A FINITE ELDIENT ANALYSISOF NON-MONOTONE ELLIPTIC BOUNDARY-VALUE PROBLEMS

J. T. OdenThe University of Texas at Austin

U. S. - Japan Seminar onInterdisciplinary Finite Element Analysis

Cornell [niversity

August 1978

,--l --)

A FINITE ELEMENT ANALYSISOF NON-~lONOTONE ELLIPTIC BOUNDARY-VALUE PROBLEMS

J. T. Oden

Texas Institute for Computational MechanicsThe University of Texas at Austin

1. Introduction

In this note, I will summarize some joint work with C.T. Reddy

and N. Kikuchi on finite element approximations of certain nonlinear

boundary-value problems which involve non-monotone nonlinear elliptic

operators. Such problems may have multiple solutions, and the study of

convergence and the estimation of errors requires some new methods.

'Complete details of the results I list here, including proofs of all of

the theorems, are given elsewhere (3,4].

As a model problem of the type described above, consider the non-

linear Dirichlet problem

A(u)

u

where, formally,

f

oin non an } (1.1)

A(u) -'V° (\'Vulp-2u) + a ~ __-v'i+ l'VuJ2

l'VulP= ('VuoVu)p/2 , 2 ~ p < 00

a = const. > 0

} (1. 2)

Cases for which a < 0 and 1 < p < 2 are discussed in [4); a related case

of parabolic problems is studied in [1]. In (1.1), n is a smooth bounded

'nopen domain in:R with a Lipschitz boundary an The "weak"

2

or variational form of (1.1) is:

(A(u),v> = '(f,v)

Find u E w1,p(n) such thato

(1. 3)

where Wl,p(n) is the usual Sobolev space of order (l,p) witho

v € w~' p (n) => v I an == O. For 2 2. p < 00, this space is reflexive when

equipped with the norm,

(1.4)

;rn (1.3), (-,-j

p' = p/ (p-l) ,

WI, p (m. Thus,o

denotes duality pairing on

with W-l,p'(n) = (Wl,p(n»'o denoting the dual of

<A(u) ,v>

(1.5)

We wish to study finite-element-Galerkin approximations of (1.5) con-

structed using piecewise linear CO-finite elements and regular mesh

refinements satisfying the usual angIe conditions (see, e.g. Ciarlet [2)

or [5]). Thus, we approximate wl,p (n) by a family of finite-dimensionalo

subs paces {Uh} , everywhere dense inO<h<1

Wl,p(n), whereo

3

(1. 6)

where G is a finite elem~nt in a partition ~ of Q and P1(G) is the

space of polynomials of degree < 1 on G. These spaces possess the following

interpolation property as h ~ 0. For uEW~,p(mnWl,p(m, there existso

a constant

such that

K > 0, independent of u and h , and an element

µ = min(l,R.-l)

(1. 7)

The finite element approximation of (1.5) consists of seeking

Uh E ~ such that

(1.8)

2. Existence and Convergence of Ga1erkin Approximations

It can be shown [4] that the operator A:w1,P(n) ~ w-1,p' (n) definedo

in (1.2) satisfies the following conditions:

(i) For P > 3n/(n+2) (n = dimW», and VU,v,w E W1,p(n)- 0

!(A(u) - A(v),w)! ~ g(u,v) Ilu-vlll,pllwl11,p

4

(2.1)

g(u,v) 8(llu111, +llvlll )p-2 + Cla(1 +Ilvlll ),p ,p ,p

..(p+l for 2~p~3

8 = 1 ; Cl = const > 0

p - I for p .: 3

(ii) <A(v) , v> -+ + 00 as Ilvll ~ 00

Ilv Ill, pl,p

(2.2)

(2.3)

(iii)

we have

v u,v t B (0) t whereµ

B (0) = {w t WI, P (n)µ 0

Ilwlll <µ},p(2.4)

2<A(u) - A(v),u-v) > p Ilu-vllP1 - Y(µ)llu-vIIPo'z ,-allu-vll (2.5)

- 0 ,p , P 0,2

in which

p = Zp-lo - E y(µ) 1p' (Ep)l/(p-l)

(acµ)p' (2.6)

p-lE is a positive constant < 2 ,C is an absolute constant depending

only on n ,n, and p, and p' = p/(p-1).

The following existence and convergence theorem governs (1.3) and its

approximation (1.8):

Theorem 1. Let conditions (i), (ii), and (iii) given above hold

5

(particularly (2.1), (2.3), and (2.5». Then

1. There exists at least one solution1 'for any data f E}-l- ,p (m

u E wI, P (n) to (1.3)o

In addition, let (1. 7) hold as h -+ 0 V U h' where U h is defined in (1. 6) .

Then

2. There exists at least one solution uhEUhCW1,PW) to (1.8)1 ' 0

for any data fEW- ,p (m.Finally, if {~} is a sequence of approximations satisfying (1.8) V h,

as h -+ 0 , there exists a subsequence {uhf} which converges strongly

to a solution u of (1.3); i.e.

limh'-+O

II U - uh' 111, P o 0

3. Local Analysis and Error Estimates

Let u be a solution of (1.3) and uh a solution of (1.8) and

suppose conditions (2.1)-(2.6) hold. Notice also the orthogonality

condition

which is obtained by setting v = vh in (1.3) and subtracting (l.8). Then,

from (2.5), (3.1), and (2.1), we have, for any vhfUh,

p p' 2pollu-~II - y(µ) Ilu-~II - al lu-vl I ~ (A(u)

l,p O,q 0,2

6

= (A(u) - A(uh) , u-vh>

~ g(u,uh)! \u-uhlIl,pl Iu-vh I Il,p

Here q 2p' or, if v f Wl, p (n) is boundedo in Lr(n) for some r > 1,

then q = rp/(rp-r-p). Thus,

p'

< g(u,~)llu-~11 +y(µ)llu-uhlll,p O,q

2+ allu-~ II 2

0,

(3.2)

The difference u-~ in (3.2), however, may not represent the error

in the finite element approximation of u: since neither (1.3) nor (1.8)

necessarily have unique solutions, there is no reason to expect that uh

is an approximation of the particular solution u. To proceed further, we

need to examine the local behavior of solutions.

Consider the mean-value formula for Gateaux derivatives:

(A(u) - A(v),w) = (DA(au+(l-a)v) (u-v),w>

(DA(u).v,w,> = lim t- <A(u+tv),w) ,V WtWl,p(Q)t~O+ t 0

aE[O,l]

and consider the auxiliary linear boundary-value problem;

(3.3)

<DA(w):~,n) = <~,~) V ~ f wl'Z(n)o

for ~ € Lq~(n), q~ 2 2. Suppose that numbers M(w), mew), A(W)

exist for each w (wl,p(n) such thato

(3.4)

(3.5)

7

Since I I~I10 2 2 C I I~I/1 2 ,where C is the constant of Poincare's, p, P

inequality, the right side of (3.5)2 is positive whenever

m(w) - C2 A(W) > 0p (1.6)

(3.7)

When (3.5) holds, (3.4) is solvable for 11, but the solutions are

not necessarily unique. When (3.5) also holds, a unique solution n in

Wl,2(n) n W2,2(n) to (2.4) exists ando

IInllz,2 ~ C(u) 11~llo,2

Continuing,

(i) Let 0.3), (3.4), and (3.5) hold

(ii) Let {~} be a sequence of finite element approximations,

satisfying (l.B), which converge strongly to a solution

u of (1.3) (the existence of such a sequence is guaranteed. 1 p 1 00

by Theorem 1), and u (Wo ' (n) nW ' (n);

(iii) For some £ > 0, let (3.6) hold for all

wEB (u) = {wE.wl,p(n): Ilu-wlll < d.£ 0 ,p

.~

Under these conditions, there exists an h and a e€ [O,l] such that£

8

for h < h£

the approximation error eh

= u-~ satisfies

sup'II

(1J>,eh>.II t/! 110, p

< sup

1J>

(DA(8u+(1-e)uh eh,n>

11'11110,2

(A(U)-A(uh),n-~n>sup

1J> 11t/!llo,2

Thus, if (1.7), (2.1), and (3.7) holda:nd q"'~p~2, then as h ~ 0,

I lehl 10 ~ g(u,uh)! lehl 11 K h,q ,p (3.8)

We can now complete the determination of (non-optimal) a-priori error

estimates.

Theorem 2. Let (1.7), (2.1), (3.7), and conditions (i), (ii)

200 1pand (iii) above hold. Let uEw' (mnH ' en) he a solution of (1.3).o

Then, as h ~ 0, the error eh

= u-~ in the finite element approxi-

rnation of u satisfies

I lehl I < G(u)h1/(p-l) + H(u)h1/(p-2)

l,p(3.9)

where G(u) and H(u) are positive constants depending on the solution u

but not on h. 0

The estimate (3.9) follows easily from (3.3), (3.8), and the fact that

since strongly, are bounded in wl,p(n).o

It is also possible to obtain reasonably sharp estimates of G(u) and

H(u) [3; 4].

9

Acknowledgement," The results communicated here were developed in the

course of research supported by the U. S. National Science Foundation

under Grant NSF-ENG-75-07846 and the Army Research Office - Durham, under

Grant DAAG 29-77-G-0087,

References

1. A1duncin, G. and Oden, J.T., "Qualitative Analysis and Ga1erkinApproximation of a Class of Nonlinear Convection-Diffusion Problems;Part I - Qualitative Analysis; Part II - A Model Problem," TICOMReport 78-7, Austin, 1978.

2. Ciar1et, P.G., The Finite Element Method for Elliptic Problems,North-Holland, Amsterdam, 1978,

3. Oden, J.T. and Reddy, C.T., "Finite Element Approximations of a Classof Highly Nonlinear Boundary-Value Problems in Finite Elasticity,Part I - Preliminaries and Qualitative Analysis; Part II - Approxi-mation Theory," J. Num. Functional Anal. Appl., Vol. 1, No.1, 1978.

4, Oden, J.T., Reddy, C.T., and Kikuchi, N., "Qualitative Analysis andFinite Element Approximation of a Class of Nonmonotone NonlinearDirichlet Problems," TICOM Report 78-8, Austin, 1978.

5. Oden, J.T. and Reddy, J.N., An Introduction to the Mathematical Theoryof Finite Elements, Wi1ey-Interscience, New York, 1976.