Numerical_Methods for Nonlinear Variable Problems Episode 2 pot

CHAPTER II
Application
of the
Finite Element Method
to the
Approximation
of
Some Second-Order
EVI
1.
Introduction
In this chapter we consider some examples of EVI of the first and second
kinds. These EVI are related to second-order partial differential operators (for
fourth-order problems, see Glowinski [2] and G.L.T. [2], [3]). The physical
interpretation and some properties of the solution are given. Finite element
approximations of these EVI are considered and convergence results are
proved. In some particular cases we also give error estimates.
Some of the results in this chapter may be found in G.L.T. [1], [2], [3].
For the approximation of the EVI of the first kind by finite element methods,
we also refer the reader to Falk [1], Strang [1], Mosco and Strang [1],
Ciarlet [1], [2], [3], and Brezzi, Hager and Raviart [1], [2].
We also describe iterative methods for solving the corresponding approxi-
mate problems (cf. Cea [1], [2] and G.L.T. [1], [2], [3]).
2.
An
Example
of EVI of the
First
Kind:
The
Obstacle Problem

Notations
All the properties of Sobolev spaces used in this chapter are proved in Lions
[2],
Necas [1], and Adams [1]. Usually we shall have
•
Q: a bounded domain in IR
2
,
•
F = <9Q,
•
x = {x
u
x
2
}, a generic point of Q,
.
V
=_{d/8x
u
d/dx
2
},
•
C
m
(Q): space of m-times continuously differentiable real valued functions
for which all the derivatives up to order m are continuous in Q,
• CQ(Q)
= {v e C

m
(Q) | Supp(u) is a compact subset of Q},
• IML,
P>
n
=
L«l<m \\D«v\\
LP(n)
{or
v e
C
m
(Q), where
a =
{cc
1
,a
2
};x
1
,oc
2
aie
non-negative integers, |a| = otj +
oc
2
and D" = d^/dxf dx
2
2
,

28 II Application of the Finite Element Method
•
W
m
'
p
(Q):
completion of C
m
(U) in the above norm,
• WS'"(Q,): completion of
CQ(Q)
in the above norm,
• H
m
(Q.)=
W
m
-
2
(Q),
2.1.
The continuous problem
Let
V = tf£(Q) = {v e H\Q)\v\
r
= trace of v on T = 0}
(cf. Lions [2] and Necas [1] for a precise definition of the trace),
where
a(u, v) = VM • Vv dx,

Jn
_ du dv du dv
• Vv = —- — +
dx
2
dx
2
L(v) = </, v) for / e V* = H~\Q) and v e V. Let W e H^O) n C°(Q) and
'PIr < 0. Define K = {u e Hj(Q)|u > i> a.e. on Q}.
Then the obstacle problem is a particular (PJ problem defined by:
Find u such that
a(u, v-u)> L(v - w), V»eK, ueK. (2.1)
The physical interpretation of this problem is as follows: Let an elastic
membrane occupy a region Q in the x
u
x
2
plane; this membrane is fixed
along the boundary T on Q. If there is no obstacle, from the theory of elasticity,
the vertical displacement u, obtained by applying a vertical force F, is given by
the solution of the following Dirichlet problem:
—AM = /' in Q,
(2.2)
where / = F/t, t being the tension of the membrane.
If there is an obstacle, we have a free boundary problem, and the displace-
ment
M
satisfies the variational inequality (2.1) with \\i being the height of the
obstacle. Similar EVI also occur, sometimes with nonsymmetric bilinear forms,
in mathematical models for the following problems:

• Lubrication phenomena (cf. Cryer [1]).
• Filtration of liquids in porous media (cf. Baiocchi [1] and Comincioli [1]).
• Two-dimensional irrotational flows of perfect fluids (cf. Brezis and
Stampacchia [1], Brezis [1], and Ciavaldini and Tournemine [1]).
• Wake problems (cf. Bourgat and Duvaut [1]).
2.
An
Example
of
EVI
of the
First Kind:
The
Obstacle Problem
29
2.2.
Existence
and
uniqueness results
For proving the existence and uniqueness of the problem (2.1), we need the
following lemmas stated below without proof (for the proofs of the lemmas,
see,
for instance, Lions [2], Necas [1], and Stampacchia [1]).
Lemma 2.1. Let Qbe a bounded domain in U
N
. Then the seminorm on H
1
(Q)
\l/2
\Vv\

2
dx\
i
/
is a norm on HQ(Q) and it is equivalent to the norm on Hj(Q) induced from
H
l
(Q).
The above Lemma 2.1 is known as the Poincare-Friedrichs lemma.
Lemma
2.2.
(Stampacchia
[1]). Let
f:U-*M
be
uniformly Lipschitz
con-
tinuous (i.e.,
3 k > 0
such that
\f(t) -
f(t')\
<k\t - £'|, V t,t' eW) and
such
that
/' has a
finite number
of
points
of

discontinuity. Then
the
induced
map /*
on H
1
(D.) defined
by v -> f(v) is a
continuous
map
into
H
1
(Q).
Similar results
hold
for #£(Q)
whenever
/(0) = 0.
Corollary
2.1. // v
+
and v~
denote
the
positive
and the
negative parts
of v
for veH

1
(Q.) (respectively,
HQ(Q)),
then
the map v -* {v
+
, v~} is
continuous
from
H\Q) -> H\Q) x H\Q)
(respectively,
Hj(Q) -+
ff£(fi)
x
Hj(Q)). Also
v
-* \v\ is
continuous.
Theorem
2.1.
Problem
(2.1) has a
unique solution.
PROOF.
In
order
to
apply Theorem
3.1 of
Chapter

I, we
have
to
prove that
a(-, •) is
F-elliptic
and
that
K is a
closed convex nonempty
set.
The F-ellipticity
of
a( •,
•)
follows from Lemma 2.1
and the
convexity
of K is
trivial.
(1)
K is
nonempty.
We
have
¥
e H
l
(Q) n C°(Q)
with

*P < 0 on F.
Hence,
by
Corollary
2.1,
>?
+
e H^Q).
Since
VP
|
r
< 0, we
have
W
+
|
r
= 0.
This implies that
I*"
1
" e
HQ(£1);
since
we have
f
+
e K.
Hence

K is
nonempty.
(2)
K is
closed.
Let v
n
-* v
strongly
in
Hj(n), where
v
n
e K and v e
HJ(Q). Hence
v
n
—>v strongly
in
L
2
(Q).
Therefore
we can
extract
a
subsequence
{v
n
.}

such that
v
n
. -* v
a.e.
on Q.
Then
v
n
. > T a.e. on Q
implies that
v
> ¥ a.e. on Q;
therefore
v e K.
Hence,
by
Theorem 3.1
of
Chapter
I,
we have
a
unique solution
for
(2.1).
•
30
II
Application

of the
Finite Element Method
2.3.
Interpretation
of (2.1) as a
free boundary problem
From
the
solution
u of
(2.1),
we
define
Q
+
=
{x\xeQ,u(x)>
»F(x)},
Q°
=
{x\xeQ,u(x)
=
V(x)},
y
= dQ
+
n
dQ°;u
+
=

w|
n+
;M
0
= u\
n0
.
Classically, problem
(2.1) has
been formulated
as the
problem
of
finding
y
(the free boundary)
and u
such that
-Au=/inQ
+
,
(2.3)
u = V on Q°, (2.4)
u = 0 on T, (2.5)
u
+
\
y
= u°\
y

. (2.6)
The physical interpretation
of
these relations
is the
following:
(2.3)
means
that
on Q
+
the
membrane
is
strictly over
the
obstacle;
(2.4)
means that
on
Q°
the
membrane
is in
contact with
the
obstacle;
(2.6) is a
transmission relation
at

the
free boundary.
Actually (2.3)-(2.6)
are not
sufficient
to
characterise
u
since there
are an
infinite number
of
solutions
for
(2.3)-(2.6). Therefore
it is
necessary
to add
other transmission properties:
for
instance,
if *P is
smooth enough
(say
*P
E
H
2
(Q)),
we

require
the
"continuity"
of
Vw
at y (we may
require
Vu e H
1
(Q)
Remark
2.1.
This kind
of
free boundary interpretation holds
for
several
problems modelled
by EVI of the
first
and
second kinds.
2.4.
Regularity
of the
solutions
We state without proof
the
following regularity theorem
for the

solution
of
problem (2.1).
Theorem
2.2
(Brezis
and
Stampacchia
[2]). Let Qbe a
bounded domain
in U
2
with
a
smooth boundary.
If
L(v)
= f fv dx
with
f E
Z/(Q),
1 < p < + oo (2.7)
and
*Pe^
2
'"(Q), (2.8)
then
the
solution
of the

problem
(2.1) is in
W
2
'
p
(il).
2.
An
Example
of
EVI
of
the First Kind:
The
Obstacle Problem
31
Remark
2.2. Let
QcR*
have
a
smooth
boundary.
We
know that
W
S
'\Q)
<= C(Q) if s > — + k (2.9)

P
(cf.
Necas
[1]). It
follows that
the
solution
u of
(2.1)
will
be in C^U) if/ e
Z/(Q),
¥
G
W
2
'"(Q)
with
p > 2
(take
s = 2,
AT
= 2, fc = 1 in
(2.9)).
The
proof
of
this regularity result will
be
given

in the
following simple
case:
L(v)
= [ fv dx, f e
L
2
(Q),
(2.10)
T-OonQ.
(2.11)
Before proving that (2.10), (2.11) imply
u e
H
2
(Q),
we
shall recall
a
classical
lemma (also very useful
in the
analysis
of
fourth-order problems).
Lemma
2.3. Let Q be a
bounded domain
of U
N

with
a
boundary
F
sufficiently
smooth. Then ||Au||
L
2
(n)
defines
a
norm
on H
2
(Q) n
Hl(il) which
is
equivalent
to
the
norm induced
by the
H
2
(Q)-norm.
EXERCISE
2.1.
Prove Lemma
2.3
using

the
following regularity result
due to
Agmon,
Doughs,
and
Nirenberg
[1]:
If
w e L
2
(Q) and if r is
sufficiently
smooth,
then
the
Dirichlet problem
—
Av = w in Q,
has
a
unique solution
in
HQ(Q)
n H
2
(Q)
(this regularity result also holds
if
Q

is a
convex domain with
F
Lipschitz
continuous).
We
shall
now
apply Lemma
2.3 to
prove
the
following theorem using
a
method
due to
Brezis
and
Stampacchia
[2].
Theorem
2.2*. // F is
smooth enough,
if *F = 0, and if L(v) =
j
Q
fv
dx
with
feL

2
(Q) then
the
solution
u of
the problem
(2.1)
satisfies
ueKn
H
2
(Q),
\\Au\\
LHa)
<
\\f\\
ma)
-
(2-12)
PROOF.
With
L and
\j/
as
above,
it
follows from Theorem
2.1
that problem
(2.1) has a

unique solution
u.
Letting
e > 0,
consider
the
following Dirichlet problem:
-eAu
£
+ u
t
= u in Q, u
t
\
Y
= 0.
(2.13)
Problem (2.13)
has a
unique solution
in
H\{Q),
and the
smoothness
of T
implies that
u
s
belongs
to

H
2
(Ci).
Since
u > 0 a.e. on Q, by the
maximum principle
for
second-order
elliptic differential operators
(cf.
Necas
[1]), we
have
u
e
> 0.
Hence
u
c
eK.
(2.14)
32
II
Application
of the
Finite Element Method
From (2.14)
and
(2.1),
we

obtain
a(u, u
c
-u)>
L{u
E
- u) = j f(u
c
- u) dx.
(2.15)
The F-ellipticity
of
o(-,
•)
implies
a(u
c
,
u
£
—
u) = a(u
t
—
u, u
z
—
u) + a(u, u
s
—

u) > a(u, u
e
—
u),
so that
a(u
e
, u
s
-u)>
\ f(u
s
- u) dx.
(2.16)
By (2.13)
and
(2.16),
we
obtain
E
Vu
£
•
V(Au
£
)
dx > s \
fAu
s
dx,

so that,
\u.
• v(A«.)
dx > f Au, dx.
(2.17)
By Green's formula, (2.17) implies
-
f(AtO
2
*c> f fAu
£
dx.
•>n
-la
Thus
IIAwJum
<
II/IIL^,,
(2.18)
using Schwarz inequality
in
L
2
(ii).
By
Lemma
2.3 and
relations (2.13), (2.18)
we
obtain

lim
u
t
= u
weakly
in
H
2
(Q),
(2.19)
(which implies that
lim u
e
= u
strongly
in
H
S
(Q),
for
every
s < 2 (cf.
Necas [1])),
so
that
u
e
H
2
(fi) with

||Au||
L2(n)
<
||/||L2(Q).
(2.20)
•
2.5. Finite element approximations of (2.1)
Henceforth we shall assume that Q is a polygonal domain of U
2
. Consider a
"classical" triangulation 2T
h
of Q, i.e. 2T
h
is a finite set of triangles T such that
TcQ VTef», U T = Q, (2.21)
^0^
= 0 V T
u
T
2
e ^ and T
x
^ T
2
. (2.22)
2.
An
Example
of EVI of the

First Kind:
The
Obstacle Problem
Figure
2.1 M
iT
m
2T
33
M
lT
m
3T
Moreover Vr
1
,r
2
eJ, and Ti # T
2
, exactly one of the following conditions
must hold
(1)
T,nT
2
= 0,
(2)
Ti and T
2
have only
one

common vertex,
(3)
T
t
and T
2
have only
a
whole common edge.
(2.23)
As usual
h
will
be the
length
of
the largest edge
of
the triangles
in the
triangula-
tion.
From
now on we
restrict ourselves
to
piecewise linear
and
piecewise
quadratic finite element approximations.

2.5.1.
Approximation
of V and K
•
P
k
:
space
of
polynomials
in x
t
and x
2
of
degree less than
or
equal
to k,
•
l.
h
= {P e U, P is a
vertex
of T e
$~
h
},
.
IJ, = {P e Q, P is the

midpoint
of an
edge
of T e ^"J,
Figure
2.1
illustrates some further notation associated with
an
arbitrary
triangle
T. We
have
m,
T
e
Sj,, M,
T
eZ
t
.
The
centroid
of the
triangle
T is
denoted
by G
T
.
The space

K = Hj(Q) is
approximated
by the
family
of
subspaces (V%)
h
with
k = 1 or 2,
where
F£
= to e
C°(O), w
fc
|
r
= 0 and
v
h
\
T
eP
k
,\/Te
3T
h
},
k = 1,2.
It
is

clear that
the V\ are
finite dimensional
(cf.
Ciarlet
[1]). It is
then quite
natural
to
approximate
K by
K\
= K e V\, v
h
(P) > Y(P), V P £
S£},
fe = 1, 2.
Proposition
2.1.
T/ien
Xj /or fc = 1, 2 are
closed convex nonempty subsets
of
34 II Application of the Finite Element Method
EXERCISE 2.2. Prove Proposition 2.1.
2.5.2. The approximate problems
For k = 1, 2, the approximate problems are defined by
a(u
k
h

, v
h
- u\) > L(v
h
- u
k
),
Vv
h
eK
k
h
,
u
k
h
sK
k
h
.
(P\
h
)
From Theorem 3.1 of Chapter I and Proposition 2.1, it follows that:
Proposition 2.2. (P\
h
) has a unique solution for k = 1 and 2.
Remark 2.3. Since the bilinear form a(-,-) is symmetric, (P*
A
) is actually

equivalent to (cf. Chapter I, Remark 3.2) the quadratic programming problem
Min rjafo,
v
h
) -
L(v
h
)l (2.24)
v
h
e
2.6. Convergence results
In order to simplify the convergence
proof,
we shall assume in this section that
¥ e C°(Q) n H\Q) and *P < 0 in a neighborhood of F. (2.25)
Before proving the convergence results, we shall describe two important
numerical quadrature schemes which will be used to prove the convergence
theorem.
EXERCISE 2.3. With notations as in Fig. 2.1, prove the following identities
for any triangle T:
w
dx =
mea
3
S(r)
f
w(M
iT
), V

w e P
ls
(2.26)
wdx =
vae
^
r
l
£
w
(m
iT
),
V
w e P
2
.
(2.27)
Formula (2.26) is called the Trapezoidal Rule and (2.27) is known as Simpson's
Integral formula. These formulae not only have theoretical importance, but
practical utility as well.
We have the following results about the convergence of u\ (solution of the
problem (P
k
lh
)) as h -» 0.
Theorem 2.3. Suppose that the angles of the triangles of 2T
h
are uniformly
bounded below by 9

0
> 0 as h -» 0; then for k = 1,2,
lim ||u\ -
M
||
H4(n)
= 0, (2.28)
where u\ and u are the solutions of(P\
h
) and (2.1), respectively-
2.
An
Example
of EVI of the
First
Kind:
The
Obstacle Problem
35
PROOF.
In this proof we shall use the following density result to be proved later:
nK = K. (2.29)
To prove (2.28) we shall use Theorem 5.2 of Chapter I. To do this we have to verify that
the following two properties hold (for k = 1,2):
(i) If (v
h
)
h
is such that v
h

e Kjj, V h and converges weakly to v as h -* 0, then v e K.
(ii) There exist x, I = K and r\: x -> K\ such that lim^o r^v = v strongly in V,
V
v e x-
Verification of (i). Using the notation of Fig. 2.1 and considering
</>
e
3>{Q)
with
<j>
> 0,
we define
<j>
h
by
cf>
h
= ^T
S
^
h
<^(G
T
)^
T
, where ^r is the
characteristic
function
1
of T and G

T
is the centroid of T. It is easy to see from the uniform continuity of
<j>
that
lim
(/>„
=
(j>
strongly in L°°(n). (2.30)
Then we approximate *P by
*P
fc
such that
(2.31)
^(P) = ¥(P), VPeZj.
This function *P
h
satisfies
lim «P
h
= ¥ strongly in L°°(Q). (2.32)
h->0
Let us consider a sequence
(v
h
)
h
,
v
h

e Kl,V h such that
lim v
h
= v weakly in V.
Then lim^
0
v
h
= v strongly in L
2
(Q) (cf. Necas [1]) which, using (2.30) and (2.32),
implies that
lim I {v
h
-
¥„)(£„
dx = I (v - VW dx, (2.33)
lim
f
(v
h
-
VM>k
dx= f (v -
(actually, since </>,, -> ^> strongly in L
co
(fi), the weak convergence of v
h
in L
2

(Q) is enough
to prove (2.33)).
We have
[
(v
h
-
V
h
)<f>
h
dx= X
<KG
T
)
f
(»*
-
Y*)
^.
(2.34)
From (2.26), (2.27), and from the definition of T
ft
, we obtain for all Te#~
h
,
f (t>* - «P») dx =
mea
'
(r)

I K(M,
T
) - WM
ff
)] if /c = 1, (2.35)
f (^ - T
ft
) dx = ^
(T)
E
[O^T)
- W«ir)] if * = 2. (2.36)
l
Xi{x)
= 1,
VxeT,
XT(X) = 0 if X
<£
T.
36
II
Application of the Finite Element Method
Using the fact that
4>
h
>
0, the definition of
K\
and relations (2.35) and (2.36),
it

follows
from (2.34) that
1
(ffi
—
^hi^h
dx > 0,
V
(j)
a
so that
as h
-»•
0
I
(t> -
4*)^
dx > 0,
V
cf>
€
&(Q), 4>>0
•In
which
in
turn implies
v > T
a.e.
in
£2. Hence (i)

is
verified.
Verification
of
(ii).
From (2.29)
it is
natural
to
take
x =
®(P) <">
K.
We define
as the "linear" interpolation operator
if k = 1
and "quadratic" interpolation operator
iffe
=
2,i.e,
r*
h
veV
k
h
,
V
v e
Hj(fi)
n

C°(fi),
(2-37)
V
P e
SJ
for
fc
=
1,
2.
On the one hand
it is
known (cf., for instance, Ciarlet [1], [2] and Strang and Fix [1])
that under the assumption made on
9~
h
in
the statement of Theorem 2.3, we have
\\r
k
h
v-v\\
v
<Ch
k
\\v\\
Hk
,
Hn)
, Voe©(QX

k =
1,
2,
with
C
independent
of
h and u. This implies that
lim
\\riv - v\\
v
= 0,
V t;
£ x,
k=l,2.
On the other hand,
it is
obvious that
r&eKl,
VveKnC°(n),
so that
r
k
h
veK
k
h
,
Vvex
for

fe
=
1, 2.
In conclusion, with
the
above
x
and
r£, (ii) is
satisfied. Hence
we
have proved
the
Theorem 2.3 modulo, the proof
of
the density result (2.29).
D
Lemma 2.4. Under
the
assumptions (2.25),
we
have 3}(£l)
n K = K.
PROOF. Let us prove the Lemma
in
two steps.
Step 1. Let us show that
Jf
=
{ti

e K n
C
0
(H),
v
has
a
compact support
in
Q} (2.38)
is dense
in K.
Let
v e K; K a
H},(Q) implies that there exists
a
sequence
{</>„}„
in
3(0.) such that
lim
4>
n
= v
strongly
in V.
2.
An Example of EVI of the First Kind: The Obstacle Problem 37
Define v
n

by
v
n
=
maxCP,
4>
n
)
(2.39)
so that
v
n
= %£¥ +
<$>„)
+
I
^
~<t>
n
\l
Since ve K, from Corollary 2.1 and relation (2.39), it follows that
lim v
n
=
\\_QV
+ u) + |
v
P-i;|] = max(*P, v) = v strongly in V. (2.40)
From (2.25) and (2.39), it follows that
each v

n
has a compact support in Q, (2.41)
v
n
6 K n C°(Q). (2.42)
From (2.40)-(2.42) we obtain (2.38).
Step 2. Let us show that:
For every v e Jf, there exists a sequence {v
m
}
m
such that
v
m
6 ®(fi) n K, V OT and lim |[u
m
— u[|
H
J(n) = 0.
From Step 1 this proves that
S>(Q)
n K is dense in K. Let p
n
be a sequence of mollifiers,
i.e.,
p
n
e
2(U
2

),
p
n
> 0, (2.43)
f
Pn
{y) dy = 1, (2.44)
Supp(p
n
) = {0}, {Supp(p
n
)} is a decreasing sequence.
n=l
Let u e JT. Let v be an extension of v defined by
(v(x) ifxefi,
then t5 e
H^U
2
).
Let {!„ = v * p
n
, i.e.,
o.W
= f P^x -
y)C(y)
dy,
(2.45)
then
5
B

e @(R
2
),
Supp(C
n
) <= Supp(y) + Supp(pJ, (2.46)
lim v
n
= v strongly in
H^R
2
).
n~*
oo
Hence from (2.41) and (2.46), we have
Supp( | »„ |) <= 12 for n large enough. (2.47)
38
II
Application
of the
Finite Element Method
We also have (since supp(0)
is
bounded)
lim
v
n
= v
strongly
in

L
CO
(R
2
). (2.48)
Define
v
n
= v
n
\
n
;
then (2.46)-(2.48) imply
v
n
e
(2.49)
lim
v
n
= v
strongly
in #o(Q) n
C°(Q);
v
e Jf and
*P
< 0 in a
neighborhood

of F
imply that there exists
a 5 > 0
such that
v
= 0, ¥ < 0 on a
d
,
(2.50)
where
Qj = {x e
fi|d(x,
F) < 5} (d(x, F) =
distance from
x to F).
From (2.48)
and
(2.50)
it
follows that V
e > 0,
there exists
an n
0
= n
o
(e)
such that
V
n > n

o
(e)
u(x)
- £ <
t)
B
(x)
< v{x) + e,
Vxefi-
Q,
/2
,
(2.51)
y
n
(x)
= t)(x) = 0 for x e
fl
s/2
Since
Q
—
Q.
s/2
is a
compact subset
of Q,
there exists
a
function

0 (cf., for
instance,
H. Cartan
[1])
such that
0
e
S>(fi),
0 > 0 in Q
(2.52)
Finally, define
w^ = u
n
+ £0.
Then from (2.49), (2.51),
and
(2.52),
we
have
w*
e
3){Q),
lim
w
c
n
= v
strongly
in
Ho(fl),

with
W^(JC)
> v(x) > T(x),
V
x e Q, so that Step 2 is proved. •
Remark 2.4. Analyzing verification (i) in the proof of Theorem 2.3, we
observe that if for k = 2 we use, instead of K\, the convex set
{v
h
eV
2
h
,v
h
{P) ±
then the convergence of u\ to u still holds provided
2T
h
obeys the same assump-
tions as in the statement of Theorem 2.3.
EXERCISE
2.4. Extend the previous analysis if
Q.
is not a polygonal domain.
EXERCISE
2.5. Let Q be a bounded domain of U
2
and let F
o
be a "nice" subset

of F (see Fig. 2.2). Define V by V = {v e
H\Q.),
v
\
TQ
= 0}. Taking the bilinear
form a(-, •) as in (2.1), and L e F*, study the following EVI:
a(u,
v
—
u) > L(v
—
u),
VueK,
u e K,
2.
An Example of EVI of the First Kind: The Obstacle Problem 39
Figure 2.2
where K = {veV, v>¥ a.e. in O} and ¥ £ C°(fi) n H\O), ¥ < 0 in a
neighborhood of F
o
. Also study the finite element approximation of the above
EVI. _
Hint: Use the fact that if F and F
o
are smooth enough, then -V = V, where
(see Fig. 2.2), iT = {v e C°°(Q), t; = 0 in a neighborhood of F
o
}.
2.7. Comments on the error estimates

We do not emphasize this subject too much since this is done in detail in
Ciarlet [1, Chapter 9], [2, Chapter 5] and G.L.T. [3, Appendix 1], at least for
piecewise linear approximations.
2.7.1.
Piecewise linear approximation
Using piecewise linear finite elements and assuming that feL
2
(Q) and
\ji, ueH
2
(Q), 0(h) estimates for \\u — u
h
\\
HHn)
have been obtained by Falk
[1],
[2], [3], Strang [1], Mosco and Strang [1], and Brezzi, Hager, and
Raviart [1]. We also refer to Ciarlet [1, Chapter 9], [2, Chapter 5] (resp.,
G.L.T. [3, Appendix 1]) in which the Falk (resp., Brezzi, Hager, and Raviart)
analysis is given.
2.7.2. Piecewise quadratic approximation
Assuming more regularity for /, *F, and u than in the previous case (also
assuming some smoothness hypotheses for the free boundary, an O(/i
3/2
~
E
)
estimate for \\u
h
— u||

H
i
(n)
has been obtained by Brezzi, Hager, and Raviart
[1] and Brezzi and Sacchi [1] for an approximation by piecewise quadratic
finite elements, similar to the one described in Sec. 2.6.
40 II Application of the Finite Element Method
2.8. Iterative solution of the approximate problem
Once the continuous problem has been approximated and the convergence
proved, it remains to effectively compute the approximate solution. In the
case of the discrete obstacle problem, this can be done easily by using an
over-relaxation method with projection as described in Cea and Glowinski
[1],
Cea [2], and also in Chapter V, Sec. 5 of this book.
Let us justify the use of this method. It follows from Remark 2.3 that the
discrete problem is of the following type:
Min &(Av, v) - (b, i>)] (2.53)
veC
where (-, •) denotes the usual inner product in U
N
and v = {v
u
, v
N
} and
A = (a
0
), 1 < i < N, 1 <;" < N (2.54)
is a symmetric positive-definite N x N matrix and C is the set given by
C={»eR

lv
,i;
i
>4'
i)
l<i< N}. (2.55)
Since C is the product of closed intervals of U, the over-relaxation method
with projection on C can be used. Let us describe it in detail:
u° e C, ii° arbitrarily chosen in C (u° =
{^*
l5
,
^JV}
may be a good guess),
(2.56)
then u" being known, we compute «"
+1
, component by component using for
\
where
P
t
(x) = Max(x, ¥,), V x e K. (2.59),
From Chapter V, Sec. 5 (see also Cea and Glowinski [1], Cea [2] ,and G.L.T.
[1],
[3]) it follows that:
Proposition 2.3. Let {«"}„ be defined by (2.56)-(2.59). Then for every u° e C
and V 0 < to < 2, we have lim,,^ u" = u, where u is the unique solution of
(2.53).
Remark 2.5. In the case of the discrete obstacle problem, the components of u

will be the values taken by the approximate solution at the nodes of t.
h
if
k = 1 and t
h
n l'
h
if k = 2. Similarly, ^ will be the values taken by ¥ at the
nodes stated above, assuming these nodes have been ordered from 1 to N.
3.
A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem 41
Remark 2.6. The optimal choice for co is a critical and nontrivial point.
However, from numerical experiments it has been observed that the so-called
Young method for obtaining the optimal value of
co
during the iterative process
itself leads to a value of co with good convergence properties. The convergence
of this method has been proved for linear equations and requires special
properties for the matrix of the system (see Young [1] and Varga [1]). However,
an empirical justification of its success for the obstacle problem can be made,
but will not be given here.
Remark 2.7. From numerical experiments it has been found that the optimal
value of
co
is always strictly greater than unity.
3.
A Second Example of EVI of the First Kind:
The Elasto-Plastic Torsion Problem
3.1.
Formulation. Preliminary results

Let SI be a bounded domain of IR
2
with a smooth boundary F. With the same
definition for V, a(-, •), and L(-) as in Sec. 2.1 of this chapter, we consider the
following EVI of the first kind:
a(u, v - u) > L(v - u), VceK, u e K, (3.1)
where
K = {ve Hj(fi), |Vv\ < 1 a.e. in Q}. (3.2)
Theorem 3.1. Problem (3.1) has a unique solution.
PROOF. In order to apply Theorem 3.1 of Chapter I, we only have to verify that K is a
nonempty closed convex subset of V. K is nonempty because 0 e K, and the convexity of
K is obvious. To prove that K is closed, consider a sequence {v
n
} in K such that v
n
-> v
strongly in V. Then there exists a subsequence {v
n
.} such that
lim Vv
n
. = Vv a.e.
Since | Vv
n
\
< 1 a.e., we get | Vv
\
< 1 a.e. Therefore v e K. Hence K is closed. •
The following proposition gives a very useful property of K.
Proposition 3.1. K is compact in C°(Q) and

I
v(x) | < d(x, F), V x e Q andM veK, (3.3)
where d(x, F) is the distance from x to F.
42 II Application of the Finite Element Method
EXERCISE 3.1. Prove Proposition 3.1.
Remark 3.1. Let us define u
x
and
M_
X
by
M^OC) = d(x, T),
M.JX) = -d(x, T).
Then w^ and w_ „ belong to K. We observe that u
x
is the maximal element
of X and
M_
x
is the minimal element of K.
Remark 3.2. Since a(-, •) is symmetric, the solution u of (3.1) is characterized
(see Sec. 3.2 of Chapter I) as the unique solution of the minimization problem
J(u) < J(v),
VveK,
ueK, (3.4)
with J(v) = ^a(v, v) - L(v).
3.2.
Physical motivation
Let us consider an infinitely long cylindrical bar of cross section Q, where Q is
simply connected. Assume that this bar is made up of an isotropic elastic

perfectly plastic material whose plasticity yields is given by the Von Mises
Criterion. (For a general discussion of plasticity problems, see Koiter [1] and
Duvaut and Lions [1, Chapter 5]). Starting from a zero-stress initial state, an
increasing torsion moment is applied to the bar. The torsion is characterised
by C, which is defined as the torsion angle per unit length. Then for all C, it
follows from the Haar-Karman Principle that the determination of the stress
field is equivalent (in a convenient system of physical units) to the solution of
the following variational problem:
Minji f
|VP|
2
dx - C f v dx\. (3.5)
This is a particular case of (3.1) or (3.4) with
L(i>) = C [vdx. (3.6)
Jn
The stress vector a in a cross section is related to u by a = Vu, so that u is a
stress potential, and we can obtain a once the solution of (3.5) is known.
Proposition 3.2. Let us denote by u
c
the solution of (3.5) and let, as before,
u
x
= d(x, F); then lim
c
^
+ x
u
c
= u
x

strongly in HQ(£1) n C°(fi).
PROOF.
Since
u
c
is the
solution
of
(3.5),
it is
characterized
by
f
V«
c
•
V(y
-u
c
)dx>C
j (u - u
c
) dx,
V
v e K, u
c
e K. (3.7)
3.
A
Second Example

of
EVI
of the
First Kind:
The
Elasto-Plastic Torsion Problem
43
Since
u
x
e K,
from
(3.7) we
have
f
Vu
c
•
V(
Uoo
- u
c
) dx > C \ {u
x
- u
c
) dx, (3.8)
i.e.
C
I («„ - u

c
) dx + ) |
Vu
c
\
2
dx< \ Vu
m
• Vu
c
dx
<
f |
Vu
w
| • |
VM
C
I
dx <
meas(Q).
(3.9)
From
(3.3) we
have
u^ — u
c
> 0, so
that
(3.9)

implies
ll
M
oo
~
U
C\\LHCI)
^ C~
l
meas(Q)
which
in
turn implies
lim
\\u
m
-
u
c
\\
LHn)
= 0.
(3.10)
C—+oo
From
the
definition
of K and
from
the

Proposition 3.1,
we
find that
K is
bounded
and
weakly closed
in V and
hence weakly compact
in
V. Furthermore,
K is
compact
in
C°(ii).
Relation (3.10) implies
lim
u
c
= u,
strongly
in
C°(Q),
lim
u
c
= u
M
weakly
in K

It follows from
(3.8)
that
Vu
m
• Viu^ - «
c
) > |V(u
M
- u
c
)\
2
dx + C \ (u
m
- u
c
) dx
*il *Q *£2
=
\\u
c
— u^Wy + C\\u
x
—
u
c
\\
L
i

(a)
.
(3.12)
It follows easily from (3.11)
and
(3.12) that
lim
C\\u^ ~
u
c
\\
LHa)
= °>
lim
\\u
m
— u
c
\\
v
= 0. D
Remark 3.3. In the case of a multiply connected cross section, the variational
formulation of the torsion problem has to be redefined (see Lanchon [1],
Glowinski and Lanchon [1], and Glowinski [1, Chapter 4]).
3.3.
Regularity properties and exact solutions
3.3.1.
Regularity results
Theorem 3.2 (Brezis and Stampacchia [2]). Let ube a solution of (3.1) or (3.4)
and L(v) = §

n
fv dx.
44 II Application of the Finite Element Method
1.
If SI is a bounded convex domain
ofU
2
with F Lipschitz continuous and if
/e L
P
(Q) with 1 < p < + oo, then we have
ueW
2
-
p
(Q).
(3.13)
2.
IfQ is a bounded domain
ofU
2
with a smooth boundary F and iffeL
p
(€i)
with 1 < p < +oo, then u e
W
2
-"(Q).
Remark 3.4. It will be seen in the next section that, in general, there is a
limit for the regularity of the solution of (3.1) even if F and / are very smooth.

Remark 3.5. It has been proved by H. Brezis that under quite restrictive
smoothness assumptions on F and /, we may have
3.3.2. Exact solutions
In this section we are going to give some examples of problems (3.1) for which
exact solutions are known.
EXAMPLE 1. We take Q = {x| 0 < x < 1} and L(v) = c JJ v dx with c > 0.
Then the explicit form of (3.1) is
f u'(v' -u')dx>c\(v-u)dx, V v e K, we K,
Jo Jo
where K = {v e Hj(O), | v'
\
< 1 a.e. on Q} and v' = dv/dx.
The exact solution of (3.14) is given by
(3.14)
u
(x) =
C
-x(l - x), Vx, if c < 2; (3.15)
if c > 2, we have
x if 0 < x < - ,
2 c
1 - x if- + - < x < 1.
2 c
EXAMPLE 2. In this example we consider a two-dimensional problem. We
take
fi= {x|xf + x\ <R
2
},
L(v) = c v dx with c > 0.
3.

A Second Example of EVI of the First Kind: The Elasto-Plastic Torsion Problem 45
Then setting r = (x\ + x|)
1/2
, the solution u of (3.1) is given by
u{x)
=
C
-(R
2
-r
1
) ifc<|;
(3.17)
4 K
if c > 2/R, then
2
u(x) = R-r ii-<r<R,
c
u(x) = - \(R
2
- r
2
) -(R- -
ifO<r< (3.18)
c
These examples illustrate Remark 3.4. We see that for c sufficiently large, we
have
u e W
2
' °°(J2) n Hj(Q), u $ H

3
(Q). (3.19)
In fact we have
u e
H
S
(Q.),
V s < |.
EXERCISE 3.2. Verify that the u given in Examples 1 and 2 are exact solutions
of the corresponding problems.
3.4.
An equivalent variational formulation
In H. Brezis and M. Sibony [1] it is proved that if Q is a bounded domain of IR
2
with a smooth boundary T and if
L(v) = c v(x) dx (c > 0 for instance),
Jsi
then the solution of (3.1) is also a solution of
a(u, v — u) < c \ (v — u) dx, VceX,
Jil
(3.20)
uek= {ve Ho(Q), | v(x) | < d(x, T)
a.e.}.
Problem (3.20) is very similar to the obstacle problem considered in Sec. 2
of this chapter. Since a(-, •) is symmetrical, (3.20) is also equivalent to
J(u) < J(v),
VveK,
uek, (3.21)
with
J(v) = ja(v, v) — c v(x) dx.

46 II Application of the Finite Element Method
The numerical solution of (3.20) and (3.21) is considered in G.L.T. [1,
Chapter 3] (see also Cea [2, Chapter 4] and Chapter V, Sec. 5.5 of this book).
EXERCISE 3.3. Study the numerical analysis of (3.21).
EXERCISE 3.4. Assume c > 0 in (3.20). Then prove that the solution u of (3.20)
is also the solution of the EVI obtained by replacing K by {v e HQ(Q.), V(X) <
d(x, T) a.e.} in (3.20).
3.5.
Finite element approximation of (3.1)
In this section we consider an approximation of (3.1) by first-order finite
elements. From the viewpoint of applications in mechanics (in which / = c),
it seems that, given the equivalence of (3.1) and (3.20), it is sufficient to approxi-
mate (3.20) (using essentially the same method as in Sec. 2). However, in view
of other possible applications, it seems to us that it would be interesting to
consider the numerical solution of (3.1), working directly with K instead of K.
For the numerical analysis of (3.20) by finite differences, see G.L.T.
[3,
Chapter 3] and Cea, Glowinski, and Nedelec [1].
3.5.1.
Approximation of V and K
We use the notation of Sec. 2.5 of this chapter. We assume that Q is a polygonal
domain of U
2
(see Remark 3.8 for the nonpolygonal case), and we consider
a triangulation
2T
h
of Q satisfying (2.21)-(2.23). Then V and K, respectively, are
approximated by
V

h
= H e C°(Q), v
h
= 0 on r, v
h
\
T
e P
u
V T e ST
h
},
K
h
= Kn V
h
.
Then one can easily prove:
Proposition 3.3. K
h
is a closed convex nonempty subset of V
h
.
Remark 3.6. If v
h
e V
h
, then Vv
h
is a constant vector on every T e ST

h
.
3.5.2. The approximate problem
The approximate problem is defined by:
Find u
h
e K
h
such that
a(u
h
, v
h
- u
h
) > L(v
h
- u
h
), V
v
h
eK.
(3.22)
3.
A
Second Example
of EVI of the
First
Kind:

The
Elasto-Plastic Torsion Problem
47
One can easily prove:
Proposition 3.4. The approximate problem (3.22) has a unique solution.
In Sec. 7 of this chapter one may find practical formulae related to finite
element approximation. Using these formulae, (3.22) and the equivalent
problem (3.23) can be expressed in a form more suitable for computation.
Remark 3.7. Since a(-, •) is symmetrical, (3.22) is equivalent to the non linear
programming problem
(3.23)
v
h
6 A'j,
The natural variables in (3.23) are the values taken by v
h
over the set t,
h
of the interior nodes of ^,. Then the number of variables in (3.23) is Card(L
h
).
The number of constraints is the number of triangles, i.e., Card (^), and each
constraint is quadratic with respect to these variables since
VuJ < 1 if and only if | Vv
h
|
2
< 1 over T. (3.24)
Remark 3.8. If O is not polygonal, it is always possible to approximate Q by
a polygonal domain Q

h
in such a way that all vertices of F
h
= dQ
h
belong to T.
Then, instead of defining (3.22) over Q, we define it over Q
h
.
3.5.3.
Remarks on the use of
higher-order
finite elements
In this book only an approximation of (3.1) by first-order finite elements has
been considered. This fact is justified by the existence of a regularity limitation
for the solution of (3.2), which implies that even with very smooth data one
may have u $ V n H
3
(Q) (see the examples of Sec. 3.3.2).
We refer to G.L.T. [3, Chapter 3] and Glowinski [1, Chapter 4, Sec. 3.5.3]
for further discussions on the use of finite elements of order < 2.
3.6.
Convergence results. General case
In this section
we
take
L(v) = </, v),
with
feH~
1

(Q)
= V*.
3.6.1.
A density lemma
In order to apply the general results of Chapter I, the following density lemma
will be very useful.
Lemma 3.1. We have
nK = K. (3.25)
48 II Application of the Finite Element Method
PROOF. We use the notation of Lemma 2.4. Let ve K and e > 0; define v
c
by
v
c
= (v- e)
+
- (v +
£)".
(3.26)
Then we have v
E
eH
1
(Q) with |V«J < 1 a.e. in Q. From the inclusion XcK =
{v e V,
\v(x)\
< d(x, F) a.e. in Q}, it follows that
v
E
(x) = 0 if d(x, T) < e,

(3.27)
I
v
F
(x) | < d(x, F) — £ elsewhere
so that from (3.27) it follows that
v
s
e K and has a compact support in Q. (3.28)
From Corollary 2.1 we have
lim v
e
= v strongly in V. (3.29)
E->0
From (3.28) and (3.29) it follows that if Jf = {v e K, v has a compact support in Q},
then 3f = K.
Thus,
to prove the lemma, it suffices to prove that any v e Jf can be approximated by a
sequence (v
n
\ of functions in ®(fl) n K. Let (p
n
)
n
be a mollifying sequence as defined in
Lemma 2.4 of this chapter. Let v e JT. Denote by v the extension of v to U
2
by taking
zero outside Q. Then 5 e H'(IR
2

).
Let v
n
= v * p
n
so that
C
n
(x) = f p
B
(x - y)i5Cy) dy, (3.30)
VD
n
(x)= I" A(x - yJVeO-) dy. (3.31)
Then
»„ 6
S>(U
2
)
and lim C
n
= v strongly in
H^tt
2
).
(3.32)
Since Supp S c Q, from (3.30) we have
Supp »„ c Q for n sufficiently large. (3.33)
Define v
n

= v
n
\
n
for n sufficiently large. Then (3.32) and (3.33) imply
v
n
e @(£2), lim v
n
= v strongly in V. (3.34)
From (3.31), p
n
> 0, J
R2
p
n
dy
= 1 and |VC(}>)| < 1 a.e. on U
2
, we obtain
| Vv
n
{x) | = | Vv
n
{x)
I
< f
I
Vt>0;) |
A

(x - y) dy < 1, V x e fl, (3.35)
which completes the proof of the lemma. •
3.
A
Second Example
of
EVI
of the
First Kind:
The
Elasto-Plastic Torsion Problem
49
3.6.2. A convergence theorem
Theorem 3.3. Suppose that the angles of the triangles of ST
h
are uniformly
bounded by 9
0
> 0 as h > 0. Then
lim u
h
= u strongly in Vn C°(U), (3.36)
where u and u
n
are, respectively, the solutions of (3.1) and (3.22).
PROOF. TO prove
the
strong convergence
in V, we use
Theorem

5.2 of
Chapter
I, Sec. 5.
To
do
this,
one has to
verify
the
following properties.
(i)
If
(v
h
)
h
,
v
h
e K
h
,
V
h,
converges weakly
to v,
then
v e K,
(ii)'
There exists

x and r
h
with
the
following properties:
1-
X = K,
2.
r
h
:
X
^K
h
,y
h;
3.
for
each
v e x, we can
find
h
0
= h
o
(v)
such that
for all h <
h
o

(v),
r
h
veK
h
and
lim^o
r
h
v = v
strongly
in V.
Verification
of
(i). Since
K
h
<= K and K is
weakly closed,
(i) is
obvious.
Verification
of
(ii)'.
Let us
define
x by
Then
by
Lemma 3.1

and
from lim
A
^
1
Iv — v
strongly
in
V,
V
v e
V,
it
follows that
x = K.
Define r
h
: V n C°(U)-+
V
h
by
r
h
veV
h
,
y
V
eVnC°(U),
(r

h
v)(P) = v(P), \/Pet
h
. (3.37)
Then
r
h
v is the
"linear" interpolate
of » on 5i.
From
the
assumptions
on
S~
h
we
have
(cf. Strang
and Fix [1] and
Ciarlet
[1], [2])
I
V(r_ v
-
v)
| <
Ch
||
v

||
m
,
„<„,
a.e.,
V
v e
&(Q), (3.38)
with
C
independent
of h and v.
This implies
lim ||r
fc
t? —
tj||
K
= 0,
Vi;e
Z
, (3.39)
\Vr
h
v(x)\
<
\Vv(x)\
+
Ch\\v\\
W

2.«,
m
a.e
(3.40)
Since
v e x, it
follows from (3.40) that
we
have
|
Vr
h
v{x)
| < 1
a.e.
for h <
h
o
(v).
This implies
r
h
v e K
h
.
This completes
the
verification
of (ii)' and
hence,

by
Theorem
5.2 of
Chapter
I, we
have
the
strong convergence
of u
h
to u in V.
The strong convergence
of u
h
to u in the
L°°-norm follows from
the
convergence
in V
and from
the
compactness
of K in C°(Q) (see
Proposition
3.1). •
50 II Application of the Finite Element Method
3.7. Error estimates
From now on we assume that / e L
p
for some p > 2.

In Sec. 3.7.1 we consider a one-dimensional problem (3.1). In this case if
/eL
2
(Q) we derive an O(h) error estimate in the F-norm. In Sec. 3.7.2 we
consider a two-dimensional case with feL",p>2, and Q convex; then we
derive an 0(/i
1/2
~
1/p
) error estimate in the F-norm.
3.7.1.
One-dimensional case
We assume here O = {x e K|0 < x < 1} and that feL
2
(Q). Then problem
(3.1) can be written as
f
J
o
o
dx \dx dx
ueK =
<veV,
< la.e. in£H. (3.41)
dx
Let
AT
be a positive integer and h = l/N. Let x
{
= ih for i =

0,1, ,
N and
e, = [x
i
-
1
,x&
i=l,2, ,N.
Let V
h
= {v
h
E C°(U), v
h
(0) = v
h
(l) = 0,
v
h
\
ei
e P
u
i = 1,2, ,N},
K
h
= K n V
h
= {v
h

e V
h
, \ v
h
(Xi) — v
h
(x
t
J| < h for i = 1,
2, ,
N}.
The approximate problem is defined by
Uh)dX
' ^
V
"
€K
"'
u
»
eK
>-
(142)
Obviously this problem has a unique solution. Now we are going to prove:
Theorem 3.4. Let u and u
h
be the solutions of (3.41) and (3.42), respectively.
If fe L
2
(Q), then we have

Ik - M||
K
= 0{h).
PROOF. Since u
h
e K
h
cz K, from (3.41) we have
a(u, u
h
-u)> f(u
h
- u) dx. (3.43)
Jo
Adding (3.42) and (3.43), we obtain
a{u
h
- u,u
h
- u) < a(v
h
- u,u
k
- u) + a(u, v
h
- u) - f(v
h
- u) dx, V v
h
e K

h
which in turn implies
-
u\\
2
v
<
i\\v
h
- uf
v
+ J
d
£ i^ - ~J
dx
- ^
f(v
h
-
u)
dx,
Vv
h
e
K
H
.
(3.44)
3.
A

Second Example
of
EVI
of
the First Kind: The Elasto-Plastic Torsion Problem
51
Since
u e K n
H
2
(0, 1),
we
obtain
du
d
—
—
(v
h
- u) dx =
0
dxdx
Ik*
-
"IIL*-
But
we
have
d~x~
2

< fl/IL-
Therefore (3.44) becomes
ill«»
-
M||K
<
JIIP*
-
M|IF
+
2||/||i21|«*
-
u\\
L
i, Vi),eK
A
.
Let
v e K.
Then
the
usual linear interpolate
r
h
v is
defined
by
r
h
veV

h
,
{r
h
v)(xd
=
ifai).
i = 0,
1, , JV.
We have
d
dx
vixd-vjx^,) 1
= = -
h h J.
1 p dv
—dx.
Hence
we
obtain
<
1
since
<
1
a.e.
in Q.
(3-45)
(3.46)
(3.47)

(3.48)
Thus
r
h
ve
K
h
.
Let
us
replace
v
h
by r
h
u in
(3.46). Then
Ilk
-
u\\
2
v
<
i\\r
h
u
- u\\l +
2||/||
L2(n)
||r

ft
u
-
u\\
ma)
.
(3.49)
From (3.45)
and
standard approximation
results,
we
have
\\r
h
u
- u\\
v
<
Ch\\u\\
HHn)
<
Ch\\f\\
LHa)
,
(3.50)
\\r
k
u
~

u\\
LHn)
<
Cfc
2
||«||
H1(
n)
^
Ch
2
\\f\\
LHai
,
(3.51)
where
C
denotes constants independent
of
u
and
h.
Combining
(3.49)-(3.51), we
get
|k
-
U\\y
=
O(h).

This proves
the
result.
•
EXERCISE
3.5.
Prove
(3.45).
3.7.2.
Two-dimensional case
In this subsection we shall assume that Q is a convex bounded polygonal
domain in IR
2
and that feL
p
(Q) with p > 2. The latter assumption is quite
reasonable since in practical applications in Mechanics we have / = constant.