-error analysis for a system of quasivariational inequalities with noncoercive operators
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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 continuous piecewise linear approximation. Copyright © 2006 M. Boulbrachene and S. Saadi. This is an open access article distributed 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 the L∞ norm of the following system of quasivariational inequalities (QVIs): find U = (u1 ,...,uJ ) ∈ (H01 (Ω))J satisfying
ai ui ,v − ui f i ,v − ui ui ≤ (MU)i ,
∀v ∈ H01 (Ω),
ui ≥ 0, v ≤ (MU)i .
(1.1)
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 (MU)i = k + inf uμ , μ=i
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 unknowns u1 ,...,uJ . 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
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System of quasivariational inequalities
find U = (u1 ,...,uJ ) ∈ (H01 (Ω))J such that
bi ui ,v − ui f i + λui ,v − ui ui ≤ (MU)i ,
∀v ∈ H01 (Ω),
ui ≥ 0, v ≤ (MU)i ,
(1.3)
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, bi (v,v) ≥ γv2H 1 (Ω) ,
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 continuous 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 solving a sequence of coercive systems of QVIs. This algorithm is shown to converge geometrically to the solution of system (1.1). It is worth mentioning th
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