L∞-ERROR ANALYSIS FOR A SYSTEM OF 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.
(cid:2)
(cid:3)
(cid:2) ui,v − ui
1. Introduction We are interested in the finite element approximation in the L∞ norm of the following 0 (Ω))J satisfying system of quasivariational inequalities (QVIs): find U = (u1,...,uJ ) ∈ (H 1 (cid:3) ai f i,v − ui (1.1)
(cid:2) ui ≤ (MU)i,
0 (Ω), ∀v ∈ H 1 ui ≥ 0, v ≤ (MU)i.
Here, Ω is a bounded smooth domain of RN , N ≥ 1, with boundary ∂Ω, (·, ·) is the inner product in L2(Ω), for i = 1,...,J, ai(u,v) is a continuous bilinear form on H 1(Ω) × H 1(Ω), and f i is a regular function.
Problem (1.1) arises in the management of energy production problems where J power generation machines are involved (see [2] and the references therein). In the case studied here, (MU)i represents a “cost function” and the prototype encountered is
uμ, i = 1,...,J. (1.2) (MU)i = k + inf μ(cid:7)=i
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 unknowns u1,...,uJ .
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 15704, Pages 1–13 DOI 10.1155/JIA/2006/15704
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:
2 System of quasivariational inequalities
0 (Ω))J such that (cid:3)
(cid:2)
(cid:3)
find U = (u1,...,uJ ) ∈ (H 1
∀v ∈ H 1
(cid:2) ui,v − ui
0 (Ω),
bi f i + λui,v − ui (1.3)
(cid:2) ui ≤ (MU)i,
ui ≥ 0, v ≤ (MU)i,
where, for λ > 0 large enough,
bi(u,v) = ai(u,v) + λ(v,v) (1.4)
is a strongly coercive bilinear form, that is,
H 1(Ω),
bi(v,v) ≥ γ(cid:8)v(cid:8)2 γ > 0, ∀v ∈ H 1(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 the L∞ 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.
jk(x), ai
k(x), ai
0(x), 1 ≤ i ≤ J,
The paper is organized as follows. In Section 2, we lay down some necessary prelim- inaries. In Section 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. In Section 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 the L∞ norm.
(cid:4)
2. Preliminaries 2.1. Assumptions and notations. We are given functions ai sufficiently smooth functions such that
1≤ j,k≤N
ζ ∈ RN , α > 0, ai jk(x)ξ jξk (cid:2) α|ζ|2, (2.1) (x ∈ Ω). ai 0(x) (cid:2) β > 0,
M. Boulbrachene and S. Saadi 3
(cid:7)
(cid:5)
0 (Ω), N(cid:4)
(cid:4)
We define the bilinear forms: for all u,v ∈ H 1 (cid:6)
0(x)uv
Ω
1≤ j,k≤N
k=1
dx. ai(u,v) = + v + ai (2.2) ai jk(x) ai k(x) ∂u ∂x j ∂v ∂xk ∂u ∂xk
We are also given right-hand sides f i such that f i ∈ L∞(Ω) and f i ≥ f0 > 0 for i = 1,...,J.
2.2. Elliptic quasivariational inequalities. Let f ∈ L∞(Ω) such that f > f0 > 0, M a non- decreasing operator from L∞(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): find u ∈ K(u) such that
b(u,v − u) (cid:2) ( f ,v − u) ∀v ∈ K(u), (2.3)
0 (Ω) such that v ≤ Mu a.e.}.
(cid:3)
= ( f ,v) ∀v ∈ H 1
(cid:10)
(cid:9)
where K(u) = {v ∈ H 1 Thanks to [2], the QVI (2.3) has a unique solution. Moreover, this solution enjoys
∀v ∈ H 1
=
(cid:8)u0,v
(cid:8)f ,v
0 (Ω), 0 (Ω).
some important qualitative properties. 2.2.1. A Monotonicity property. Let f , (cid:8)f in L∞(Ω) and u = σ( f ,MU), (cid:8)u = σ( (cid:8)f ,M (cid:8)u) be the corresponding solutions of (2.3). Then we have the following comparison principle. Proposition 2.1. If f ≥ (cid:8)f then u ≥ (cid:8)u. Proof. Let u0 and (cid:8)u0 be the respective solutions to equations (cid:2) u0,v (cid:3) b (cid:2) (2.4) b
(cid:2)
(cid:2)
Now let us associate with u and (cid:8)u the respective decreasing sequences
(cid:8)un+1 = σ
(cid:3) .
(cid:3) ,
(cid:8)f ,M (cid:8)un
un+1 = σ f ,Mun (2.5)
Then the following assertion holds:
if f ≥ (cid:8)f then un ≥ (cid:8)un. (2.6)
Indeed, since f ≥ (cid:8)f and M is nondecreasing, we have u0 ≥ (cid:8)u0. So, MU 0 ≥ M (cid:8)u0, and thus applying standard comparison results in elliptic variational inequalities, we get
u1 ≥ (cid:8)u1. (2.7)
Now assume that un−1 ≥ (cid:8)un−1. Then, as f ≥ (cid:8)f , applying the same comparison argument as before, we get
un ≥ (cid:8)un. (2.8)
(cid:3)
Finally, passing to the limit (n → ∞) as in [2, pages 342–358], we get u ≥ (cid:8)u.
4 System of quasivariational inequalities
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,
(cid:8) f − (cid:8)f (cid:8)L∞(Ω).
(cid:8)u − (cid:8)u(cid:8)L∞(Ω) ≤ 1 λ + β
(2.9)
Proof. Let us set
(cid:8) f − (cid:8)f (cid:8)L∞(Ω).
Φ = 1 (2.10) λ + β
0(x) (cid:2) β > 0, we get
Then, since ai
≤ (cid:8)f +
f ≤ (cid:8)f + (cid:8) f − (cid:8)f (cid:8)L∞(Ω)
(cid:8) f − (cid:8)f (cid:8)L∞(Ω) (cid:3) Φ.
≤ (cid:8)f +
(2.11)
a0(x) + λ λ + β (cid:2) a0(x) + λ
So, due to Proposition 2.1, we obtain
u ≤ (cid:8)u + Φ. (2.12)
(cid:8)u ≤ u + Φ
Likewise, interchanging the roles of f and (cid:8)f , we similarly get
(2.13)
(cid:3)
which completes the proof.
Remark 2.3. The above monotonicity and Lipschitz continuity results stay true in the discrete case provided a discrete maximum principle is satisfied (see Section 3).
3. The continuous problem
(cid:11) (cid:11)
+ (Ω))J equipped with the norm (cid:11) (cid:11)vi
3.1. The continuous system of QVIs. The existence of a unique solution to system (1.1) + (Ω) denote the positive cone of can be proved as in [2, pages 342–358]. Indeed, let L∞ L∞(Ω) and consider H+ = (L∞
L∞(Ω)
(cid:8)V (cid:8)∞ = max 1≤i≤J
. (3.1)
Consider the mapping
(cid:2) ζ 1,...,ζ J
(cid:3) ,
T : H+ −→ H+, (3.2) W −→ TW = ζ =
M. Boulbrachene and S. Saadi 5
0 (Ω) solves the following variational inequality (VI): (cid:2)
(cid:3)
(cid:3)
where ζ i = σ( f i + λwi,(MW)i) ∈ H 1
(cid:2)
(cid:2) ζ i,v − ζ i
0 (Ω),
∀v ∈ H 1 v ≤ (MW)i.
bi (3.3) f i + λwi,v − ζ i ζ i ≥ 0, ζ i ≤ (MW)i,
(cid:3)
(cid:2)
(cid:3)
∀v ∈ H 1
Problem (3.3), being a coercive VI, thanks to [1], has one and only one solution. Consider now ¯U 0 = ( ¯u1,0,..., ¯uJ,0), where ¯ui,0 is solution to the following variational equation:
=
(cid:2) ¯ui,0,v
0 (Ω).
ai f i,v (3.4)
∞.
Thanks to [2], problem (3.4) has a unique solution. Moreover, ui,0 ∈ W 2,p(Ω); 2 ≤ p <
The mapping T 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 ¯U 0 defined in (3.4) (resp., U 0 = (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 n and U n are
jk(x) in C1,α( ¯Ω), ai(x), ai
0(x), and f i in C0,α( ¯Ω),
solutions of VIs. Theorem 3.3. Let Proposition 3.1 hold; then, the sequences ( ¯U n) and (U n) remain in the sector (cid:10)0, ¯U 0(cid:11). Moreover, they converge monotonically to the unique solution of system (1.1). (cid:3) Proof. See [2, pages 342–358].
3.1.1. Regularity of the solution of system (1.1). Theorem 3.4 [2, page 453]. Assume ai α > 0. Then (u1,...,uJ ) ∈ (W 2,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+, W −→ TW = Z,
(3.7)
6 System of quasivariational inequalities
(cid:3)
(cid:3)
∀v ∈ H 1
(cid:2) zi,v − zi
0 (Ω),
where Z = (z1,...,zJ ) is solution to the coercive system of QVIs below: (cid:2) bi (3.8)
(cid:2) zi ≤ (MZ)i,
f i + λwi,v − zi zi ≥ 0, v ≤ (MZ)i.
Thanks to [2], problem (3.8) has one and only one solution.
(cid:8)TW − T (cid:12)W (cid:8)∞ ≤
(cid:8)W − (cid:12)W (cid:8)∞.
Theorem 3.5. Under conditions of Proposition 2.2, the mapping T is a contraction on H+, that is,
(3.9) λ λ + β
(cid:9)
(cid:2)
(cid:2)
(cid:3)i
(cid:8)zi = σ
Therefore, T admits a unique fixed point which coincides with the solution U of the system of QVIs (1.1). Proof. Let W, (cid:12)W ∈ H+, and let Z = TW, (cid:8)Z = T (cid:12)W be the corresponding solutions to sys- tem of QVIs (3.8) with right-hand sides Fi = f i + λwi and (cid:8)Fi = f i + λ (cid:8)wi, respectively. Let us also denote
(cid:10) .
(cid:3) ,
(cid:8)Fi,
M (cid:8)Z zi = σ Fi,(MZ)i (3.10)
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11)zi − (cid:8)zi
≤
(cid:11) (cid:11)wi − (cid:8)wi
Then, making use of Proposition 2.2, we immediately get
L∞(Ω)
L∞(Ω)
(3.11) λ λ + β
(cid:8)TW − T (cid:12)W (cid:8)∞ = (cid:8)Z − (cid:8)Z(cid:8)∞
(cid:11) (cid:11)
L∞(Ω)
= max 1≤i≤J
(cid:11) (cid:11)
(cid:11) (cid:11)zi − (cid:8)zi (cid:14) (cid:13) (cid:11) (cid:11)zi − (cid:8)zi
L∞(Ω)
and, consequently,
(cid:11) (cid:11)
(cid:11) (cid:11)zi − (cid:8)zi
≤
L∞(Ω)
(3.12) λ λ + β (cid:14)
≤
(cid:8)W − (cid:12)W (cid:8)∞,
≤ max 1≤i≤J (cid:13) λ λ + β λ λ + β
max 1≤i≤J
(cid:3)
which completes the proof.
1,..., (cid:15)u0
J ) such that (cid:15)u0 i solves the equation (cid:3)
(cid:2)
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. Let (cid:15)U 0 = ((cid:15)u0
i ,v (cid:15)u0
= ( f ,v) ∀v ∈ H 1
0 (Ω).
b (3.13)
M. Boulbrachene and S. Saadi 7
(cid:15)U n = T (cid:15)U n−1,
Algorithm 3.6. Starting from (cid:15)U 0 (resp., ˇU0 = 0), we define a decreasing sequence
n = 1,2,..., (3.14)
(resp., an increasing sequence)
J ) and ˇU n =
1,... , (cid:15)un
J ) solve coercive QVIs
ˇU n = T ˇU n−1, n = 1,2,.... (3.15)
(cid:3)
(cid:2)
(cid:3)
( ˇun Note that unlike sequences (3.5), (3.6), the components of (cid:15)U n = ((cid:15)un 1,... , ˇun
(cid:2)
0 (Ω),
∀v ∈ H 1 (cid:3)i
(cid:2) M (cid:15)U n
bi
(cid:2) M (cid:15)U n (cid:2) (cid:3)
; (3.16)
(cid:2)
0 (Ω),
∀v ∈ H 1 (cid:3)i.
(cid:2) M ˇU n
(cid:2) i ,v − (cid:15)un (cid:15)un i (cid:15)un i ≤ (cid:2) i ,v − ˇun ˇun i (cid:2) ˇun M ˇU n i ≤
bi
,v − (cid:15)un f i + λ(cid:15)un−1 i i (cid:3)i (cid:15)un i ≥ 0, v ≤ , (cid:3) i ,v − ˇun f i + λ ˇun i (cid:3)i ˇun i ≥ 0, v ≤ ,
Theorem 3.7. Let ρ = λ/(λ + β). Then, under conditions of Theorem 3.5, the sequences ( (cid:15)U n) and ( ˇU n) remain in the sector (cid:10)0, (cid:15)U 0(cid:11) and converge geometrically to the unique solution U of (1.1), that is,
(cid:11) (cid:11) (cid:15)U 0 − U (cid:11) (cid:11) (cid:15)U 0 − U
(cid:11) (cid:11) (cid:15)U n − U (cid:11) (cid:11) ˇU n − U
(3.17)
(cid:11) (cid:11) ∞ ≤ ρn (cid:11) (cid:11) ∞ ≤ ρn
(cid:11) (cid:11) ∞, (cid:11) (cid:11) ∞.
(3.18)
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11)
Proof. Let us prove (3.17). The proof of (3.18) is similar.
(cid:11) (cid:11)T (cid:15)U 0 − U
(cid:11) (cid:11)T (cid:15)U 0 − TU
(cid:11) (cid:11) (cid:15)U 0 − U
∞ ≤ ρn
∞.
∞ =
∞ =
For n = 1, we have (cid:11) (cid:11) (cid:15)U 1 − U (3.19)
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11) (cid:15)U n−1 − U
(cid:11) (cid:11) (cid:15)U 0 − U
Assume
∞ ≤ ρn−1
∞.
(3.20)
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11) (cid:15)U n − U
(cid:11) (cid:11)T (cid:15)U n−1 − TU
(cid:11) (cid:11) (cid:15)U n−1 − U
Then,
∞ ≤ ρ
(cid:11) (cid:11) ∞.
∞ =
(3.21)
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11) (cid:15)U n − U
(cid:11) (cid:11) (cid:15)U 0 − U
(cid:11) (cid:11) (cid:15)U 0 − U
∞ ≤ ρρn−1
∞ ≤ ρn
∞.
Thus
(3.22) (cid:3)
4. 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 that the family τh is regular and quasi-uniform.
8 System of quasivariational inequalities
Let Vh denote the standard piecewise linear finite element space, and let Bi, 1 ≤ i ≤ J, be the matrices with generic coefficients bi(ϕl,ϕs), where ϕs, s = 1,2,..., and m(h) are the nodal basis functions. Let also rh be the usual interpolation operator. Definition 4.1. A real n × n matrix B = [bi j] with bi j ≤ 0 for all i (cid:7)= j is an M-matrix if B is nonsingular and B−1 ≥ 0. The discrete maximum principle assumption (d.m.p.). We assume that the matrices Bi are M-matrices (cf. [6]).
(cid:3)
(cid:2)
4.1. Discrete elliptic quasivariational inequalities. The discrete counterpart of QVI (2.3) reads as follows: find uh ∈ Kh(uh) such that (cid:3) (cid:2)
(cid:2)
∀v ∈ Kh
(cid:2) uh
(cid:3) ,
b uh,v − uh f ,v − uh (4.1)
where Kh(uh) = {v ∈ Vh such that v ≤ rhMUh}.
Next we will state properties for the solution of (4.1) which are the direct discrete counterparts of those given in Propositions 2.1 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 , (cid:8)f be in L∞(Ω) and uh = σh( f ,MUh), (cid:8)uh = σh( (cid:8)f ,M (cid:8)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 ≥ (cid:8)f , then σh( f ,MUh) ≥ σh( (cid:8)f ,M (cid:8)uh).
4.1.2. A discrete Lipschitz dependence property.
(cid:11) (cid:11)
≤ 1
(cid:11) (cid:11)uh − (cid:8)uh
Proposition 4.3. Let Proposition 4.2 hold. Then,
(cid:8) f − (cid:8)f (cid:8)L∞(Ω).
L∞(Ω)
(4.2) λ + β
h,...,uJ
h) ∈ (Vh)J such that
(cid:2)
(cid:3)
(cid:3)
4.2. The discrete system of QVIs. We define the discrete system of QVIs as follows: find Uh = (u1
ai (4.3)
(cid:2) (cid:3)i
(cid:2) h,v − ui ui h (cid:2) MUh
∀v ∈ Vh, (cid:3)i. (cid:2) MUh
, ui h ≤ rh f i,v − ui h ui h ≥ 0, v ≤ rh
h) ∈ (Vh)J solution to the equivalent system
(cid:3)
(cid:2)
(cid:3)
Similarly to the continuous problem, the above problem can be transformed into the
h,v − ui h
bi (4.4)
(cid:2) (cid:3)i
∀v ∈ Vh, (cid:3)i. (cid:2) MUh
h,...,uJ following: find Uh = (u1 (cid:2) h,v − ui ui h (cid:2) ui MUh h ≤ rh
v ≤ rh , f i + λui ui h ≥ 0,
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
M. Boulbrachene and S. Saadi 9
(cid:3)J
key idea consists of associating with the above system the following fixed point mapping:
(cid:3) ,
(cid:2) Vh , (cid:2) h ,...,ζ J ζ 1
h
Th : H+ −→ (4.5) W −→ ThW = ζh =
h = σh( f i + λwi,(MW)i) is the solution of the following discrete VI:
(cid:3)
(cid:2)
(cid:3)
where ζ i
bi
(cid:2)
∀v ∈ Vh,
(cid:2) h,v − ζ i ζ i h ζ i h ≤ rh(MW)i,
(4.6) f i + λwi,v − ζ i h ζ i h ≥ 0, v ≤ rh(MW)i.
h ,..., ¯uJ,0
h = ( ¯u1,0
h ) be the discrete analogue of ¯U 0 defined in (3.4): (cid:3)
(cid:2)
(cid:3)
Let ¯U 0
=
∀v ∈ Vh.
(cid:2) ¯ui,0 h ,v
h solution of (4.7), (resp., U 0
ai f i,v (4.7)
Then, Th possesses analogous properties to those enjoyed by mapping T (see Proposition 3.1). Proposition 4.4. Th is increasing, concave on H+ and satisfies ThW ≤ ¯U 0 for all W ≤ ¯U 0 h . h = (0,...,0)), we define a Algorithm 4.5. Starting from ¯U 0 discrete decreasing sequence
h = Th ¯U n h ,
¯U n+1 n = 0,1,..., (4.8)
(resp., a discrete increasing sequence)
h = ThU n h,
h ) and (U n
h) remain in the h (cid:11). Moreover, they converge monotonically to the unique solution Uh of system of
U n+1 n = 0,1,.... (4.9)
Theorem 4.6. Let Proposition 4.4 hold, then, the sequences ( ¯U n sector (cid:10)0, ¯U 0 QVIs (4.3).
(cid:3)J
Th : H+ −→
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:
(cid:3) ,
(cid:2) Vh , (cid:2) h,...,zJ z1
h
(4.10) W −→ ThW = Zh =
h,...,z J
h ) is solution to the discrete coercive system of QVIs:
(cid:2)
(cid:3)
(cid:3)
where Zh = (z1
bi
(cid:2)
∀v ∈ Vh,
(cid:2) zi h,v − zi h zi h ≤ rh(MZ)i,
(4.11) f + λwi,v − zi h zi h ≥ 0, v ≤ rh(MZ)i.
Then, making use of Proposition 4.3, we get the following.
10 System of quasivariational inequalities
(cid:11) (cid:11)
(cid:11) (cid:11)ThW − Th (cid:12)W
(cid:8)W − (cid:12)W(cid:8)∞.
Theorem 4.7. The mapping Th is a contraction on H+. That is,
∞ ≤
(4.12) λ λ + β
Therefore, there exists a unique fixed point which coincides with the solution Uh of the system of QVI (4.3).
(cid:3)
Proof. It is very similar to that of the continuous case.
h = ((cid:15)u1,0
h ,..., (cid:15)uJ,0
h solves the equation
h ) such that (cid:15)ui,0
(cid:3)
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, let (cid:15)U 0
= ( f ,v) ∀v ∈ Vh.
(cid:2) (cid:15)ui,0 h ,v
bi (4.13)
h (resp., ˇU0h = 0), we define a decreasing sequence
(cid:15)U n
Algorithm 4.8. Starting from (cid:15)U 0
h = Th (cid:15)U n−1
h
, n = 1,2,..., (4.14)
(resp., an increasing sequence)
h = Th ˇU n−1,
ˇU n n = 1,2,.... (4.15)
h = ((cid:15)u1,n
h ,... , (cid:15)uJ,n h )
h = ( ˇu1,n
(cid:2)
(cid:3)
(cid:3)
Note that unlike sequences (4.8), (4.9), the components of both (cid:15)U n and ˇU n
(cid:2)
∀v ∈ Vh, (cid:3)i
(cid:2)
bi
(4.16) bi
(cid:2)
h
(cid:3)i
h ,..., ˇuJ,n h ) solve discrete coercive QVIs, which are (cid:2) h ,v − (cid:15)ui,n (cid:15)ui,n h (cid:2) (cid:15)ui,n M (cid:15)U n h ≤ rh h (cid:3) (cid:2) ˇui,n h ,v − ˇui,n ˇui,n h ≤ rh
h (cid:2) M ˇU n h
(cid:2) M (cid:15)U n ; h ∀v ∈ Vh, (cid:3)i. (cid:2) M ˇU n h
h ) remain in the sector (cid:10)0, (cid:15)U 0
, f i + λ(cid:15)ui,n−1 ,v − (cid:15)ui,n h h (cid:3)i (cid:15)ui,n h ≥ 0, v ≤ rh , (cid:3) h ,v − ˇui,n f i + λ ˇui,n ˇui,n h ≥ 0, v ≤ rh
Theorem 4.9. Let ρ = λ/(λ + β). Then, under conditions of Theorem 4.7, the sequences ( (cid:15)U n h ) and ( ˇU n h (cid:11) and converge geometrically to the unique solution Uh of (4.3), that is,
(cid:11) (cid:11) (cid:15)U 0 (cid:11) (cid:11) (cid:15)U 0
(cid:11) (cid:11) (cid:15)U n (cid:11) (cid:11) ˇU n
h − Uh h − Uh
h − Uh h − Uh
(cid:11) (cid:11) ∞ ≤ ρn (cid:11) (cid:11) ∞ ≤ ρn
(cid:11) (cid:11) ∞, (cid:11) (cid:11) ∞.
(4.17)
(cid:3)
Proof. The proof is similar to that of the continuous case.
5. L∞-error analysis We now turn to the L∞-error analysis. For that purpose, we will give two different ap- proaches.
M. Boulbrachene and S. Saadi 11
h,..., ¯zJ
h) solution to
(cid:2)
(cid:3)
(cid:2)
(cid:3)
5.1. The contraction approach. It stands on the characterization of the solutions of both the continuous and discrete systems (1.1) and (4.3) as fixed points of contractions. First, let us introduce the following intermediate discrete coercive system of QVIs: find Zh = ( ¯z1
b
(cid:2)
(cid:3)i
∀v ∈ Vh, (cid:3)i. (cid:2) M ¯Zh
f + λui,v − ¯zi h (5.1) , ¯zi h,v − ¯zi h (cid:2) ¯zi M ¯Zh h ≤ rh ¯zi h ≥ 0, v ≤ rh
Clearly, (5.1) is a coercive system whose right-hand side depends on U = (u1,...,uJ ), the solution of system (1.1). So, in view of (4.10), (4.11), we readily have
¯Zh = ThU. (5.2)
(cid:11) (cid:11) ¯Zh − U
Therefore, using the result of [5], we get the following error estimate:
(cid:11) (cid:11) ∞ ≤ Ch2|Logh |3.
(5.3)
(cid:11) (cid:11)
(cid:11) (cid:11)U − Uh
Theorem 5.1. Let U and Uh be the solutions of systems (1.1) and (4.3), respectively. Then,
∞ ≤ Ch2|Logh |3.
(5.4)
Proof. In view of (5.2) and Theorems 3.5 and 4.7, we clearly have
¯Zh = ThU. U = TU; Uh = ThUh; (5.5)
(cid:11) (cid:11)
(cid:11) (cid:11)ThU − TU
(cid:11) (cid:11) ¯Zh − U
Then, using estimation (5.3), we get
(cid:11) (cid:11) ∞ ≤ Ch2|Logh |3.
∞ =
(5.6)
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11)Uh − U
(cid:11) (cid:11)ThU − TU
∞ ≤
∞ + (cid:11) (cid:11)
∞ (cid:11) (cid:11)
≤
(cid:11) (cid:11)ThU − TU
∞ +
∞
Therefore
(cid:11) (cid:11)
≤ ρ
(cid:11) (cid:11)Uh − ThU (cid:11) (cid:11)ThUh − ThU (cid:11) (cid:11)U − Uh
∞ + Ch2|Logh |3.
(5.7)
(cid:11) (cid:11)
Thus
(cid:11) (cid:11)U − Uh
∞ ≤
. Ch2|Logh |3 (1 − ρ) (5.8) (cid:3)
5.2. The algorithmic approach. It combines the error estimate between the nth iterate of (3.14) and its discrete counterpart (4.15), and the geometrical convergence of those algorithms.
h ,..., (cid:8)uJ,n
12 System of quasivariational inequalities
(cid:3)
(cid:2)
(cid:3)
Let us first introduce the following sequence of discrete coercive systems of QVIs: find h = ((cid:8)u1,n (cid:8)U n
(cid:2)
∀v ∈ Vh, (cid:3)i
(cid:2) M (cid:8)U n h
h ) such that (cid:2) h ,v − (cid:8)ui,n (cid:8)ui,n h (cid:2) (cid:8)ui,n M (cid:8)U n h ≤ rh h
bi (5.9) , , f i + λ(cid:15)ui,n−1,v − (cid:8)ui,n h (cid:3)i (cid:8)ui,n h ≥ 0, v ≤ rh
h = ((cid:15)u1,n
h = (cid:15)U 0 h .
h ,..., (cid:15)uJ,n
h ) is the continuous sequence defined in (3.14), and (cid:8)U 0
where (cid:15)U n The following lemma plays a crucial role in the present approach.
(cid:6)
(cid:7) n(cid:4)
(cid:11) (cid:11)
(cid:11) (cid:11)
Lemma 5.2.
(cid:11) (cid:11) (cid:15)U n − (cid:15)U n h
(cid:11) (cid:11) (cid:15)U p − (cid:8)U p h
∞ ≤
∞.
p=0
(cid:11) (cid:11)
(cid:11) (cid:11)
(5.10) 1 − ρn+1 1 − ρ
∞
∞ +
(cid:11) (cid:11)
∞ ≤ ≤
∞
Proof. Th being a contraction, we have (cid:11) (cid:11) (cid:15)U 1 − (cid:15)U 1 h
(cid:11) (cid:11) (cid:11) (cid:11) (cid:11) (cid:11)
≤
∞
(cid:10)
(cid:11) (cid:11) (cid:8)U 1 (cid:11) (cid:11)Th (cid:8)U 0 (cid:11) (cid:11) (cid:8)U 0 (cid:11) (cid:11)
(cid:11) (cid:11)
h − (cid:15)U 1 h h − Th (cid:15)U 0 h (cid:11) (cid:11) h − (cid:15)U 0 h (cid:11) (cid:11) (cid:8)U 0
(5.11)
≤ (1 + ρ)
h − (cid:15)U 0 h
(cid:11) (cid:11) (cid:15)U 1 − (cid:8)U 1 h (cid:11) (cid:11) (cid:15)U 1 − (cid:8)U 1 ∞ + h (cid:11) (cid:11) (cid:15)U 1 − (cid:8)U 1 ∞ + ρ h (cid:9)(cid:11) (cid:11) (cid:15)U 1 − (cid:8)U 1 h
∞ +
∞
.
(cid:6)
(cid:7) n−1(cid:4)
(cid:11) (cid:11) (cid:15)U n−1 − (cid:15)U n−1
(cid:11) (cid:11).
Now assume that
(cid:11) (cid:11) (cid:15)U p − (cid:8)U p h
h
(cid:11) (cid:11) ∞ ≤
p=0
(5.12) 1 − ρn 1 − ρ
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11) (cid:15)U n − (cid:15)U n h
∞
∞ +
(cid:11) (cid:11)
∞ ≤ ≤
∞
h (cid:11) (cid:11)
(cid:11) (cid:11) (cid:11) (cid:11) (cid:11) (cid:11)
(cid:11) (cid:11) (cid:8)U n h − (cid:15)U n h (cid:11) (cid:11)Th (cid:15)U n−1 − Th (cid:15)U n−1 (cid:11) (cid:11) (cid:15)U n−1 − (cid:15)U n−1
≤
h
(cid:11) (cid:11) (cid:15)U n − (cid:8)U n h (cid:11) (cid:11) (cid:15)U n − (cid:8)U n h (cid:11) (cid:11) (cid:15)U n − (cid:8)U n h
∞
∞ + ∞ + ρ
(cid:2)
(cid:3) n(cid:4)
(cid:11) (cid:11)
(cid:11) (cid:11)
≤
Then, using, again, the fact that Th is a contraction, we get
(cid:11) (cid:11) (cid:15)U p − (cid:8)U p h
(cid:11) (cid:11) (cid:15)U n − (cid:8)U n h
∞ + ρ
(cid:3) n(cid:4)
(cid:11) (cid:11)
(cid:11) (cid:11)
≤
(cid:2) 1 + ρ + · · · + ρn
(cid:11) (cid:11) (cid:15)U n − (cid:8)U n h
p=0 (cid:11) (cid:11) (cid:15)U p − (cid:8)U p h
p=0
(cid:6)
∞ + (cid:7) n(cid:4)
(cid:11) (cid:11)
≤
(cid:11) (cid:11) (cid:15)U p − (cid:8)U p h
1 + ρ + · · · + ρn−1 (5.13)
p=0
1 − ρn+1 1 − ρ
(cid:3)
(cid:11) (cid:11)
(cid:11) (cid:11)U − Uh
which completes the proof. Theorem 5.3. Let U and Uh be the solutions of systems (1.1) and (4.3), respectively. Then,
∞ ≤ Ch2|Logh |4.
(5.14)
M. Boulbrachene and S. Saadi 13
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11)U − (cid:15)U n
(cid:11) (cid:11) (cid:15)U n
(cid:11) (cid:11)U − Uh
∞ ≤
∞
(cid:6)
∞ + (cid:7) n(cid:4)
Proof. We have
∞ + (cid:11) (cid:11)
(cid:11) (cid:11)
(cid:11) (cid:11)
≤ ρn
(cid:11) (cid:11) (cid:15)U 0 − U
(cid:11) (cid:11) (cid:15)U 0
h − Uh (cid:11) (cid:11) (cid:15)U p − (cid:8)U p h
h − Uh
∞ +
∞ + ρn
∞.
(cid:11) (cid:11) (cid:15)U n − (cid:15)U n h 1 − ρn+1 1 − ρ
p=0
(5.15)
Now, taking
ρn ≤ h2, (5.16)
(cid:11) (cid:11)
(cid:11) (cid:11)U − Uh
∞ ≤ Ch2|Logh |4.
we get
(5.17) (cid:3)
Remark 5.4. Clearly, the first approach provides a better approximation as the second one leads to a convergence order with an extra logarithmic factor.
[1] A. Bensoussan and J.-L. Lions, Applications des in´equations variationnelles en contrˆole stochas-
tique, M´ethodes Math´ematiques de l’Informatique, no. 6, Dunod, Paris, 1978.
, Impulse Control and Quasivariational Inequalities, Gauthier-Villars, Montrouge, 1984. [2] [3] M. Boulbrachene, Pointwise error estimate for a noncoercive system of quasi-variational inequali- ties related to the management of energy production, Journal of Inequalities in Pure and Applied Mathematics 3 (2002), no. 5, 9, article 79.
, L∞-error estimate for a noncoercive system of elliptic quasi-variational inequalities: a sim-
[4]
ple proof, Applied Mathematics E-Notes 5 (2005), 97–102.
[5] M. Boulbrachene, M. Haiour, and S. Saadi, L∞-error estimate for a system of elliptic quasi- variational inequalities, International Journal of Mathematics and Mathematical Sciences 2003 (2003), no. 24, 1547–1561.
[6] P. G. Ciarlet and P.-A. Raviart, Maximum principle and uniform convergence for the finite element method, Computer Methods in Applied Mechanics and Engineering 2 (1973), no. 1, 17–31.
Messaoud Boulbrachene: Department of Mathematics & Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Muscat 123, Sultanate of Oman E-mail address: boulbrac@squ.edu.om
Samira Saadi: Departement de Math´ematiques, Facult´e des Sciences, Universit´e d’Annaba, BP 12, Annaba 23000, Algeria E-mail address: signor 2000@yahoo.fr
References
