zbMATH — the first resource for mathematics

Pointwise error estimates for a class of elliptic quasi-variational inequalities with nonlinear source terms. (English) Zbl 1065.65082
The aim of the paper is to show that the class of elliptic quasi-variational inequalities (QVIs) with nonlinear source terms can be properly approximated by a finite element method, provided some realistic assumptions are made on both the nonlinearity $$f(\cdot)$$ and the operator $$S$$.
Main results: The author show that the class of QVIs $$(a(u, v- u)\geq(f (u), v- u)$$ for any $$v\in{\mathbf K}(u)= \{v\in H^1_0(\Omega): v\leq S(u)$$ a.e. in $$\Omega\}$$, $$S(u)= \psi+\phi(u)$$, $$\psi$$ is a regular function, $$\phi(u)$$ is a nonlinear operator from $$L_\infty(\Omega)$$ into itself such that: $$\phi(w)\leq\phi(\widetilde w)$$ for $$w\leq\widetilde w$$ a.e. in $$\Omega$$, $$\phi(w+ C)\leq\phi(w)+ C$$ $$(C> 0)$$, $$f(\cdot)$$ is a Lipschitz nondecreasing nonlinear source term with rate $$\alpha$$, $$(\alpha/\beta) <1)$$ can be properly approximated by a finite element method, when the approach rests on a discrete $$L_\infty$$-stability result and $$L^\infty$$-error estimates for elliptic QVIs.
The continuous QVI problem is associated with a fixed point mapping and the existence of a unique continuous solution is proved. A discrete stability result for elliptic linear QVIs is established and an abstract $$L_\infty$$-error estimate is proposed. The extension to the noncoercive case is considered, too. Finally, the applications to an obstacle type problem and to an impulse control problem are given.

MSC:
 65K10 Numerical optimization and variational techniques 49J40 Variational inequalities 49M15 Newton-type methods
Full Text:
References:
 [1] Cortey-Dumont, P., Approximation numerique d’une inequation quasi-variationnelle liee a des problemes de gestion de stock, RAIRO anal. numer, 14, 4, 335-346, (1980) · Zbl 0462.65045 [2] Boulbrachene, M., The noncoercive quasi-variational inequalities related to impulse control problems, Comput. math. applic, 35, 12, 101-108, (1998) · Zbl 0995.49009 [3] Boulbrachene, M.; Cortey-Dumont, P.; Miellou, J.C, L∞-error estimates for a class of semi-linear elliptic variational inequalities and quasi-variational inequalities, Ijmms, 27, 6, 309-319, (2001) · Zbl 1002.49015 [4] Boulbrachene, M., Optimal L∞-error estimate for variational inequalities with nonlinear source terms, Appl. math. lett, 15, 1013-1017, (2002) · Zbl 1057.65038 [5] Cortey-Dumont, P., On the finite element approximation in the L∞-norm of variational inequalities with nonlinear operators, Numer. math, 47, 1, 45-57, (1985) · Zbl 0574.65064 [6] A. Bensoussan, J.L. Lions, Applications des inequations variationnelles en controle stochastique, Dunod Paris, 1978 · Zbl 0411.49002 [7] Bensoussan, A.; Lions, J.L., Impulse control and quasi-variational inequalities, (1984), Gauthier Villars Paris · Zbl 0324.49005 [8] Ciarlet, P.G.; Raviart, P.A., Maximum principle and uniform convergence for the finite element method, Comput. meth. appl. mech. eng, 2, 17-31, (1973) · Zbl 0251.65069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.