Numerical_Methods for Nonlinear Variable Problems Episode 2 pot

An Example of EVI of the First Kind: The Obstacle Problem • CmQ: space of m-times continuously differentiable real valued functions for which all the derivatives up to order m are contin

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 secondkinds These EVI are related to second-order partial differential operators (forfourth-order problems, see Glowinski [2] and G.L.T [2], [3]) The physicalinterpretation and some properties of the solution are given Finite elementapproximations of these EVI are considered and convergence results areproved 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 mate problems (cf Cea [1], [2] and G.L.T [1], [2], [3])

approxi-2 An Example of EVI of the First Kind: The Obstacle Problem

• Cm(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 Cm(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 and D" = d^/dxf dx ,

• W m ' p (Q): completion of C m (U) in the above norm,

• WS'"(Q,): completion of CQ(Q) in the above norm,

Then the obstacle problem is a particular ( P J problem defined by:

Find u such that

a(u, v-u)> L(v - w), V » e K , 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 andStampacchia [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\2dx\

i /

is a norm on HQ(Q) and it is equivalent to the norm on Hj(Q) induced from Hl(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 H1(D.) defined by v -> f(v) is a continuous map into H1(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 veH1(Q.) (respectively, HQ(Q)), then the map v -* {v + , v~} is continuous from H\Q) -> H\Q) x H\Q) (respectively, H j ( 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 vne K and v e HJ(Q) Hence vn—>v strongly in L2(Q) Therefore we can extract a subsequence {vn.} such that vn -* 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) •

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+;M0 = u\ n0 Classically, problem (2.1) has been formulated as the problem of finding y (the free boundary) and u such that

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 addother transmission properties: for instance, if *P is smooth enough (say

*P E H2(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 ofproblem (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

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 Q c R * 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),

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||L2(n) defines a norm on H2(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 toAgmon, Doughs, and Nirenberg [1]:

If w e L2(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 amethod 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 H2(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, ut\Y = 0 (2.13)

Problem (2.13) has a unique solution in H\{Q), and the smoothness of T implies that

us 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

From (2.14) and (2.1), we obtain

a(u, uc-u)> L{uE - u) = j f(uc - u) dx (2.15)

The F-ellipticity of o(-, •) implies

a(uc, u£ — u) = a(u t — u, u z — u) + a(u, u s — u) > a(u, ue — u),

\u • v(A«.) dx > f Au, dx (2.17)

By Green's formula, (2.17) implies

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

T c Q V T e f » , U T = Q, (2.21)

^ 0 ^ = 0 V T u T2 e ^ and T x ^ T2 (2.22)

2 An Example of EVI of the First Kind: The Obstacle Problem

(2) Ti and T2 have only one common vertex,

(3) T t and T 2 have only a whole common edge

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 ,Te Zt 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), wfc |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

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 kh , v h - u\) > L(v h - u k ), Vv h eK kh , u kh sK kh (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

EXERCISE 2.3 With notations as in Fig 2.1, prove the following identitiesfor any triangle T:

w dx = m e a 3 S ( r ) f w(MiT), V w e Pl s (2.26)

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 klh )) 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 he 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

(actually, since </>,, -> ^> strongly in L co(fi), the weak convergence of v h in L 2 (Q) is enough

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*hveVkh, 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

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

J f = {ti e K n C0(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> = v strongly in V.

2 An Example of EVI of the First Kind: The Obstacle Problem 37

Define v n by

vn = maxCP, 4>n) (2.39)

so that

vn = %£¥ + <$>„) + I ^ ~<t>n\l

Since ve K, from Corollary 2.1 and relation (2.39), it follows that

lim v n = \\_QV + u) + |vP - 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)

vn 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

vm 6 ®(fi) n K, V OT and lim |[um — 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.,

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^R2).

n~* oo

Hence from (2.41) and (2.46), we have

Supp( | »„ |) <= 12 for n large enough (2.47)

We also have (since supp(0) is bounded)

lim v n = v strongly in LCO (R 2 ) (2.48)

Define v n = vn\n; then (2.46)-(2.48) imply

vn 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 ad, (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 > no (e)

u(x) - £ < t) B(x) < v{x) + e, V x e f i - Q,/ 2 ,

(2.51)

y n(x) = t)(x) = 0 for x e fls/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 cn = 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

{vheV2h,vh{P) ±

then the convergence of u\ to u still holds provided 2T h obeys the same tions as in the statement of Theorem 2.3.

assump-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 Fo Also study the finite element approximation of the aboveEVI _

Hint: Use the fact that if F and Fo are smooth enough, then -V = V, where (see Fig 2.2), iT = {v e C°°(Q), t; = 0 in a neighborhood of Fo}

2.7 Comments on the error estimates

We do not emphasize this subject too much since this is done in detail inCiarlet [1, Chapter 9], [2, Chapter 5] and G.L.T [3, Appendix 1], at least forpiecewise 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, andRaviart [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(/i3/2~E)

estimate for \\u h — u||Hi(n) has been obtained by Brezzi, Hager, and Raviart[1] and Brezzi and Sacchi [1] for an approximation by piecewise quadraticfinite elements, similar to the one described in Sec 2.6

2.8 Iterative solution of the approximate problem

Once the continuous problem has been approximated and the convergenceproved, it remains to effectively compute the approximate solution In thecase 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 thediscrete problem is of the following type:

Min &(Av, v) - (b, i>)] (2.53)

u° e C, ii° arbitrarily chosen in C (u° = {^*l5 , ^JV} may be a good guess),

Proposition 2.3 Let {«"}„ be defined by (2.56)-(2.59) Then for every u° e C

and V 0 < to < 2, we have l i m , , ^ 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 specialproperties 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 IR2 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), V c e K , 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 {vn} in K such that vn -> 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.

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 J X ) = -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 elasticperfectly plastic material whose plasticity yields is given by the Von MisesCriterion (For a general discussion of plasticity problems, see Koiter [1] andDuvaut and Lions [1, Chapter 5]) Starting from a zero-stress initial state, anincreasing 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 stressfield is equivalent (in a convenient system of physical units) to the solution ofthe following variational problem:

Proposition 3.2 Let us denote by u c the solution of (3.5) and let, as before,

u x = d(x, F); then limc^ + 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« • V(y -u )dx>C j (u - u ) dx, V v e K, u 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

lim \\u m - uc\\LHn) = 0 (3.10)

It follows from (3.8) that

Vum • Viu^ - «c ) > |V(u M - u c)\2 dx + C \ (um - u c ) dx

*il *Q *£2

= \\u c — u^Wy + C\\ux — uc\\Li(a) (3.12)

It follows easily from (3.11) and (3.12) that

lim C\\u^ ~ u c\\LHa) = °>

lim \\u m — uc\\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.

1 If SI is a bounded convex domain ofU 2 with F Lipschitz continuous and if

/ e LP(Q) with 1 < p < + oo, then we have

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 whichexact 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, w e 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

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

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 IR2

with a smooth boundary T and if

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.

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 finiteelements 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 mate (3.20) (using essentially the same method as in Sec 2) However, in view

approxi-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 , v - u ) > L(v - u ), V v eK (3.22)