| 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 |
| . |
