# zbMATH — the first resource for mathematics

$$L^{\infty }$$-asymptotic behavior for a finite element approximation in parabolic quasi-variational inequalities related to impulse control problem. (English) Zbl 1211.65083
Summary: The parabolic quasi-variational inequalities are transformed into a noncoercive elliptic quasi-variational inequalities. A new iterative discrete algorithm is proposed to show the existence and uniqueness, and a simple proof to asymptotic behavior in uniform norm is also given using the theta time scheme combined with a finite element spatial approximation. The proposed approach stands on a discrete $$L^{\infty }$$-stability property with respect to the right-hand side and obstacle defined as an impulse control problem.

##### MSC:
 65K15 Numerical methods for variational inequalities and related problems 49J40 Variational inequalities 35K86 Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators 35J86 Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 49M25 Discrete approximations in optimal control
Full Text:
##### References:
 [1] Achdou, Y.; Hecht, F.; Pommier, D., A posteriori error estimates for parabolic variational inequalities, J. sci. comput., 37, 336-366, (2008) · Zbl 1203.65096 [2] Boulbrachene, M., Pointruise error estimates for a class of elliptic quasi-variational inequalities with non linear source terms, Comput. math. appl., 161, 129-138, (2005) · Zbl 1065.65082 [3] Boulbrachene, M., Optimal L∞-error estimate for variational inequalities with nonlinear source terms, Appl. math. lett., 15, 1013-1017, (2002) · Zbl 1057.65038 [4] Boulbrachene, M., L∞-error estimate for a system of elliptic quasi-variational inequalities with noncoercive operators, Comput. math. appl., 45, 983-989, (2003) · Zbl 1060.65065 [5] Boulbrachene, M.; Haiour, M.; Saadi, S., L∞-error estimate for a system of elliptic quasi-variational inequalities, Int. J. math., 1547-1561, (2003) · Zbl 1163.35399 [6] Boulbrachene, M., L∞-error estimate for a noncoercive of elliptic quasi-variational inequalities: a simple proof, Appl. math. E-notes, 5, 97-102, (2005) · Zbl 1123.35322 [7] Ciarlet, P.; Raviart, P., Maximum principle and uniform convergence for the finite element method, Comput. math. appl. mech. eng., 2, 1-20, (1973) · Zbl 0251.65069 [8] Ph. Cortey-Dumont, On finite element approximation in the L∞-norm parabolic obstacle variational and quasivariational inequalities, Rapport interne no. 112, CMA, Ecole Polytechnique, Palaiseau, France. · Zbl 0574.65064 [9] Cortey-Dumont, Ph., On finite element approximation in the L∞-norm of variational inequalities, Numer. math., 47, 45-57, (1985) · Zbl 0574.65064 [10] Lions, J.; Stampacchia, G., Variational inequalities, Commun. pure appl. math., 20, 493-519, (1967) · Zbl 0152.34601 [11] Kornhuber, R., A posteriori error estimates for elliptic variational inequalities, Comput. math. appl., 31, 8, 49-60, (1996) · Zbl 0857.65071 [12] Nochetto, R.H.; Siebert, K.G.; Veeser, A., Pointwise a posteriori error control for elliptic obstacle problems, Numer. math., 95, 1, 163-195, (2003) · Zbl 1027.65089 [13] Perthame, B., Some remarks on quasi-variational inequalities and the associated impulsive control problem, Annales de l’I.H.P section C, 2, 3, 237-260, (1985) [14] Quarteroni, A.; Valli, A., Numerical approximation of partial differential equations, (1994), Springer Berlin and Heidelberg · Zbl 0852.76051 [15] Zhenhai, L., Existence results for quasi linear parabolic hemi variational inequalities, J. differ. equat., 244, 1395-1409, (2008) · Zbl 1139.35006
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.