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
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.
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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.
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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