zbMATH — the first resource for mathematics

Optimal \(L^{\infty}\)-error estimate for variational inequalities with nonlinear source terms. (English) Zbl 1057.65038
Summary: We establish optimal \(L^{\infty}\)-error estimate for a class of variational inequalities (VIs) with nonlinear source term, using a very simple argument mainly based on the discrete \(L^{\infty}\)-stability property with respect to the right-hand side in elliptic VIs. We also show that the same approach extends to the corresponding noncoercive problems and optimal uniform convergence order is obtained as well.

65K10 Numerical optimization and variational techniques
49J40 Variational inequalities
49M25 Discrete approximations in optimal control
Full Text: DOI
[1] Brezzi, F.; Caffarelli, L.A., Convergence of the discrete free boundary for finite element approximations, R.A.I.R.O anal., 17, 385-395, (1983)
[2] Nochetto, R.H., A note on the approximation of free boundaries by finite element methods, R.A.I.R.O anal., 20, 355-368, (1986) · Zbl 0596.65092
[3] Nitsche, J., L∞-convergence of finite element approximations, Mathematical aspects of finite element methods, Lect. notes math., Volume 606, 261-274, (1977)
[4] Baiocchi, C., Estimation d’erreur dans L∞ pour LES inequations a obstacle, (), 27-34 · Zbl 0374.65053
[5] Cortey-Dumont, P., On the finite element approximation in the L∞-norm of variational inequalities with nonlinear operators, Numer. math., 47, 45-57, (1985) · Zbl 0574.65064
[6] Nochetto, R.H., Sharp L∞-error estimates for semilinear elliptic problems with free boundaries, Numer. math., 54, 243-255, (1988) · Zbl 0663.65125
[7] Cortey-Dumont, P.; Conrad, F., Sur l’ analyse numerique d’un probleme de bifurcation dans LES inequations variationnelles, Numerical functional analysis and optimization, (1987)
[8] Cortey-Dumont, P., Sur LES inequations variationnelles a operateurs non coercifs, M^{2}, an, 19, 195-212, (1985)
[9] Bensoussan, A.; Lions, J.L., ()
[10] Bensoussan, A.; Lions, J.L., ()
[11] P.G. Ciarlet and P.A. Raviart, Maximum principle and uniform convergence for the finite element method, Comp. Meth. in Appl. Mech. and Eng. {\bf1973}, 1-20. · Zbl 0251.65069
[12] Cortey-Dumont, P.; Loinger, E., Sur l’ approximation d’une IQV liee a des problemes d’infiltration en milieu poreux, Calcolo, XIX, 125-152, (1982), (Fasc. II) · Zbl 0501.76085
[13] Verdi, C., Convergence of the approximate free boundary for the multidimensional one-Stefan problem, J. comp. mech., (1985)
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.