Báo cáo hóa học: "L∞ -ERROR ANALYSIS FOR A SYSTEM OF QUASIVARIATIONAL INEQUALITIES WITH NONCOERCIVE OPERATORS" potx

To handle such a situation, one can transform problem 1.1 into the following auxiliary system of QVIs: Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006

QUASIVARIATIONAL INEQUALITIES WITH

NONCOERCIVE OPERATORS

MESSAOUD BOULBRACHENE AND SAMIRA SAADI

Received 11 July 2005; Revised 14 November 2005; Accepted 18 December 2005

This paper deals with a system of elliptic quasivariational inequalities with noncoercive operators Two different approaches are developed to prove L∞-error estimates of a con-tinuous piecewise linear approximation

Copyright © 2006 M Boulbrachene and S Saadi This is an open access article distrib-uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

We are interested in the finite element approximation in theL ∞norm of the following system of quasivariational inequalities (QVIs): findU =(u1, ,u J)∈(H1(Ω))Jsatisfying

a i

u i,v − u i

f i,v − u i

∀ v ∈ H1(Ω),

u i ≤(MU) i, u i ≥0,v ≤(MU) i (1.1)

Here,Ω is a bounded smooth domain ofRN,N ≥1, with boundary∂Ω, ( ·,·) is the inner product inL2(Ω), for i =1, ,J, a i(u,v) is a continuous bilinear form on H1(Ω)×

H1(Ω), and fiis a regular function

Problem (1.1) arises in the management of energy production problems whereJ power

generation machines are involved (see [2] and the references therein) In the case studied here, (MU) irepresents a “cost function” and the prototype encountered is

(MU) i = k + inf

μ = i u μ, i =1, ,J. (1.2)

In (1.2),k represents the switching cost It is positive when the unit is “turn on” and

equal to zero when the unit is “turn off.” Note also that operator M provides the coupling

between the unknownsu1, ,u J

In the present paper we are interested in the noncoercive problem To handle such a situation, one can transform problem (1.1) into the following auxiliary system of QVIs: Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2006, Article ID 15704, Pages 1 13

DOI 10.1155/JIA/2006/15704

findU =(u1, ,u J)∈(H1(Ω))Jsuch that

b i

u i,v − u i

f i+λu i,v − u i

∀ v ∈ H1(Ω),

u i ≤(MU) i, u i ≥0,v ≤(MU) i, (1.3) where, forλ > 0 large enough,

b i(u,v) = a i(u,v) + λ(v,v) (1.4)

is a strongly coercive bilinear form, that is,

b i(v,v) ≥ γ  v 2

H1 ( Ω), γ > 0, ∀ v ∈ H1(Ω) (1.5) Naturally, the structure of problem (1.1) is analogous to that of the classical obstacle problem where the obstacle is replaced by an implicit one depending on the solution sought The term quasivariational inequality being chosen is a result of this remark

In [5], a quasi-optimal L ∞-error estimate was established for the coercive problem This result was then extended to the noncoercive case (cf [3,4])

In this paper two new approaches are proposed to prove theL ∞convergence order for the noncoercive problem The first approach consists of characterizing both the

continu-ous and the finite element solutions as fixed points of contractions in L ∞

The second one which is of algorithmic type stands on an algorithm generated by

solv-ing a sequence of coercive systems of QVIs This algorithm is shown to converge geomet-rically to the solution of system (1.1)

It is worth mentioning that the second approach may be very useful for computational

purposes

It should also be mentioned that none of [3,4] provides a computational scheme, even though they both contain the same approximation order as the one derived by the first approach presented in this paper

The paper is organized as follows InSection 2, we lay down some necessary prelim-inaries InSection 3, we state the continuous problem, recall existence, uniqueness, and regularity of a solution, and characterize the solution as the unique fixed point of a con-traction InSection 4, we give analogous qualitative properties for the discrete problem, and characterize its solution as the unique fixed point of a contraction In Section 5,

we develop, separately, the two approaches and show that they both converge quasi-optimally in theL ∞norm

2 Preliminaries

2.1 Assumptions and notations We are given functionsa i jk(x), a i k(x), a i0(x), 1 ≤ i ≤ J,

sufficiently smooth functions such that



1≤ j,k ≤ N

a i jk(x)ξ j ξ kα | ζ |2, ζ ∈ R N,α > 0,

a i(x)β > 0, (x ∈ Ω).

(2.1)

We define the bilinear forms: for allu,v ∈ H1(Ω),

a i(u,v) =



Ω

1≤ j,k ≤ N

a i

jk(x) ∂u

∂x j

∂v

∂x k +

N



k =1

a i

k(x) ∂u

∂x k v + a i0(x)uv



dx. (2.2)

We are also given right-hand sides f isuch that f i ∈ L ∞(Ω) and f i ≥ f0 > 0 for i =

1, ,J.

2.2 Elliptic quasivariational inequalities Letf ∈ L ∞(Ω) such that f > f0> 0, M a

non-decreasing operator fromL ∞(Ω) into itself, and b(u,v) a bilinear form of the same form

as those defined in (1.4) The following problem is called an elliptic quasivariational in-equality (QVI): findu ∈ K(u) such that

b(u,v − u)(f ,v − u) ∀ v ∈ K(u), (2.3) whereK(u) = { v ∈ H1(Ω) such that v ≤ Mu a.e }

Thanks to [2], the QVI (2.3) has a unique solution Moreover, this solution enjoys some important qualitative properties

2.2.1 A Monotonicity property Let f , f in L ∞(Ω) and u = σ( f ,MU), u= σ(f ,M u) be the corresponding solutions of (2.3) Then we have the following comparison principle

Proposition 2.1 If f ≥  f then u ≥  u.

Proof Let u0andu 0be the respective solutions to equations

b

u0,v

=(f ,v) ∀ v ∈ H1(Ω),

b



u0,v

= f ,v ∀ v ∈ H1(Ω). (2.4)

Now let us associate withu and u the respective decreasing sequences

u n+1 = σ

f ,Mu n , un+1 = σ f ,M un

Then the following assertion holds:

iff ≥  f thenu n ≥  u n (2.6) Indeed, since f ≥  f and M is nondecreasing, we have u0≥  u0 So,MU0≥ M u 0, and thus applying standard comparison results in elliptic variational inequalities, we get

Now assume thatu n −1≥  u n −1 Then, as f ≥  f , applying the same comparison argument

as before, we get

Finally, passing to the limit (n → ∞) as in [2, pages 342–358], we getu ≥  u. 

The solution of QVI (2.3) is Lipschitz continuous with respect to the right-hand side.

2.2.2 A Lipschitz dependence property

Proposition 2.2 Let Proposition 2.1 hold Then,

 u −  u  L ∞(Ω)≤ 1

λ + β  f −  f  L ∞(Ω). (2.9)

Proof Let us set

Φ= 1

λ + β  f −  f  L ∞( Ω). (2.10) Then, sincea i0(x)β > 0, we get

f ≤  f +  f −  f  L ∞(Ω)

≤  f + a0(x) + λ

λ + β  f −  f  L ∞(Ω)

≤  f +

a0(x) + λ

Φ.

(2.11)

So, due toProposition 2.1, we obtain

Likewise, interchanging the roles of f and f , we similarly get



Remark 2.3 The above monotonicity and Lipschitz continuity results stay true in the

discrete case provided a discrete maximum principle is satisfied (seeSection 3)

3 The continuous problem

3.1 The continuous system of QVIs The existence of a unique solution to system (1.1) can be proved as in [2, pages 342–358] Indeed, letL ∞+(Ω) denote the positive cone of

L ∞(Ω) and considerH +=(L ∞+(Ω))J equipped with the norm

 V ∞ =max

1≤ i ≤ J

v i

Consider the mapping

T :H +−→ H+,

W −→ TW = ζ =ζ1, ,ζ J

whereζ i = σ( f i+λw i, (MW) i)∈ H1(Ω) solves the following variational inequality (VI):

b i

ζ i,v − ζ i

f i+λw i,v − ζ i

∀ v ∈ H1(Ω),

ζ i ≤(MW) i, ζ i ≥0, v ≤(MW) i (3.3)

Problem (3.3), being a coercive VI, thanks to [1], has one and only one solution Consider now ¯U0=( ¯u1,0, , ¯u J,0), where ¯u i,0is solution to the following variational equation:

a i

¯

u i,0,v

=f i,v

Thanks to [2], problem (3.4) has a unique solution Moreover,u i,0 ∈ W2,p(Ω); 2≤ p <

∞

The mappingT possesses the following properties.

Proposition 3.1 (cf.[2]) T is increasing, and concave and satisfies TW ≤ U¯0 such that

W ≤ U¯0.

Algorithm 3.2 Starting from ¯ U0defined in (3.4) (resp.,U0=(0, ,0)), we define a de-creasing sequence

¯

U n+1 = T ¯ U n, n =0, 1, , (3.5)

(resp., an increasing sequence)

U n+1 = TU n, n =0, 1, (3.6)

It is clear that in view of (3.2), (3.3), the components of the vectors ¯U nandU nare solutions of VIs

Theorem 3.3 Let Proposition 3.1 hold; then, the sequences ( ¯ U n ) and ( U n ) remain in the sector 0, ¯U0 Moreover, they converge monotonically to the unique solution of system (1.1).

3.1.1 Regularity of the solution of system (1.1).

Theorem 3.4 [2, page 453] Assumea i jk(x) in C1,α( ¯Ω), a i(x), a i0(x), and f i in C0,α( ¯Ω),

α > 0 Then (u1, ,u J)∈(W2,p(Ω))J ; 2 ≤ p < ∞

3.2 Characterization of the solution of system ( 1.1 ) as a fixed point of a contraction.

Consider the following mapping:

T:H +−→ H+,

whereZ =(z1, ,z J) is solution to the coercive system of QVIs below:

b i

z i,v − z i

f i+λw i,v − z i

∀ v ∈ H1(Ω),

z i ≤(MZ) i, z i ≥0, v ≤(MZ) i (3.8)

Thanks to [2], problem (3.8) has one and only one solution

Theorem 3.5 Under conditions of Proposition 2.2, the mappingTis a contraction onH +, that is,

T W − T W ∞ ≤ λ

Therefore,Tadmits a unique fixed point which coincides with the solution U of the system

of QVIs (1.1).

Proof Let W, W∈ H+, and letZ = T W, Z= T W be the corresponding solutions to sys-tem of QVIs (3.8) with right-hand sidesF i = f i+λw iandFi = f i+λ wi, respectively Let us also denote

z i = σ

F i, (MZ) i

, zi = σ



F i,

M Zi

Then, making use ofProposition 2.2, we immediately get

z i −  z i

L ∞(Ω)≤ λ

λ + β w i −  w i

and, consequently,

T W − T W ∞ =  Z −  Z ∞

=max

1≤ i ≤ J

z i −  z i

L ∞( Ω)

≤max

1≤ i ≤ J

λ

λ + β



z i −  z i

L ∞(Ω)

λ + β

 max

1≤ i ≤ J

z i −  z i

L ∞( Ω)

λ + β  W − W ∞,

(3.12)

3.3 Another iterative scheme for system ( 1.1 ) In view of the above result, it is natural

to associate with the solution of system of QVIs (1.1) the following algorithm

LetU 0=(u01, , u0J) such thatu0i solves the equation

b



u0

i,v

Algorithm 3.6 Starting from U 0(resp., ˇU0 = 0), we define a decreasing sequence



(resp., an increasing sequence)

ˇ

U n = T Uˇn −1, n =1, 2, (3.15)

Note that unlike sequences (3.5), (3.6), the components ofUn =(un1, , un J) and ˇU n =

( ˇu n1, , ˇu n J ) solve coercive QVIs

b i



u n i,v −  u n i

f i+λ un i −1,v −  u n i

∀ v ∈ H1(Ω),



u n i ≤M Uni

, un i ≥0,v ≤M Uni

;

b i

ˇu n

i,v − ˇu n i



f i+λ ˇu n

i,v − ˇu n i



∀ v ∈ H1(Ω),

ˇu n

i ≤M ˇ U ni

, ˇu n

i ≥0,v ≤M ˇ U ni

.

(3.16)

Theorem 3.7 Let ρ = λ/(λ + β) Then, under conditions of Theorem 3.5, the sequences ( Un)

and ( ˇ U n ) remain in the sector 0,U 0 and converge geometrically to the unique solution U

of (1.1), that is,

U n − U

∞ ≤ ρ n U0− U

Uˇn − U

∞ ≤ ρ n U0− U

Proof Let us prove (3.17) The proof of (3.18) is similar

Forn =1, we have

U1− U

∞ = T U0− U

∞ = T U0− T U

∞ ≤ ρ n U0− U

∞ (3.19) Assume

U n −1− U

∞ ≤ ρ n −1 U0− U

Then,

U n − U

∞ = T U n −1− T U

∞ ≤ ρ U n −1− U

Thus

U n − U

∞ ≤ ρρ n −1 U0− U

∞ ≤ ρ n U0− U



4 The discrete problem

LetΩ be decomposed into triangles and let τ h denote the set of all those elements;h > 0

is the mesh size We assume that the familyτ his regular and quasi-uniform

LetVhdenote the standard piecewise linear finite element space, and letBi, 1≤ i ≤ J,

be the matrices with generic coefficients bi(ϕ l,ϕ s), whereϕ s,s =1, 2, , and m(h) are the

nodal basis functions Let alsor hbe the usual interpolation operator

Definition 4.1 A real n × n matrix B =[b i j] withb i j ≤0 for alli = j is an M-matrix if B

is nonsingular andB −1≥0

The discrete maximum principle assumption (d.m.p.) We assume that the matricesBiare

M-matrices (cf [6])

4.1 Discrete elliptic quasivariational inequalities The discrete counterpart of QVI

(2.3) reads as follows: findu h ∈ K h(u h) such that

b

u h,v − u h

f ,v − u h

∀ v ∈ K h

u h

whereKh(u h)= { v ∈ V hsuch thatv ≤ r h MU h }

Next we will state properties for the solution of (4.1) which are the direct discrete counterparts of those given in Propositions2.1 and 2.2 We will omit their respective proofs as these are very similar to those of the continuous case

4.1.1 A discrete monotonicity property Let f , f be in L ∞(Ω) and u h = σ h(f ,MU h),uh =

σ h(f ,M uh) the corresponding solutions to (4.1) Then, under the d.m.p., we have the following discrete comparison result

Proposition 4.2 If f ≥  f , then σ h(f ,MU h)≥ σ h(f ,M uh ).

4.1.2 A discrete Lipschitz dependence property.

Proposition 4.3 Let Proposition 4.2 hold Then,

u h −  u h

L ∞( Ω)≤ 1

λ + β  f −  f  L ∞( Ω). (4.2)

4.2 The discrete system of QVIs We define the discrete system of QVIs as follows: find

U h =(u1

h, ,u J h)∈(Vh)J such that

a i

u i

h,v − u i h



f i,v − u i

h



∀ v ∈ V h,

u i

h ≤ r h

MU hi , u i

h ≥0, v ≤ r h

MU hi

Similarly to the continuous problem, the above problem can be transformed into the following: findU h =(u1

h, ,u J h)∈(Vh)J solution to the equivalent system

b i

u i h,v − u i h

f i+λu i h,v − u i h

∀ v ∈ V h,

u i h ≤ r h



MU h

i , u i h ≥0, v ≤ r h



MU h

i

The existence of a unique solution to system (4.3) can be shown very similarly to that

of the continuous case provided the discrete maximum principle (d.m.p.) is satisfied The

key idea consists of associating with the above system the following fixed point mapping:

T h:H +−→Vh

J ,

W −→ T h W = ζ h =ζ h1, ,ζ h J

whereζ h i = σ h(f i+λw i, (MW) i) is the solution of the following discrete VI:

b i

ζ i

h,v − ζ i h



f i+λw i,v − ζ i

h



∀ v ∈ V h,

ζ h i ≤ r h(MW) i, ζ h i ≥0,v ≤ r h(MW) i (4.6)

Let ¯U0

h =( ¯u1,0h , , ¯u J,0 h ) be the discrete analogue of ¯U0defined in (3.4):

a i

¯

u i,0 h ,v

=f i,v

Then,T hpossesses analogous properties to those enjoyed by mappingT (seeProposition 3.1)

Proposition 4.4 T h is increasing, concave onH +and satisfies T h W ≤ U¯0for all W ≤ U¯0

h Algorithm 4.5 Starting from ¯ U0

h solution of (4.7), (resp.,U0

h =(0, ,0)), we define a discrete decreasing sequence

¯

U h n+1 = T h U¯h n, n =0, 1, , (4.8)

(resp., a discrete increasing sequence)

U n+1 h = T h U n h, n =0, 1, (4.9)

Theorem 4.6 Let Proposition 4.4 hold, then, the sequences ( ¯ U h n ) and ( U n h ) remain in the sector 0, ¯U h0 Moreover, they converge monotonically to the unique solution U h of system of QVIs (4.3).

4.3 Characterization of the solution of system ( 4.3 ) as a fixed point of a contraction.

Similarly to the continuous problem, the solution of system (4.3) can be characterized as the unique fixed point of a contraction

Indeed, consider the following mapping:

Th:H +−→Vh

J ,

W −→ T h W = Z h =z1h, ,z h J

whereZ h =(z1h, ,z h J) is solution to the discrete coercive system of QVIs:

b i

z i

h,v − z i h



f + λw i,v − z i

h



∀ v ∈ V h,

z h i ≤ r h(MZ) i, z h i ≥0,v ≤ r h(MZ) i (4.11)

Then, making use ofProposition 4.3, we get the following

Theorem 4.7 The mappingTh is a contraction onH + That is,

Th W − T h W

Therefore, there exists a unique fixed point which coincides with the solution U h of the system

of QVI (4.3).

Proof It is very similar to that of the continuous case. 

4.4 Another iterative scheme for system ( 4.3 ) In view of the above result, it is natural

to associate with the solution of system of QVIs (1.1) the following algorithm

First, letU 0

h =(u1,0h , , uJ,0 h ) such thatui,0 h solves the equation

b i



u i,0 h ,v

Algorithm 4.8 Starting from U 0

h (resp., ˇU0 h = 0), we define a decreasing sequence



U h n = T h Un −1

(resp., an increasing sequence)

ˇ

U n

h = T h Uˇn −1, n =1, 2, (4.15) Note that unlike sequences (4.8), (4.9), the components of bothUn

h =(u1,h n, , uh J,n) and ˇU n

h =( ˇu1,h n, , ˇu J,n h ) solve discrete coercive QVIs, which are

b i



u i,n h ,v −  u i,n h 

f i+λ ui,n h −1,v −  u i,n h 

∀ v ∈ V h,



u i,n h ≤ r h

M Un h

i , ui,n h ≥0,v ≤ r h

M Un h

i

;

b i

ˇu i,n h ,v − ˇu i,n h 

f i+λ ˇu i,n h ,v − ˇu i,n h 

∀ v ∈ V h,

ˇu i,n h ≤ r h



M ˇ U h ni , ˇu i,n h ≥0,v ≤ r h



M ˇ U h ni

.

(4.16)

Theorem 4.9 Let ρ = λ/(λ + β) Then, under conditions of Theorem 4.7, the sequences ( Un

h)

and ( ˇ U h n ) remain in the sector 0,U 0

h and converge geometrically to the unique solution U h

of (4.3), that is,

U n

h − U h

∞ ≤ ρ n U0

h − U h

∞,

Uˇn

h − U h

∞ ≤ ρ n U0

h − U h

Proof The proof is similar to that of the continuous case. 

5.L ∞-error analysis

We now turn to theL ∞-error analysis For that purpose, we will give two different ap-proaches

LetVhdenote... discrete maximum principle (d.m.p.) is satisfied The

key idea consists of associating with the above... onH +and satisfies T h W ≤ U¯0for all W ≤ U¯0

h Algorithm 4.5 Starting