# Article

 Title: Approximations of parabolic variational inequalities (English) Author: Ženíšek, Alexander Language: English Journal: Aplikace matematiky ISSN: 0373-6725 Volume: 30 Issue: 1 Year: 1985 Pages: 11-35 Summary lang: English Summary lang: Czech Summary lang: Russian . Category: math . Summary: The paper deals with an initial problem of a parabolic variational inequality whichcontains a nonlinear elliptic form $a(v,w)$ having a potential $J(v)$, which is twice $G$-differentiable at arbitrary $v\in H^1(\Omega)$. This property of $a(v,w)$ makes it possible to prove convergence of an approximate solution defined by a linearized scheme which is fully discretized - in space by the finite elements method and in time by a one-step finite-difference method. Strong convergence of the approximate solution is proved without any regularity assumptions on the exact solution. An error bound is also derived under the assumption that the exact solution is sufficiently smooth. (English) Keyword: parabolic variational inequalities Keyword: one-step finite difference method Keyword: finite element method Keyword: convergence Keyword: error bound MSC: 49A29 MSC: 49J40 MSC: 65K10 MSC: 65M60 idZBL: Zbl 0574.65066 idMR: MR0779330 DOI: 10.21136/AM.1985.104124 . Date available: 2008-05-20T18:26:33Z Last updated: 2020-07-28 Stable URL: http://hdl.handle.net/10338.dmlcz/104124 . Reference: [1] I. Bock J. Kačur: Application of Rothe's method to parabolic variational inequalities.Math. Slovaca 31 (1981), 429-436. MR 0637970 Reference: [2] J. Céa: Optimization.Dunod, Paris 1971. Zbl 0231.94026, MR 0298892 Reference: [3] P. G. Ciarlet: The Finite Element Method for Elliptic Problems.North-Holland, Amsterdam 1978. Zbl 0383.65058, MR 0520174 Reference: [4] H. Gajewski K. Gröger K. Zacharias: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen.Akademie-Verlag, Berlin 1974. MR 0636412 Reference: [5] J. Haslinger: Finite element analysis for unilateral problems with obstacles on the boundary.Apl. mat. 22 (1977), 180-188. Zbl 0434.65083, MR 0440956 Reference: [6] I. Hlaváček J. Haslinger J. Nečas J. Lovíšek: Solving Yariational Inequalities in Mechanics.Alfa-SNTL, Bratislava-Prague, 1982. (In Slovak.) MR 0755152 Reference: [7] V. Jarník: Integral Calculus II.Nakladatelství ČSAV, Prague 1955. (In Czech.) Reference: [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, MR 0483545, 10.1137/0713050 Reference: [9] J. Kačur: On an approximate solution of variational inequalities.(To appear in Math. Nachr.) MR 0809346 Reference: [10] A. Kufner O. John S. Fučík: Function Spaces.Academia, Prague 1977. MR 0482102 Reference: [11] J. L. Lions: Quelques Méthodes de Résolution des Problèmes aux Lirnites Non Linéaires.Dunod and Gauthier - Villars, Paris 1969. MR 0259693 Reference: [12] J. Nečas: Les Méthodes Directes en Théorie des Équations Elliptiques.Academia, Prague 1967. MR 0227584 Reference: [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. MR 0284016, 10.1002/nme.1620020302 Reference: [14] M. Zlámal: Curved elements in the finite element method I.SIAM J. Numer. Anal. 30 (1973), 229-240. MR 0395263, 10.1137/0710022 Reference: [15] M. Zlámal: Finite element solution of quasistationary nonlinear magnetic field.R.A.I.R.O. Anal. Numer. 16 (1982), 161-191. MR 0661454 Reference: [16] M. Zlámal: A linear scheme for the numerical solution of nonlinear quasistationary magnetic fields.Math. Соmр. 41 (1983), 425-440. MR 0717694 .

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