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