×

zbMATH — the first resource for mathematics

Approximations of parabolic variational inequalities. (English) Zbl 0574.65066
This paper considers the problem of finding a function \(u: I\to V\) where \(I=[0,T]\), \(0<T<\infty\) and V is a Hilbert space (satisfying other conditions) such that \((\dot u,w-u)+a(u,w-u)\geq (f,w-u)\) for all w belonging to a closed, convex subset of V and for all t in I except for a set of measure zero. The initial function u(0) is given and a(u,v) is a nonlinear elliptic form having a potential J(v) which is twice G- differentiable. The basic problem is discretized in time by a one-step finite difference method and in space by a finite element method which triangularises the region. The discretization in time are such that I is divided into r equally spaced points and the solution produces \(U^ i\) at \(t=t_ i\), \(i=1,2,...,r\) and \(U^ 0\) is a finite element approximation to u(0). From these abstract functions are constructed: \(U_ r=U^{i-1}+((t-t_{i-1})/\Delta t)(U^ i-U^{i-1}),\quad t\in [t_{i-1},t_ i].\) The form of a(u,v) and further conditions lead to convergence theorems and an error bound providing u(t) is sufficiently smooth.
Reviewer: B.Burrows

MSC:
65K10 Numerical optimization and variational techniques
49J40 Variational inequalities
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] I. Bock J. Kačur: Application of Rothe’s method to parabolic variational inequalities. Math. Slovaca 31 (1981), 429-436. · Zbl 0477.49008
[2] J. Céa: Optimization. Dunod, Paris 1971. · Zbl 0231.94026
[3] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam 1978. · Zbl 0383.65058
[4] H. Gajewski K. Gröger K. Zacharias: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin 1974. · Zbl 0289.47029
[5] J. Haslinger: Finite element analysis for unilateral problems with obstacles on the boundary. Apl. mat. 22 (1977), 180-188. · Zbl 0434.65083
[6] I. Hlaváček J. Haslinger J. Nečas J. Lovíšek: Solving Yariational Inequalities in Mechanics. Alfa-SNTL, Bratislava-Prague, 1982. · Zbl 0654.73019
[7] V. Jarník: Integral Calculus II. Nakladatelství ČSAV, Prague 1955. · Zbl 1160.26300
[8] C. Johnson: A convergence estimate for an approximation of a parabolic variational inequality. SIAM J. Numer. Anal. 13 (1976), 599-606. · Zbl 0337.65055
[9] J. Kačur: On an approximate solution of variational inequalities. · Zbl 0615.73119
[10] A. Kufner O. John S. Fučík: Function Spaces. Academia, Prague 1977.
[11] J. L. Lions: Quelques Méthodes de Résolution des Problèmes aux Lirnites Non Linéaires. Dunod and Gauthier - Villars, Paris 1969. · Zbl 0189.40603
[12] J. Nečas: Les Méthodes Directes en Théorie des Équations Elliptiques. Academia, Prague 1967. · Zbl 1225.35003
[13] A. Ženíšek M. Zlámal: Convergence of a finite element procedure for solving boundary value problems of the fourth order. Int. J. Numer. Meth. Engng. 2 (1970), 307-310. · Zbl 0256.65055
[14] M. Zlámal: Curved elements in the finite element method I. SIAM J. Numer. Anal. 30 (1973), 229-240. · Zbl 0285.65067
[15] M. Zlámal: Finite element solution of quasistationary nonlinear magnetic field. R.A.I.R.O. Anal. Numer. 16 (1982), 161-191.
[16] M. Zlámal: A linear scheme for the numerical solution of nonlinear quasistationary magnetic fields. Math. Comp. 41 (1983), 425-440.
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.