© Hindawi Publishing Corp. L ∞-ERROR ESTIMATE FOR A SYSTEM OF ELLIPTIC QUASIVARIATIONAL INEQUALITIES

We deal with the numerical analysis of a system of elliptic quasivariational inequalities (QVIs). Under W 2 , p ( Ω ) -regularity of the continuous solution, a quasi-optimal L ∞ -convergence of a piecewise linear finite element method is established, involving a monotone algorithm of Bensoussan-Lions type and standard uniform error estimates known for elliptic variational inequalities (VIs).


Introduction.
In this paper, we are concerned with the L ∞ -convergence of the standard finite element approximation for the following system of quasivariational inequalities (QVIs): find U = (u 1 ,...,u M ) ∈ (H 1 0 (Ω)) J satisfying where Ω is a bounded smooth domain of R N , N ≥ 1, with boundary ∂Ω, a i (u, v) are J-elliptic bilinear forms continuous on H 1 (Ω) × H 1 (Ω), (•, •) is the inner product in L 2 (Ω), and f i are J-regular functions.This system, introduced by Bensoussan and Lions (see [3]), arises in the management of energy production problems where J-units are involved (see [4] and the references therein).In the case studied here, Mu i represents a "cost function" and the prototype encountered is where 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 the operator M provides the coupling between the unknowns u 1 ,...,u J .
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 upon the solution sought.The terminology QVI being chosen is a result of this remark.
The L ∞ -error estimate is a challenge not only for its practical reasons but also due to its inherent difficulty of convergence in this norm.Moreover, the interest in using such a norm for the approximation of obstacle problems is that they are a type of free boundary problems.This fact has been validated by the paper of Brezzi and Caffarelli [7] and later by that of Nochetto [15] on the convergence of the discrete free boundary to the continuous one.
A lot of results on error estimates for the classical obstacle problems and variational inequalities (VIs) were achieved in this norm, (cf.[1,11,14,16]).However, very few works are known on this subject concerning QVIs (cf.[5,10]) and especially the case of systems (see [6]).
Our primary aim in this paper is, precisely, to show that problem (1.1) can be properly approximated by a finite element method which turns out to be quasi-optimally accurate in L ∞ (Ω).The approximation is carried out by first introducing a monotone iterative scheme of Bensoussan-Lions type which is shown to converge geometrically to the continuous solution.Similarly, using the standard finite element method and a discrete maximum principle (d.m.p.), the solution of the discrete system of QVIs is in its turn approximated by an analogue discrete monotone iterative scheme, and a geometric convergence to the discrete solution is given as well.An L ∞ -error estimate is then established combining the geometric convergence of both the continuous and discrete iterative schemes with known uniform error estimates in elliptic VIs.
An outline of the paper is as follows.We lay down some necessary notations, assumptions, and preliminaries in Section 2. We consider the continuous problem and prove some related qualitative properties in Section 3. Section 4 deals with the discrete problem for which an analogue study to that of the continuous problem is achieved.Finally, in Section 5, we prove a fundamental lemma and give the main result.

Preliminaries
2.1.Assumptions and notation.We are given functions a i jk (x), a i k (x), and ) We define the variational forms, for any u, v ∈ H 1 (Ω), and the differential operator associated with the bilinear form a i (•, •) We are also given right-hand sides

Elliptic VIs
where K = {v ∈ H 1 0 (Ω) such that v ≤ ψ a.e.} and a(•, •) is a bilinear form of the same type as those defined in (2.3).
Proof.Clearly σ (ψ) + c = u + c is solution to the VI with right-hand side f + a 0 c and obstacle ψ + c whereas σ (ψ + c) is solution to the VI with righthand side f and obstacle ψ + c.Then, as a 0 (x) ≥ β > 0 (see (2.2)) and c > 0, it follows that f < f +a 0 c and thanks to Theorem 2.4 we get σ (ψ+c) ≤ σ (ψ)+c.

The continuous problem
3.1.Existence, uniqueness, and regularity.The existence of a unique solution to system (1.1) can be proved adapting the approach developed in [3, pages 343-358].
Let L ∞ + (Ω) denote the positive cone of L ∞ (Ω), and consider We define the following fixed-point mapping: where is a solution to the following VI: 3) being a coercive VI, thanks to [2,13] it has one and only one solution.
For all ϕ ∈ H 1 (Ω), we let ϕ + = max(ϕ, 0).By the fact that both of ζ i and u i,0 belong to H 1 0 (Ω), we clearly have ) + as a trial function in (3.3).This gives So, by addition, we obtain which, by (2.4), yields that is, (2) T W ≥ 0, for all W ∈ H + .This follows immediately from standard comparison results in elliptic VIs since f i ≥ 0.
Remark 3.5.The discrete version of Proposition 3.4 plays an important role in the finite element error analysis part of this work.Remark 3.6.We notice that the solutions of system (1.1) correspond to fixed points of mapping T , that is, U = T U.Then, in this view, it is natural to consider the following iterative scheme.

A continuous iterative scheme of Bensoussan-Lions type. Starting from U
0 defined in (3.4) and U 0 = (0,...,0), we define the sequences ) The convergence analysis of these sequences rests upon the following results.
Proof.The proof is very similar to that of [3, page 351].
Then, under the conditions of Lemma 3.7, Proof.From (3.21), we have (1−γ)W ≤ W .Then, applying Proposition 3.3, we get and, due to Lemma 3.7, the desired result follows.

Rate of convergence of the continuous iterative scheme
Proposition 3.11.Let the conditions of Proposition 3.8 hold.Then Proof.By Theorem 3.9, we have and inductively, We prove estimation (3.25) as estimation (3.24).

The discrete problem.
Let Ω 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 family τ h is regular and quasi-uniform.
Let V h denote the standard piecewise linear finite element space and A i , 1 ≤ i ≤ J, be the matrices with generic coefficients a i (ϕ l ,ϕ s ), where ϕ s , s = 1, 2,...,m(h), are the nodal basis functions.Let also r h be the usual interpolation operator.
The d.m.p.We assume that A i are M-matrices (cf.[9]).Let u h ∈ V h be the finite element approximation of u defined in (2.7), that is, Now, let σ h be a mapping from L ∞ (Ω) into V h , defined by The mapping σ h possesses analogous properties to those of the mapping σ (see Proposition 2.5) provided the d.m.p is satisfied.
Proposition 4.1.The mapping σ h is increasing, concave, and Lipschitz continuous with respect to ψ.
4.1.The discrete system of QVIs.We define the discrete system of QVIs as follows: find (4.3)

Existence and uniqueness.
The existence and uniqueness of a solution to system (4.3) can be shown similarly to that of the continuous case provided the d.m.p is satisfied.Indeed, the key idea for proving that consists in associating with this system the following discrete fixed point mapping: where is the solution of the following discrete VI: Remark 4.2.Under the d.m.p, the mapping T h possesses analogous properties to that of mapping T (see Propositions 3.1, 3.2, 3.3, 3.4, and 3.8).The proofs of such properties will not be given as they are very similar to those of the continuous case.We just list them below.

Some properties of the mapping T h . Let
h ) be the discrete analogue to U 0 defined in (3.4): Then, we have the discrete analogues to Propositions 2.6, 3.1, 3.2, and 3.3, respectively.
Proposition 4.4.The mapping T h is increasing and concave on H + .

Proposition 4.5. The mapping T h is Lipschitz continuous on H + , that is,
Remark 4.6.It is not hard to see that the solution of system of QVIs (4.3) is a fixed point of T h , that is, U h = T h U h .Therefore, as in the continuous problem, one can associate with T h the following iterative scheme.

A discrete iterative scheme of Bensoussan-Lions type. Starting from U
0 h solution of (4.6) (resp., from U 0 h = (0,...,0)), we define respectively Then, by analogy with the continuous problem, using the following intermediate results, we are able to prove the convergence of the discrete iterative scheme to the solution of system (4.3).

.10)
Then  and (U n h ) are monotone and well defined in C h .Moreover, they converge, respectively, from above and below to the unique solution of system (4.3).
Proof.Very similar to that of Theorem 3.9.

Rate of convergence of the discrete iterative scheme
) Proof.It is exactly the same as that of Proposition 3.11.

5.
The finite element error analysis.This section is devoted to demonstrate that the proposed method is quasi-optimally accurate in L ∞ (Ω).For this purpose, we need first to introduce an auxiliary sequence of discrete VIs and next prove a fundamental lemma.
From now on, C will denote a constant independent of both h and n.

An auxiliary sequence of discrete VIs.
Let U n = (u 1,n ,...,u n,J ) be the sequence defined in (3.19).We then introduce the following discrete sequence: where U 0 h is defined in (4.6) and for any n 1, ũi,n h is solution to the following discrete VI: ( We notice that ũi,n h , solution of (5.2), represents the piecewise finite element approximation of u i,n , the ith component of U n .Therefore, using the regularity result provided by Lemma 5.1 and next adapting [11], we have the optimal uniform error estimate given below.
Lemma 5.1.For any i = 1,...,J, where C is a constant independent of n.
Proof.We know that u i,1 = σ (Mu i,0 ) is a solution to the VI with obstacle ψ = k+inf u µ,0 , µ ≠ i and u i,0 ∈ W 2,p (Ω).So, ψ W 1,∞ (Ω) ≤ C 1 and, therefore, as in [3, Lemma 2.3, page 372], we get Ꮽ i ψ ≥ −c 1 in the sense of H −1 (Ω).Hence, by Lemma 2.2 and Theorem 2.3, it follows that So, using the same arguments as before, we get Ꮽ i ψ ≥ −c 2 in the sense of H −1 (Ω) with c independent of n, and therefore u i,n W 2,p (Ω) ≤ C, where C is a constant independent of n.(The proof of u i,n W 2,p (Ω) ≤ C is exactly as above.) Theorem 5.2.Under the conditions of Lemma 5.1, (5.4) The following lemma plays a crucial role in proving the main result.

Conclusion. (1)
We have established a convergence order in the L ∞ -norm for a coercive system of QVIs.A future paper will be devoted to the noncoercive case for which a different approach will be developed and analyzed.
(2) It is also important to notice that the error estimate obtained in this paper contains an extra power in log h than expected.We believe that this is due to the approach followed.
(3) The same approach may also be extended to other important problems such as the system of QVIs related to games theory [3].

Call for Papers
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Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
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. 11 ) 1 )
which completes the proof of Lemma 5.3.Now, guided by Lemma 5.3, Propositions 3.11 and 4.10, and Theorem 5.2, we are in a position to demonstrate our main result.5.2.L ∞ -error estimate for the system of QVIs (1.Theorem 5.4.

First
Round of ReviewsMarch 1, 2009 It follows immediately from the increasing property of the mapping σ (see Proposition 2.5).The mapping T is concave on H + .The mapping T is Lipschitz continuous on H + , that is, Proposition 3.2.The mapping T is increasing on H + .Proof.