Journal of Inequalities in Pure and
Applied Mathematics
POINTWISE ERROR ESTIMATE FOR A NONCOERCIVE SYSTEM
OF QUASI-VARIATIONAL INEQUALITIES RELATED TO THE
MANAGEMENT OF ENERGY PRODUCTION
volume 3, issue 5, article 79,
2002.
MESSAOUD BOULBRACHENE
College of Science
Department of Mathematics and Statistics
Sultan Qaboos University
P.O. Box 36 Muscat 123
Sultanate of Oman.
EMail: boulbrac@squ.edu.om
Received 27 May, 2002;
accepted 24 July, 2002.
Communicated by: R. Verma
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
058-02
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Abstract
This paper is devoted to the approximation by a piecewise linear finite element
method of a noncoercive system of elliptic quasi-variational inequalities arising
in the management of energy production. A quasi-optimal L∞ error estimate is
established, using the concept of subsolution.
Pointwise Error Estimate for a
Noncoercive System of
Quasi-Variational Inequalities
Related to The Management of
Energy Production
2000 Mathematics Subject Classification: 49J40, 65N30, 65N15.
Key words: Quasi-Variational Inequalities, Subsolutions, Finite Elements, L∞ -Error
Estimate.
Messaoud Boulbrachene
Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Continuous Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Notations, Assumptions . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Existence, Uniqueness and Regularity . . . . . . . . . . . .
2.3
A Continuous Iterative Scheme . . . . . . . . . . . . . . . . . .
2.4
A Monotonicity Property . . . . . . . . . . . . . . . . . . . . . . .
2.5
A Continuous L∞ Stability Property . . . . . . . . . . . . . .
2.6
Characterization of the solution of system (1.1) as
the least upper bound of the set of sub-solutions . . . .
The Discrete Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
A Discrete Iterative Scheme . . . . . . . . . . . . . . . . . . . . .
3.2
A Discrete Monotonicity Property . . . . . . . . . . . . . . . .
4
6
6
7
8
9
10
11
13
14
15
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J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
A Discrete L∞ Stability Property . . . . . . . . . . . . . . . .
Characterization of the solution of system (3.2) as
the least upper bound of the set of discrete subsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
The Finite Element Error Analysis . . . . . . . . . . . . . . . . . . . . . .
4.1
Definition of Two Auxiliary Coercive System of QVIs
4.2
L∞ - Error Estimate For System (1.1) . . . . . . . . . . . . .
References
3.3
3.4
15
15
17
17
18
Pointwise Error Estimate for a
Noncoercive System of
Quasi-Variational Inequalities
Related to The Management of
Energy Production
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J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
1.
Introduction
A lot of results on error estimates in the L∞ norm for the classical obstacle
problem in particular and variational inequalities (VIs) in general have been
achieved in the last three decades. (cf., e.g [6], [7], [8], [9]). However, very few
works are known in this area when it comes to quasi-variational inequalities
(QVIs) (cf., [10], [11]), and especially the case of systems which is the subject
of this paper.(cf. e.g [3]
Indeed, we are concerned with the numerical approximation in the L∞ norm
for the noncoercive problem associated with the following system of QVIs: Find
U = (u1 , . . . , uJ ) ∈ (H01 (Ω))J satisfying
 i i
 a (u , v − ui ) = (f i , v − ui ) ∀v ∈ H01 (Ω)
(1.1)
 i
u ≤ M ui ; ui ≥ 0;
v ≤ M ui
Pointwise Error Estimate for a
Noncoercive System of
Quasi-Variational Inequalities
Related to The Management of
Energy Production
Messaoud Boulbrachene
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in which, Ω is a bounded smooth domain of RN , N ≥ 1, ai (·, ·) are J- elliptic
bilinear forms continuous on H 1 (Ω)×H 1 (Ω), assumed to be noncoercive, (·, ·)
is the inner product in L2 (Ω) and f i are J- regular functions.
This system arises in the management of energy production problems, where
J-units are involved (see e.g. [1], [2] and the references therein). In the case
studied here, M ui represents a “cost function” and the prototype encountered
is
(1.2)
M ui = k + inf uµ .
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µ6=i
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In (1.2) k represents the switching cost. It is positive when the unit is “turned
J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
on” and equal to zero when the unit is “turned off”. Note also that operator M
provides the coupling between the unknowns u1 , . . . , uJ .
The L∞ -error estimate for the proposed system is a challenge not only for the
practical motivation behind the problem, but also due to the inherent difficulty
of convergence in this norm. Moreover, the interest in using such a norm for the
approximation of VI and QVIs is that they are types of free boundary problems
(cf. [4], [5]).
The coercive version of (1.1) is mathematically well understood. The numerical analysis study has also been considered in [3] and a quasi-optimal L∞ -error
estimate established.
In this paper we propose to demonstrate that the standard finite element approximation applied to the noncoercive problem corresponding to system (1.1)
is quasi-optimally accurate in L∞ (Ω). For that purpose we shall develop an
approach mainly based on both the L∞ - stability of the solution with respect to
the right hand side and its characterization as the least upper bound of the set of
subsolutions.
It is worth mentioning that the method presented in this paper is entirely
different from the one developed for the coercive problem.
The paper is organized as follows. In Section 2 we state the continuous problem and study some qualitative proerties. In Section 3 we consider the discrete
problem and achieve an analogous result to that of the continuous problem. In
Section 4, we prove the main result.
Pointwise Error Estimate for a
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J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
2.
2.1.
The Continuous Problem
Notations, Assumptions
We are given functions aijk (x) in C 1,α (Ω̄), aik (x), ai0 (x) in C 0,α (Ω) such that:
X
aijk (x)ξj ξk = α |ζ|2 ; ζ ∈ RN ; α > 0,
(2.1)
1≤j,k≤N
ai0 (x) = β > 0 (x ∈ Ω).
(2.2)
We define the second order differential operators
Pointwise Error Estimate for a
Noncoercive System of
Quasi-Variational Inequalities
Related to The Management of
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N
X
i
Aϕ=
(2.3)
1≤j,k≤N
∂ i ∂ϕ X i ∂ϕ
ajk
+
ak
+ ai0 ϕ
∂xj
∂xk k=1 ∂xk
and the associated variational forms: for any u, v ∈ H01 (Ω)
(2.4) ai (u, v)
Z
=
Ω
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!
N
X
∂u
∂v
∂u
+
aik (x)
v + ai0 (x)uv)dx .
aijk (x)
∂x
∂x
∂x
j
k
k
1≤j,k≤N
k=1
X
We are also given right hand side f 1 , . . . , f
(2.5)
Messaoud Boulbrachene
i
f ∈C
0,α
i
J
such that
0
(Ω); f ≥ f > 0.
Throughout the paper U = ∂(F, M U ) will denote the solution of system (1.1)
where F = (f 1 , . . ., f J ) and M U = (M u1 , . . ., M uJ ).
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J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
2.2.
Existence, Uniqueness and Regularity
To solve the noncoercive problem, we transform (1.1) into the following auxiliary system: find U = (u1 , . . ., uJ ) ∈ (H01 (Ω))J such that:
 i i
 b (u , v − ui ) = (f i + λui , v − ui ) ∀v ∈ H01 (Ω)
,
(2.6)
 i
u ≤ M ui ; ui ≥ 0; v ≤ M ui ,
where
(2.7)
bi (u, v) = ai (u, v) + λ(v, v)
and λ > 0 is large enough such that:
Pointwise Error Estimate for a
Noncoercive System of
Quasi-Variational Inequalities
Related to The Management of
Energy Production
bi (v, v) ≥ γ kvk2H 1 (Ω) γ > 0; ∀v ∈ H 1 (Ω).
Messaoud Boulbrachene
Let us recall just the main steps leading to the existence of a unique solution
to system (1.1). For more details, we refer the reader to ([1]).
J
1
J
i
∞
Let H+ = (L∞
+ (Ω)) = {V = (v , . . ., v ) such that v ∈ L+ (Ω)}, equipped
with the norm:
(2.9)
kV k∞ = max v i L∞ (Ω) ,
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(2.8)
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1≤i≤J
∞
where L∞
+ (Ω) is the positive cone of L (Ω). We introduce the following mapping
(2.10)
T : H+ −→ H+
W −→ T W = (ζ 1 , . . ., ζ J )
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J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
where ∀ i = 1, . . ., J, ζ i = σ(f i + λwi ; M wi ) is solution to the following VI:
(2.11)
 i i
 b (ζ , v − ζ i ) = (f i + λwi , v − ζ i ) ∀v ∈ H01 (Ω)

ζ i ≤ M wi ,
.
v ≤ M wi
Problem (2.11), being a coercive variational inequality, thanks to [12], it has
a unique solution.
Let us also define the vector Û 0 = (û1,0 , . . ., ûJ,0 ), where ∀i = 1, . . ., J, ûi,0
is solution to the equation
(2.12)
ai (ûi,0 , v) = (f i , v) ∀v ∈ H01 (Ω).
Since f i ≥ 0, there exists a unique positive solution to problem (2.12). Moreover, ûi,0 ∈ W 2,,p (Ω), p < ∞ (Cf. e.g., [1]).
Proposition 2.1. (Cf. [1]) Under the preceding notations and assumptions, the
mapping T is increasing, concave and satisfies: T W ≤ Û 0 , ∀ W ∈ H+ such
that W ≤ Û 0 .
The mapping T clearly generates the following iterative scheme.
Pointwise Error Estimate for a
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2.3.
A Continuous Iterative Scheme
Starting from Û 0 defined in (2.12) (resp. Ǔ 0 = (0, . . ., 0)) , we define the
sequences below
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(2.13)
Û n+1 = T Û n ; n = 0, 1, . . .
J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
(resp.)
Ǔ n+1 = T Ǔ n ; n = 0, 1, . . ..
(2.14)
Theorem 2.2. (cf. [1]) Let C = {W ∈ H+ such that W ≤ Û 0 }. Then,
under conditions of Proposition 2.1 the sequences (Û n ) and (Ǔ n ) remain in C
. Moreover, they converge monotonically to the unique solution of system (1.1).
Theorem 2.3. (cf. [1]) Under the preceding assumptions, the solution (u1 , . . ., uJ )
of system (1.1) belongs to (W 2,p (Ω))J ; 2 ≤ p < ∞.
In what follows, we shall give a monotonicity and an L∞ stability property
for the solution of system (1.1). These properties together with the notion of
subsolution will play a crucial role in proving the main result of this paper.
2.4.
A Monotonicity Property
Let F = (f 1 , . . ., f J ) ; F̃ = (f˜1 , . . ., f˜J ) be two families of right hands side
and U = ∂(F, M U ) = (u1 , . . ., uJ ); Ũ = ∂(F̃ , M Ũ ) = (ũ1 , . . ., ũJ ) the
corresponding solutions to system (1.1), respectively.
Theorem 2.4. If F ≥ F̃ then ∂(F, M U ) ≥ ∂(F̃ , M Ũ ).
Pointwise Error Estimate for a
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Proof. We proceed by induction. For that let us associate with U and Ũ the
following iterations
n
Û = (û
1,n
, . . ., û
J,n
en
ě1,n , . . ., u
ěJ,n )
) and Û = (u
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respectively. Then, from (2.10), (2.11), (2.13) we clearly have
ei,n+1 = σ(f i + λû
ei,n , M û
ei,n ),
ûi,n+1 = σ(f i + λûi,n , M ûi,n ) and û
f
e1,0 . . ., û
eJ,0 ) are solutions to equation
where Û 0 = (û1,0 , . . ., ûJ,0 ) and Û 0 = (û
(2.12) with right hand sides F and F̃ , respectively.
ei,0 . So, f i +λ ûi,0 ≥ f˜i +λû
ei,0 and M ûi,0 ≥
Clearly, f i ≥ f˜i implies ûi,0 ≥ û
ei,0 . Therefore, using standard comparison results in coercive variational
M û
ei,1 .
inequalities, we get ûi,1 ≥ û
ei,n−1 . Then, as f i ≥ f˜i , applying the same
Now assume that ûi,n−1 ≥ û
ei,n . Finally, by Theorem 2.2,
comparison argument as before, we get ûi,n ≥ û
making n tend to ∞, we get U ≥ Ũ . This completes the proof.
2.5.
A Continuous L∞ Stability Property
Using the above notations we have the following result.
Theorem 2.5. Under conditions of Theorem 2.4, we have
1
(2.15)
∂(F, M U ) − ∂(F̃ , M Ũ ) ≤ F − F̃ .
β
∞
∞
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Proof. Let us denote by ui = σ(f i , M ui ); ũi = σ(f˜i , M ũi ) the ith components
1
of U and Ũ , respectively. Then, setting Φi = f i − f˜i , using (2.2) it
β
L∞ (Ω)
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J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
is easy to see that ∀i = 1, 2, . . ., J
f i ≤ f˜i + f i − f˜i L∞ (Ω)
ai (x) i ˜i ≤ f˜i + 0
f − f ∞ ≤ f˜i + (ai0 (x)Φi ).
β
L (Ω)
Hence, making use of Theorem 2.4, it follows that
σ(f i , M ui ) ≤ σ(f˜i + (ai0 (x)Φi , M (ũi + Φi ))
≤ σ(f˜i , M ũi ) + Φi .
Thus,
ui − ũi ≤ Φi .
Interchanging the roles of f i and f˜i , we similarly get
Pointwise Error Estimate for a
Noncoercive System of
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Related to The Management of
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Messaoud Boulbrachene
ũi − ui ≤ Φi .
This completes the proof.
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2.6.
Characterization of the solution of system (1.1) as the
least upper bound of the set of sub-solutions
Definition 2.1. ([1]) W = (w1 , .., wJ ) ∈ (H01 (Ω))J is said to be a subsolution
for the system of QVIs (1.1) if
 i i
 b (w , v) ≤ (f + λwi , v) ∀v ∈ H01 (Ω) v ≥ 0,
(2.16)
 i
w ≤ M wi ;
i = 1, . . ., J.
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Let X be the set of such subsolutions.
Theorem 2.6. The solution of system of QVIs (1.1) is the maximum element of
the set X.
Proof. It is a straightforward adaptation of ([1, p.358])
Pointwise Error Estimate for a
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3.
The Discrete Problem
Let Ω be decomposed into triangles and let τh denote the set of all those elements; h > 0 is the mesh size. We assume the family τh is regular and quasiuniform.
Let Vh denote the standard piecewise linear finite element space and Bi , 1 ≤
i ≤ J be the matrices with generic entries:
(3.1)
(Bi )ls = bi (ϕl , ϕs ); 1 ≤ l, s ≤ m(h),
where ϕs , s = 1, 2, . . .m(h) are the nodal basis functions and rh is the usual
interpolation operator.
The discrete maximum principle assumption (dmp): We assume that the
i
B are M -matrices (cf. [13]).
In this section, we shall see that the discrete problem below inherits all the
qualitative properties of the continuous problem, provided the dmp is satisfied.
Their respective proofs shall be omitted, as they are very similar to their continuous analogues.
Let Vh = (Vh )J . The noncoercive system of QVIs consists of seeking
Uh = (u1h , . . ., uJh ) ∈ Vh such that
 i i
 a (uh , v − uih ) = (f i , v − uih ) ∀v ∈ Vh
(3.2)
 i
uh ≤ rh M uih ,
v ≤ rh M uih ,
or equivalently
 i i
 b (uh , v − uih ) = (f i + λwi , v − uih ) ∀v ∈ Vh
(3.3)
 i
uh ≤ rh M uih ,
v ≤ rh M uih .
Pointwise Error Estimate for a
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J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
Let Ûh0 be the piecewise linear approximation of Û 0 defined in (2.12):
(3.4)
i
ai (ûi,0
h , v) = (f , v) ∀v ∈ Vh ; 1 ≤ i ≤ J
and consider the following discrete mapping
(3.5)
Th : H+ −→ Vh
W −→ T W = (ζh1 , . . ., ζhJ )
where, ∀ i = 1, . . ., J, ζhi is the solution of the following discrete VI:
 i i
 b (ζh , v − ζhi ) = (f i + λwi , v − ζhi ) ∀v ∈ Vh ,
(3.6)
 i
ζh ≤ rh M wi ,
v ≤ rh M w i .
Pointwise Error Estimate for a
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Proposition 3.1. Let the dmp hold. Then Th is increasing, concave and satisfies
Th W ≤ Ûh0 ∀W ∈ H+ , W ≤ Ûh0 .
3.1.
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A Discrete Iterative Scheme
We associate with the mapping Th the following discrete iterative scheme:
starting from Ûh0 defined in (3.4), and Ǔh0 = 0, we define:
(3.7)
Ûhn+1 = Th Ûhn n = 0, 1, . . .
(3.8)
Ǔhn+1 = Th Ǔhn
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and
n = 0, 1, . . .
respectively.
Similar to the continuous case, the following theorem establishes the monotone convergence of the above discrete sequences to the solution of system (3.2).
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J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
Theorem 3.2. Let Ch = {W ∈ H+ such that W ≤ Ûh0 }. Then, under the
dmp, the sequences (Ûhn ) and (Ǔhn ) remain in Ch . Moreover, they converge
monotonically to the unique solution of system (3.2).
3.2.
A Discrete Monotonicity Property
Let F = (f 1 , . . ., f J ) and F̃ = (f˜1 , . . ., f˜J ) be two families of right hand
sides, and Uh = ∂h (F, M Uh ), Ũh = ∂h (F̃ , M Ũh ) the corresponding solutions
to system (3.2), respectively.
Theorem 3.3. Under the dmp, if F ≥ F̃ then ∂h (F, M Uh ) ≥ ∂h (F̃ , M Ũh ).
3.3.
A Discrete L∞ Stability Property
Theorem 3.4. Under conditions of Theorem 3.3, we have
1
(3.9)
∂h (F, M Uh ) − ∂h (F̃ , M Ũh ) ≤ F − F̃ .
β
∞
∞
3.4.
Characterization of the solution of system (3.2) as the
least upper bound of the set of discrete sub-solutions
Definition 3.1. W = (wh1 , .., whJ ) ∈ Vh is said to be a subsolution for the system
of QVIs (3.2) if
 i i
 b (wh , ϕs ) ≤ (f i + λwhi , ϕs ) ∀ϕs ; s = 1, . . ., m(h);
(3.10)
 i
wh ≤ rh M whi .
Pointwise Error Estimate for a
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J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
Let Xh be the set of discrete subsolutions.
Theorem 3.5. Under the dmp, the solution of system of QVIs (3.2) is the maximum element of the set Xh .
Pointwise Error Estimate for a
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4.
The Finite Element Error Analysis
This section is dedicated to prove that the proposed method is quasi-optimally
accurate in L∞ (Ω), according to the approximation theory. To achieve that, we
first introduce two auxiliary coercive systems of QVIs and give some intermediate error estimates.
4.1.
Definition of Two Auxiliary Coercive System of QVIs
1. A Continuous system of QVIs: Find Ū (h) = (ū1(h) , . . ., ūJ(h) ) ∈ (H01 (Ω))J
solution to:
 i i(h)
 b (ū , v − ūi(h) ) ≥ (f i + λuih , v − ūi(h) ) ∀v ∈ H01 (Ω);
(4.1)
 i(h)
ū
≤ M ūi(h) ;
v ≤ M ūi(h) ,
where Uh = (u1h , . . ., uJh ) is the solution of the discrete system of QVIs (3.2).
Lemma 4.1. (cf. [3])
(4.2)
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(h)
Ū − Uh ≤ Ch2 |Logh|3 .
∞
2. A Discrete System of Coercive QVIs: Find Ūh = ū1h , . . ., ūJh
solution to:
 i i
 b (ūh , v − ūih ) ≥ (f i + λui , v − ūih ) ∀v ∈ Vh ;
(4.3)

u ≤ rh M ūih ;
v ≤ rh M ūih ,
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∈ Vh
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J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
where U = (u1 , . . ., uJ ) is the solution of the continuous system of QVIs
(1.1).
Lemma 4.2. (cf. [3])
(4.4)
Ūh − U ≤ Ch2 |Logh|3 .
∞
4.2.
L∞ - Error Estimate For System (1.1)
Theorem 4.3. Let U and Uh be the solutions of the noncoercive problems
(1.1) and (3.2), respectively. Then, then under conditions of Theorem 2.3, and
Lemmas 4.1, 4.2, we have the error estimate
Pointwise Error Estimate for a
Noncoercive System of
Quasi-Variational Inequalities
Related to The Management of
Energy Production
kU − Uh k∞ ≤ Ch2 |Logh|3 .
Messaoud Boulbrachene
(4.5)
Proof. The proof will be carried out in three steps.
Step 1. It consists of constructing a vector of continuous functions β (h) =
(β 1(h) , . . ., β J(h) ) such that:
(4.6)
β (h) ≤ U and β (h) − Uh ∞ ≤ Ch2 |Logh|3
Indeed, Ū (h) being solution to system (4.1) it is easy to see that Ū (h) is also a
subsolution, i.e., ∀i = 1, . . ., J
 i i(h)
 b (ū , v) ≤ (f i + λuih , v) ∀v ∈ H01 (Ω), v ≥ 0,

ūi(h) ≤ M ūi(h) ;
v ≤ M ūi(h) .
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J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
This implies

i
i
i(h)
i(h) i
i(h)

b
ū
,
v
≤
f
+
λ
u
−
ū
+
λū
,
v
∀v ∈ H01 (Ω), v ≥ 0,

h
L∞ (Ω)


ūi (h) ≤ M ūi(h) ;
v ≤ M ūi(h) ,
and, from Theorem 2.6, it follows that
Ū (h) ≤ Ũ = ∂(F̃ , M Ũ )
with F̃ = F + λ Ū (h) − Uh ∞ . Therefore, using both the stability Theorem
2.5 and estimate (4.2) we get
3
(4.8)
U − Ũ ≤ λ Ū (h) − Uh ∞ ≤ Ch2 |Logh|
(4.7)
∞
Pointwise Error Estimate for a
Noncoercive System of
Quasi-Variational Inequalities
Related to The Management of
Energy Production
Messaoud Boulbrachene
which combined with (4.7) yields:
Ū (h) ≤ U + Ch2 |Logh|3 .
Finally, taking β (h) = Ū (h) − Ch2 |Logh|3 , (4.6) follows.
Step 2. Similarly to Step 1., we construct a vector of discrete functions αh =
(αh1 , . . ., αhJ ) satisfying
αh ≤ Uh and kαh − U k∞ ≤ Ch2 |Logh|3 .
(4.9)
Indeed, Ūh being solution to system (4.3), it is also a subsolution, i.e.
 i i
 b (ūh , ϕs ) ≤ (f i + λui , ϕs ) ∀ϕs ; s = 1, . . ., m(h);

u ≤ M ūih ;
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v ≤ M ūih ,
J. Ineq. Pure and Appl. Math. 3(5) Art. 79, 2002
which implies
 i i
 b (ūh , ϕs ) ≤ (f i + λ kui − ūih kL∞ (Ω) + λūih , ϕs ) ∀ϕs ; s = 1, . . ., m(h)

u ≤ M ūih ;
v ≤ M ūih .
Hence, letting F̃ = F + λ Ūh − U ∞ and applying Theorem 3.5, we obtain
that
Ūh ≤ Ũh = ∂h (F̃ , M Ũh ).
(4.10)
Therefore, using both Theorem 3.4 and estimate (4.4), we get
3
(4.11)
Uh − Ũh ≤ λ Ūh − U ∞ ≤ Ch2 |Logh|
∞
which combined with (4.10), yields
Pointwise Error Estimate for a
Noncoercive System of
Quasi-Variational Inequalities
Related to The Management of
Energy Production
Messaoud Boulbrachene
Ūh ≤ Uh + Ch2 |Logh|3 .
Finally, taking αh = Ūh − Ch2 |Logh|3 , we immediately get (4.9).
Step 3. Now collecting the results of Steps 1 and 2., we derive the desired error
estimate (4.5) as follows:
Uh ≤ β
(h)
2
3
+ Ch |Logh|
≤ U + Ch2 |Logh|3
≤ αh + Ch2 |Logh|3 ≤ Uh + Ch2 |Logh|3 .
Thus
kU − Uh k∞ ≤ Ch2 |Logh|3 .
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References
[1] A. BENSOUSSAN AND J.L. LIONS, Impulse Control and Quasivariational Inequalities, Gauthier Villars, Paris (1982).
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[3] M. BOULBRACHENE, M. HAIOUR AND S. SAADI, L∞ - Error estimates for a system of quasi-variational inequalities, to appear in Int. J.
Math. Math. Sci.
[4] F. BREZZI AND L.A. CAFFARELLI, Convergence of the discrete free
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[5] R.H. NOCHETTO, A note on the approximation of free boundaries by
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[6] C. BAIOCCHI, Estimation d’erreur dans L∞ pour les inequations a obstacle, in: I.Galligani, E. Magenes (eds.) Mathematical Aspects of Finite
Element Methods in Mathematics, 606 (1977), 27–34.
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261–274.
Pointwise Error Estimate for a
Noncoercive System of
Quasi-Variational Inequalities
Related to The Management of
Energy Production
Messaoud Boulbrachene
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[8] P. CORTEY-DUMONT, On the finite element approximation in the L∞
norm of variational inequalities with nonlinear operators, Numer. Num.,
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[9] R.H. NOCHETTO, Sharp L∞ −Error estimates for semilinear elliptic
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[10] P. CORTEY-DUMONT, Approximation numerique d’une inequation
quasi-variationnelles liees a des problemes de gestion de stock, RAIRO
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[11] M. BOULBRACHENE, The noncoercive quasi-variational inequalities related to impulse control problems, Comput. Math. Appl., (1998), 101–
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[12] A. BENSOUSSAN AND J.L. LIONS, Applications des Inequations Variationnelles en Controle Stochastique, Dunod , Paris, (1978).
[13] P.G. CIARLET AND P.A. RAVIART, Maximum principle and uniform
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Pointwise Error Estimate for a
Noncoercive System of
Quasi-Variational Inequalities
Related to The Management of
Energy Production
Messaoud Boulbrachene
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