inequality, we obtain

< 83(llulll,p + Ilvlll,p)P-2 Ilu-vl/l,p IIw111,p

(5.10)

For 1 < p < 2 , we make use of the inequality (4.7) of Lemma 4.2 to obtain

< IS lIu-vlli~~ Ilwlll,p (5.11)

18

We now consider operator Ao

(A (u) -A (v), w)o 0

Clearly,

(5.l2)

I (A (u) - A (v), w) I =o 0

< Ivl Iwl dx ]

By the inequality (4.8) of Lemma (4.3), we have

I (Ao(u) - Ao(v). w> I ~ 101 [In lu-vl Iwl dx + I n I~(u-v) I Ivl Iwl dj(5.13)

We recall here the Sobolev embedding theorem (see, for example,

ADAMS[1]):

1 p nSuppose z E W ' (n) , n C]R , theno

o(a) for p > n , u E C (n) and

sup Iz(~)1 ~ canst. IIVzl/xEn LP(n)

(5.l4) ..

19

where the constant depends on mes n but not on z.

(b) for p ~ nand (n-p)q ~ np , and

IlzIILq(n)< const. IIVz 1/

LP (n)q < co (5.15)

where the constant depends on mes n, q, p, n, but not on z.

Now, returning to (5.13), we observe that when we

need v, w E Lq (n) ,where q ~ Zp' = Zpl (p-l). Then, according to the

Sobolev embedding results given above, we need the condition (5.1). When

(5.1) holds, we use Holder's inequality and the inequalities (5.14) or

(5.l5) to obtain

I<A (u)-A (v), w)1o 0<

(5.16)

where Cl is a constant dependent on mes n .

Combining results (5.10), (5.ll), and (5.16) gives (5.2) and (5.4).:J

Theorem 5.2. Let the conditions of Theorem 5.1 hold. Then the

of (2.5) is strongly continuous and bounded from W1,p(n)ooperator A

. W-l,p' ( )l.nto n for any p , 1 < P < co , satisfying (5.l).

Proof: Let {uk} be a sequence in wl,p(n) converging strongly to u E0

Wl,p(n) with p satisfying (5.1). Then IIuklll,p and IIulll,p are0

bounded. Hence

II A (u) - A (uK) 11_1, p , = sup

v E Wl,p (n)o

(A(u) - A(uk) ,v>

Ilvlll,p

20

< M

p ~ 2

1 < P < 2

where M is a constant depending on the functions gl and g2 of (5.3)

and (5.5) and on the bound of lIuklll . Hence A(uk) + A(u) strongly,p

as k + co. Boundedness of A also follows; indeed, since A(O) = 0 , we

have

(I gl (v) Ilvlll,p p ~ 2

lIA(v)lI_l,p' < ~ V v E Wl,p(n)- 0

g2(v) Ilvl/i~~ 1 < p < 2 (5.17)

where

(5.18)

= IS + Cllo.l (1+ Ilvlll ) Ilvl/2l-P

, P , P

and Cl is the constant appearing in (5.3) and (5.5). []

We note that the continuity of A implies that A is also hemi-

continuous at every

,.

Theorem 5.3. Let

is coercive from Wl,p(n)o

tions hold:

21

A be the operator defined in (2.5). Then A

into W-l,p' (n) whenever the following condi-

(i) 0. > 0, 1 2 p 2 co

(ii) a. < 0, p > 2

(iii) 0. < a , p = 2 , and 1 - 10.1 C2 > 0

C2 = mes(n)1-2/p (d(n»)2/2

where den) = max dist (~ 'y) is the diameter of n .x,y E n

Proof: Clearly

(5.19)

(A(v) , v) =

(i) If a. > 0 , then

(A(v), v) ~ Ilvlli,p

Hence <A(v), v) + + co as Ilvlll,p + co for any p, l2 p 2 co •

(ii) If a < 0 , P > 2. Then

(A(v), v) ~ IIvllP1 - 10.1 IIvll~ 2, p ,

For p ~ 2 , and

22

<1_1

(mes(n»)2

p IIvlll,p

Also, from Poincar~'s inequality,

Ilvll 0,2 < d (n) II VV II0 , 212

where d(n) is the diameter of n. Hence

(A(v), v) > Ilvlli,p - lal mes (Q/ -;

so that the positive term dominates the growth of (A(v), v) as

IIv III + co if p > 2 .,p

(iii) 0. < 0, p = 2. In this case

so that coerciveness is obtained whenever the coefficient on the right

side of this inequality is positive. :1

Theorem 5.4. Let A be the operator defined in (2.5). Let B (0)µ

be the ball of radius µ in

B (0)µ Ilwl/l < µ},p (5.21)

23

For 0. I 0 , let condition (5.1) of Theorem 5.1 hold. Then for every

u , v E B (0) and arbitrary e: > 0 , there exist positive constantsµ

Y1 (e: ,µ) and y 2 (e: , µ) , dependent on e: and µ, such that

(i) for p ~ 2

(A(u)-A(v), u-v) ~ U~P-l_j Ilu-vlli,p

- lal C0.

2 p'Ilu-vllo,2 - yl(e:,µ) lIu-vllo,2p'

(5.22)

where

and

Co = t if 0. > 0

if 0. < a(5.23)

(5.24)

In (5.24), C is the constant appearing in (5.15) and p' = p/(p-l) .

(ii) for 1 < P < 2 ,

(A(u) - A(v), u - v) > [(p-l)(2µ)p-2_e:] Ilu-vlli,p

where

- 10.\ C0. II u - v II~,2 - y 2 (£ , µ) IIu - v II ~, 2p ,

(5.25)

(5.26)

24

Proof: We make use of the decomposition given in (5.6) for u, v E ~,p(n)o

and observe that

(A(u) -A(v), u-v) = (A (u) -A (v), u-v) + (A (u) -A (v), u-v)p p 0 0

(5.27)

We consider the two cases: p ~ 2 and 1 < p < 2 .

Case (i). P ~ 2. Application of the inequality (4.3) of Lemma

4.1 yields

(5.28)

For the operator A ,we haveo

(A (u) -A (v), u-v)o 0

(u - v) dx

=

(5.29)

We now apply the inequality (4.8) and observe that the first term

need not be considered when 0. > O. Moreover, for p ~ 2 ,

2-;> z E L (n) .

Thus,

..

•

+ f n Iv (u - v) I Iv I Iu - v I dx)

where C = 0 if 0. > 0 and C = 1 if 0. < a •a. - 0.

25

(5.30)

For is embedded in L2p' (n) , p' = p/(p-l) .

Applying the Holder inequality to (5.30), we get

+ Ilv(u-v)lIo,p Ilvllo,2P' Ilu-v11o,2p'

(5.31)

By the imbedding theorems (recall (5.15», there exists a constant

C such that

Ilvllo,2p' ~ C IIvlll,p

For u, v E B (0) , we then haveµ

(5.32)

(A (u)-A (v), u-v)o 0> - 10.1 2

C Ilu-vllo 20. ,

- \0.1 C µ lIu-vlll,p Ilu-vllo,2p'

We now apply the following form of Young's inequality: For

a ,b E ]R+ , and arbitrary £ > 0 ,

(5.33)

ab < £ aP + 1 pI'I bp'(£p)p p

Hence

26

(5.34 )

- £ Ilu-vlli,p - Yl (£,µ) Ilu-vll~:2P'

where Yl(£,µ) is precisely (5.24).

We obtain (5.22) by combining (5.28) and (5.35).

(5.35)

Case (ii). 1 < p < 2. By making use of inequality (4.4), follow-

ing the steps used by Glowinski and Marroco [ ], we have

2(p-l) Ilu-vlll,p < (A (u) -A (v), u-v) (1lulll + Ilvlll )2-pp p ,p,p

(5.36)

For u, v E B (0) , we haveµ

(5.37)

The arguments for

p > 3n_ n+2 ' we get

A follow those given in case (i).o For

2 2- £ Ilu-vl/l,p - Y2(£,µ) lIu-vl/o,2' (5.38)

where Co. is given by (5.23) and Y2(£,µ) is given by (5.26). (5.25)

follows by combining (5.37) and (5.38). I'"l..J

27

An examination of the last steps in the above proof lead to the

following useful result.

Corollary 5.4.1. Let the conditions of Theorem 5.4 hold and in addi-

tion let v be bounded in wO,q(n) . Then 11·1l0,2p' in (5.22) and (5.25)

can be replaced by II ·110 ,where,r

1 (5.39)o

Corollary 5.4.2. If the condition

p > 3nn+2 (5.40)

holds instead of (5.1), then the operator A is of the G~rding type.

If (5.40) holds, 2 'is compactly imbedded in L p (n) • []

6. Existence Theorem

We can now collect the results of Section 5 and state conditions

under which problem (2.6) has a solution:

1,PC) -l,p'()Theorem 6.1. Let A: W n + W f2 be the operator definedo

in (2.5) and let 0. and p satisfy the following conditions.

(i) p > 3nn+2

(ii)

(iii)

(iv)

0. > a

0. < 0

0. < 0

1 < p < co

p > 2

p = 2

21--1 - t \0.1 (mes n) p (d(n»)2 > a

28

where n is the dimension of n and d (n) is the diameter of n .

Then for every f E w-l,p' (n) there exists a solution u of the

problem (2.6).

.'Proof: By the application of condition (i) and Theorems 5.l and 5.2, we

show that A is strongly continuous. A is therefore bounded and hemicon-

Theorem 5.3. By Theorem 5.4, we have a G~rding inequality with

tinuous. Coercivity of A follows from the conditions (ii) - (iv) and

Wl,p (n)o

2 'compactly embedded in L p (n). Thus by Theorem 3.1, for every

-1 ' .fEW ,p (n) there exists at least one u such that (2.6) holds. LJ

We emphasize that the condition (i)

order to assure that Wl,p(n) is compact in

p > 3n/(n+2) is needed in

L2p' (n) , so that application

of Holder's inequality in (5.30) yields the Garding inequality (5.22) or

(5.25).

7. Finite Element Galerkin Approximations

Let us first consider a general approximation theorem for problems

of the type covered by Theorem 3.1. Again, let (3.l) hold and consider the

abstract problem of finding u E U such that

(A(u), v) (f ,v) V v E U (7.l)

where f is given in U' . Let h be a real parameter, 0 < h ~ 1 , and

suppose that {Uh} is a family of finite dimensional subspaces of U with

29

the property that

is everywhere dense in U (7.2)

For any particular h, we can consider a Ga1erkin approximation of (7.1)

by seeking ~ E Uh such that

(7.3)

Assuming (7.3) is solvable for each h, there exists a sequence of Galer-

kin approximations {uh} which, we hope, converges in some sense to a solu-

tion of (7.1). Sufficient conditions for convergence of such approximations

are listed in the following theorem due to ODEN and REDDY [12].

Theorem 7.1. Let conditions (3.l) hold and let A be a bounded,

hemicontinuous, coercive operator from U into U'

that

In addition, suppose

(A(u) -A(v), u-v) ~ F(llu-vllu) - H(µ, Ilu-vllv)

If u , v E B (0) C Uµ

(7.4)

where F:]R + + JR+ is a continuous real-valued function with the property

that

F(x) > 0 F(x) o ==> x a (7.5)

H(·,·) is a continuous, non-negative real-valued function satisfying (3.6),

and B (0) is an open ball of radius µ in U as in (3.4). Thenµ

30

(i) There exists at least one solution u E U to problem (7.1)

for the operator A.

Moreover, let (7.3) define a Galerkin approximation to (7.l) on a subspace

Uh belonging to a family satisfying (7.2). Then,

(ii) There exists at least one solution uh E Uh to (7.3) V h ,

o < h < 1 .

(iii) If {uh} is a sequence of solutions to the approximate prob-

lem (7.3) obtained as h + 0 , there exists a subsequence {uh,} such that

weakly in U

as h' + 0 , where u is a solution of (7.1).

(iv) If F ~ 0 in (7.4), and {uh} is a sequence of solutions of

the approximate problems (7.3) obtained as h + a , then there is a subse-

quence {uh,,} such that

strongly in U

as h + 0 , where u is a solution of (7.1). []

We remark that (i) and (ii) follow immediately from Theorem 3.l.

If F = 0 in (7.4), the Galerkin approximation (7.3) is still solvable, but

only weak convergence of the approximate solutions can be guaranteed.

Let us now turn to the specific problem (2.6). We wish to study

Galerkin approximations to this problem which are generated using finite

element methods. Let us suppose, for simplicity, that n is a bounded

convex polygon in ]Rn.

isfying

31

We introduce a (finite) partition ~ of n sat-

GC n

l) GGE~

G is closed, V G C ~

-n

G ,G' E ~ ~ int G n int G'

aG is Lipschitzian V G E Q.h

The angles between faces of G are bounded below

(7.6)

by

We take

e > 0o

h = max dia (G)GE~ (7.7)

and approximate by a family of subspaces

r -{vh :vh E c (n) , r > a

V G E ~} (7.8)

where Pk(G) is the space of polynomials of degree < k on G. For

second-order problems of the type under investigation, we generally take

(7.9)

32

Under conditions (7.6) - (7.8), and regular refinements of the mesh, it is

known (see, e.g., CIARLET and RAVIART [5], CIARLET [4], or ODEN and REDDY

[13]) that the spaces S~,r(n) have the following interpolation property:

If u E Ws,p(n) n Wl,p(Q) , there exists ao

~h E S~,r(n) such that as h + a ,

C hV

Ilul1s,p

(7.10)

min (k, s-l)

where C is a positive constant independent of u and h.

In general, ~h can be taken to be the projection of u onto

S~,r(n) . We will assume that (7.10) holds in subsequent discussions.

Returning to (2.6), suppose that is a subspace of

satisfying (7.8) (and (7.l0». A finite element approximation of (2.6) con

sists of seeking uh E Uh such that

fri p 2 2 -l/2 J

n ~ VUh I - Vuh• VVh + a. (l+ IVUh I) vh dx

We have:

(7.11)

Theorem 7.2. Let (7.8) and (7.l0) hold with v > 0 , and let the

conditions of Theorem 6.l hold. Then there exists at least one solution to

33

(7.11) for every h > O. Moreover, if {uh} denotes a sequence of finite

element approximations to (2.6) obtained from (7.11) as h tends to zero,

then there exists a subsequence {uh,} which converges strongly to a solu-

tion u of (2.6).

Proof: This follows immediately from Theorems 6.1 and 7.1 and from the fact

that the operator A in (2.6) satisfies (7.4) (by virtue of (5.22) and

(5.25» with F(x) for p > 2 andp-2F(x) = [(p-l)(2µ)

2 0- £] X for 1 < p < 2 .

We remark that solutions to (2.6) and (7.11) are not, in general,

unique. Indeed, suppose ul I u2 are solutions, i.e., A(ul)

A(u2) = f for given f in w-l,p' (n) . Then, for example,

f and

o

Then

from which we cannot conclude that ul = u2 When the right side of the

Garding inequality is strictly positive, of course, we then have ul = u2 .

34

8. A Linear Auxiliary Problem

As a preliminary to our study of error estimates in the next section,

we consider here an auxiliary linear boundary value problem that leads to

some useful estimates. Most of the present analysis is based on the mean-

value formula,

(A(u) - A(v), w> (DA(eu+(l-e)v) • (v-u), w)

for some e E [0,1], (8.1)

into W-l,p' (n) defined by

Here, for any u F- Wl,p(n) ,- 0

DA(u) is a linear operator from

<DA(u) • v, w) lim aat <A(u + tv), w>t+O

(8.2)

In the case of the operator A of (2.5), it is easily shown that

the limit in (8.2) exists and that

(DA (u) • v, w> J n {I VU Ip-2Vv .'Vw+ (p-2) IVu Ip-4 (Vu • Vv) (Vu •Vw)

- am (u)3u Vu •'Vvw + a.m(u) vw} dx

where, for simplicity in notation, we have denoted

-1/22m(u) = (l + IVul )

(8.3)

35

To further simplify notation, let ~ = (~l' ~3' ..., ~n) and

n be vectors in JRn and let a ..(u) , b. (u) , 1 < i,j1J 1 -

< n , denote

n

Li,j=l

a .. (u) en.1J 1 J

n

Ii,j=l

+ (p - 2) Ivulp-4 u,.c u, .n.111 JjJ (8.4)

nL bi(u) ~i

i=l

n

Li=l

3a.m(u) u u,. ~ .

1 1

where u,.J

au/ax.J

Then

<DA (u) • v, w) = f {I a ..(u) v,.W,. + I b.(u) v,.w+o.m(u) vw} dx. : 1 1J 1 J ill 1n 1,J= =

(8.5)

Our local analysis will be based on the following assumptions:

co, and, in particular, aij(u), bi(u) E L (n) , 1 ~ i,

for all u in a neighborhood N(uo) of uOj < n

ou is a solution of (2.5) such that U

O E wl,co(n) ()

(ii) If p > 2 , then there exists a constant µ > 0 such that(8.6)

a.e. in n for all ou E N(u ) .

(continued)

2 , then there exists a constant(iii) If p

µ < o.(m(u) - % u2

Ivul2) < co

µ > 0 such

36

that 1 .I (8.6)II

a.e. in n for all u E N(uo) j

We first describe some algebraic results for the linearized opera-

tor DA(u) • Using Young's inequality,

we have

ab <212

£a + 4£ b V £ > 0

By assumption (i), £ can be taken so that

(8.7)

p-4(p-2)IVul - 0.£ OLe., £

1~ (p-2) Ivu1P-4

for p > 2. If condition (ii) of (8.6) holds, then there exists a constant

p such that

For p = 2 ,

(DA(u) •v, v) ~ p J n I Vv 12

dx (8.8)

37

Thus, if condition (iii) of (8.6) holds, there exists a positive constantA

µ > a such that

< DA (u) • V, V) ~ 0 f n I Vv I 2 dx

Moreover, if condition (i) of (8.6) holds,

(DA(u) •v, w) ..::.c(lvu1o,co' lu1o,co'p , 0. , n) I\vlll,2 Ilwlll,2

(8.10)

(8.ll)

where C is a bounded continuous function of its arguments. Then the bi-

linear form

B(v , w) (DA (u) • v, w > (8.12)

.12 1 2is continuous on the H~lbert space w' (n) x W ' (n) .o 0

the bilinear form B is coercive on the Hilbert space

By (8.8) and (8.10),

Wl,2(n) x Wl,2(n)o 0

, under the assumptions (8.6). Thus, by the Lax-Milgram theorem, there

exists a unique solution n E Wl,2(n) to the problemo

B (w , n) few) (8.13)

for every functional f defined on Wl,2(n) .o

Moreover, by standard regu-v

larity results (e.g., NECAS [8])

22 n 12n belongs to W' (n) w' (n)o

for coercive bilinear forms, the solution

whenever

That is,

few) In IjJ w dx (8.l4)

Iln112,2 2 c(n, µ) 111jJ11o,2 (8.l5)

38

where the constant C depends upon only the domain n and the coercivity

constants p or µ in (8.8) or (8.10). Here the boundary of the domain

n is assumed to be sufficiently smooth; for example, the boundary of class

Theorem 8.1. Let conditions (8.6) be satisfied and suppose that the

boundary of the domain n is C2 Then the problem (8.13) has a unique

solution n E Wl,2(n) n W2,2(n) , which satisfies the estimate (8.15), foro

2 1--1every ~ ~ L (n). LJ

Let us now consider the local behavior of the error in the finite

element approximation under the assumptions of Theorem 8.1. Suppose that

is a solution to (2.6) and thato

uo

uh is a solution of the finite element

approximation (7.11). Then, from the mean value theorem (8.1),

00,-<A (u ) - A (uh), W) ,

o 0e u + (1 - e) Uh

ou

e E [0,1]

J

(8.16)

for any w E wl,p(n) . Then, for ~ E Lq' (n) , q' > 2 , we know that thereo

. 1,2( ) n 2,2() heX1sts a unique nEW n w n such t ato

In ~ w dx (8.l7)

if conditions (i), (ii), and (iii) in (8.6) are satisfied for zSh

+ e(u~ - u) , where e is an arbitrary number in [0,1] . If

ou

sufficiently closesatisfies condition

1coA lp, zeh E W ' I 'Wo' (n) for any

. 0(1i) and (iii) in (8.6), for uh

e E [0,1] . Ifo

u

39

too

u also satisfies condition (ii) and (iii) for any e E [O,l] .

Thus, we may proceed as follows:

(1)

E > 0

For a solution uO of (2.5) satisfying (i) of (8.6), let

exist such that every wEB (uo) = {v E Wl,p(n) :E 0

(2) {~} is a sequence of finite element approximations con-

IIv-uolll < d,p

(8.6) if p = 2;

satisfies (ii) of (8.6) if

coa .. (w) , b. (w) E L (n) •1J 1

p > 2 or (iii) of

(8.l8)

verging strongly too

u as h + 0 , (the existence

of such a sequence being guaranteed by Theorem under the condi-

tions of Theorem 6.1).

Then there exists an h > a such that for all h < h ,Z hE E e

f B (u) for any e E [0,1] . Hence, whenever (8.18) holds, we can takeE 0

l,p 1,2(w = eh E W (n) c w n), p > 2 , and set00-

If q' ~ p , from (8.16),

o 0)< (A(u) - A(uh), n

Using the orthogonality condition,

,0 0 )~A(u ) - A(uh), nh o ,

40

we get

for every nh

E Uh

.

By the continuity of A (recall (5.3»

1 p 2 2 1 pHere we have used the fact that W' (n) ~ W ' (n) () W ' (n)

o 0

nh as the interpolatant IThn of n, yields

Then, using (8.15) and (8.19),

(8.19)

Taking

I (tJJ , eh) I

IItJJllo,q'[l+....!...=l),q q'

< gl(u0

, u~) "eh "l.p II n 112 ,2 h

II tJJ II 0,2

(8.20)

. 0 0 1,p( )We note that 1f uh converges strongly to u in Wo

n ,then

is bounded and there exists a positive number g(uo) such that

og(u ) >

41

(8.21)

Theorem 8.2. Let the conditions (8.18) hold, and let q' ~ p ~ 2 ,

q = q'I (q'- 1) . Then,

(8.22)

is a constant independent of0 0 0

where C u , uh ' and h , g(u ) satis-

fies (8.2l), and0 0

is the error in the finite element approxi-eh = u -~mation of the solution

0 of (2.6). 0u

9. Error Estimates

In this section we make use of Theorems 5.1, 5.4 and 8.2 in estab-

lishing the error estimates for the finite element approximations. We make

the following assumptions.

(i) Let u be the solution of (2.6), then

u E w~,p(n) (J Ws,p(n) , s > 2

(ii) Let uh be the solution of the approximate problem

(7.11), then

exists a bound -µand there

such that for every h,

(9.1)

-< µ

42

1 -1 'Theorem 9.1. Let A: W 'P (n) + w ' p (n) be the operator definedo

in (2.5). Let the assumptions (9.1) and the conditions of Theorem 8.2

p-lhold. Then for arbitrary e:, 0 > 0 , e: + 0 < (l/2) there exist constants

Cl (p , 0 , e: ,u), C2(p , 0 , e: ,u) and C3(p , 0 , e: , u , it) such that as h + 0,

lie 1/ < C I\ulll/(p-l)hv/(P-l) + C h2/(p-2) + C h1/(p-2)h l,p - 1 s,p 2 3

(9.2)

where v is defined in (7.10) and

(9.3)

[ ( J pI (p-2) (P-l )11L 10.1 ca.(Cg(u»)2 l(p_2)(op/2)2/(p-2) (;) -e:-o~

(9.4)

(9.5)

1,

p' (EP)P' /p (1"1 i/ (9.6)

C , C and C' are generic constants appearing in (7.10), (8.22). g(u)

is a positive number defined in (8.2l).

43

Proof: From the step (5.30), for v = uh ' under assumption (9.1), we get

(9.7)

By the application of Young's inequality (5.34), we get

(9.8)

- -where Yl(E, µ) is defined by (9.6).

By the orthogonality property,

Thus choosing vh = uh ' the interpolant of u, and using (5.2), we get

(9.10)

Rearranging the terms and introducing (7.10) and (8.22),

44

+ 10.1 Ca ( C g (u) h) 2 \I eh II i,p(9.ll)

where v = min (k, s-l) as given in (7.10).

We use the Young's inequality (5.34). Thus, for arbitrary 0 > 0 ,

+ p "(p-2)( op/2) 2/ (p-2) ~"l C. (c g(u)) 2]P/ (p-2) h2p/ (p-2)

+ (p-2)(p-l) (0 (p-l)) 1/ (p-2) ~1 (0, µ) (C g(u)) p '] (p-l) / (p-2)

hP/(P-2)

where p' = p/(p-1) .

Then

Ilehlll

2 Cl

Ilul/l/(P-l) hvl(P-l) + C h2/(p-2) + C hl/(p-2),p s,p 2 3

(9.12)

where Cl

(p , 0 , E , u) , C2 (p , 0 , E ,u) and C3 (p , 0 , E , µ ,u) are con-

stants defined by (9.3), (9.4) and (9.5), respectively. []

45

Remark 9.1: In case of piecewise linear finite element approxima-

tions for u bounded in w2,p(n) , we have

o hl/(p-l) (9.13)

Remark 9.2: LP-estimates can be obtained by introducing the Wl,p_

estimate of (9.2) into (8.22).

Acknowledgement: The support of this work by U.S. Air Force Office of

Scientific Research under Contract F-49620-78-C-0083 is gratefully

acknowledged.

References

l. ADAMS, R.A., Sobolev Spaces, Academic Press, New York, 1975.

2. BABUSKA, 1., "Singularity Problems in the Finite Element Method,"Formulations and Computational Algorithms in Finite Element Analy-sis, Edited by J. Bathe, J.T. aden, and W. Wunderlich, M.I.T.Press, Cambridge, 1977.

3. BREZIS, H., "Equations et Inequations Non-Lineaires dans 1es EspacesVectorieles en Dualite," Ann. Inst. Fourier, Grenoble, 18, 1968,pp. ll5-175.

4. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland,Amsterdam, 1978.

5. CIARLET, P.G., and RAVIART, P.A., "General Lagrange and Hermite Inter-polation in ]Rn with Application to the Finite Element Method,"Arch. Ration. Mech. Anal., Vol. 46, 1972, pp. l77-l99.

46

6. GLOWINSKI, R. and MARROCO, A., "Sur L'approximation par ElementsFinis D'ordre un, et la Resolution, par Penalization - Dualited'une Classe de Problemes de Derechlet Non-Lineaires," RevueFrancaise d'Automatique Informatique et Recherche Operationnelle,R-2, 1975, pp. 4l-76.

7. LIONS, J.L., Quelques Methodes de Resolution des Problemes aus LimitesNon Lineaires, Dunod, Paris, 1969.

v8. NECAS, J., Les Methodes Directes en Theorie des Equations Elliptiques,

Masson, Paris, 1968.

9. ODEN, J.T., "Existence Theorems for a Class of Problems in Non-LinearElasticity," Journal of Mathematical Analysis and Applications,(to appear).

lO. ODEN, J.T. and NICOLAU DEL ROURE, R., "A Theory of Finite ElementApproximations for a Class of Nonlinear Two-Point Boundary-ValueProblems in Finite Elasticity," International Journal of Engineer-ing Science, Vol. 15, No. 12, 1977, pp. 671-692.

ll. ODEN, J.T. and REDDY, C.T., "Finite Element Approximation of a Two-Point Boundary-Value Problem in Nonlinear Elasticity," Journal ofElasticity, Vol. 7, 1977, pp. 243-263.

12. ODEN, J.T. and REDDY, C.T., "Finite Element Approximations of a Classof Highly Nonlinear Boundary-Value Problems in Nonlinear Elasticity,I. Qualitative Analysis. II. Approximation Theory," Journal ofNumerical Functional Analysis and Optimization, Vol. 1, No.1, 1978.

l3. ODEN, J.T. and REDDY, J.N., An Introduction to the Mathematical Theoryof Finite Elements, John Wiley and Sons, New York, London, Sydney,Toronto, 1976.

J.T. aden and N. KikuchiTexas Institute for Computational MechanicsThe University of Texas at AustinAustin, Texas 78712

C.T. ReddyDepartment of Engineering MechanicsThe University of KentuckyLexington, Kentucky 40506