| Title:
|
Analysis of approximate solutions of coupled dynamical thermoelasticity and related problems (English) |
| Author:
|
Kačur, Jozef |
| Author:
|
Ženíšek, Alexander |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
31 |
| Issue:
|
3 |
| Year:
|
1986 |
| Pages:
|
190-223 |
| Summary lang:
|
English |
| Summary lang:
|
Russian |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| Summary:
|
The authors study problems of existence and uniqueness of solutions of various variational formulations of the coupled problem of dynamical thermoelasticity and of the convergence of approximate solutions of these problems.
First, the semidiscrete approximate solutions is defined, which is obtained by time discretization of the original variational problem by Euler's backward formula. Under certain smoothness assumptions on the date authors prove existence and uniqueness of the solution and establish the rate of convergence $O(\Delta t^{1/2})$ of Rothe's functions in the spaces $C(I;W\frac12(\Omega))$ and $C(I;L_2(\Omega))$ for the displacement components and the temperature, respectively. Regularity of solutions is discussed.
In Part 2 the authors define the fully discretized solution of the original variational problem by Euler's backward formula and the simplest finite elements. Convergence of these approximate solutions is proved.
In Part 3, the weakest assumptions possible are imposed onto the data, which corresponds to a different definition of the variational solution. Existence and uniqueness of the variational solution, as well as convergence of the fully discretized solutions, are proved. (English) |
| Keyword:
|
Euler's backward formula |
| Keyword:
|
regularity |
| Keyword:
|
Rothe's method |
| Keyword:
|
coupled consolidation of clay |
| Keyword:
|
coupled dynamical thermoelasticity |
| Keyword:
|
convergence |
| Keyword:
|
semidiscrete approximate solutions |
| Keyword:
|
time discretization |
| Keyword:
|
discretization in space |
| Keyword:
|
finite element method |
| Keyword:
|
weakest assumptions |
| MSC:
|
35M05 |
| MSC:
|
49J20 |
| MSC:
|
65K10 |
| MSC:
|
65M20 |
| MSC:
|
65M60 |
| MSC:
|
65N30 |
| MSC:
|
73U05 |
| MSC:
|
74F05 |
| MSC:
|
74G30 |
| MSC:
|
74H25 |
| MSC:
|
74S30 |
| idZBL:
|
Zbl 0627.73002 |
| idMR:
|
MR0837733 |
| DOI:
|
10.21136/AM.1986.104199 |
| . |
| Date available:
|
2008-05-20T18:29:58Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104199 |
| . |
| Reference:
|
[1] A. Bermúdez J. M. Viaño: Étude de deux schémas numériques pour les équations de la thermoélasticité.R.A.I.R.O. Numer. Anal. 17 (1983), 121-136. MR 0705448 |
| Reference:
|
[2] B. A. Boley J. H. Weiner: Theory of Thermal Stresses.John Wiley and Sons, New York-London-Sydney, 1960. MR 0112414 |
| Reference:
|
[3] S.-I. Chou C.-C. Wang: Estimates of error in finite element approximate solutions to problems in linear thermoelasticity, Part 1, Computationally coupled numerical schemes.Arch. Rational Mech. Anal. 76 (1981), 263-299. MR 0636964 |
| Reference:
|
[4] P. G. Ciarlet: The Finite Element Method for Elliptic Problems.North-Holland, Amsterdam, 1978. Zbl 0383.65058, MR 0520174 |
| Reference:
|
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| Reference:
|
[6] J. Kačur: Application of Rothe's method to perturbed linear hyperbolic equations and variational inequalities.Czech. Math. J. 34 (109) (1984), 92-105. Zbl 0554.35086, MR 0731982 |
| Reference:
|
[7] J. Kačur: On an approximate solution of variational inequalities.Math. Nachr. 123 (1985), 205-224. MR 0809346, 10.1002/mana.19851230119 |
| Reference:
|
[8] J. Kačur A. Wawruch: On an approximate solution for quasilinear parabolic equations.Czech. Math. J. 27 (102) (1977), 220-241. MR 0605665 |
| Reference:
|
[9] A. Kufner O. John S. Fučík: Function Spaces.Academia, Prague, 1977. MR 0482102 |
| Reference:
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[10] J. L. Lions E. Magenes: Problèmes aux limites non homogènes et applications.Vol. 2. Dunod, Paris, 1968. MR 0247244 |
| Reference:
|
[11] J. Nečas: Les méthodes directes en théorie des équations elliptiques.Academia, Prague 1967. MR 0227584 |
| Reference:
|
[12] M. Zlámal: Curved elements in the finite element method. I.SIAM J. Numer. Anal. 10 (1973), 229-240. MR 0395263, 10.1137/0710022 |
| Reference:
|
[13] A. Ženíšek: Curved triangular finite $C^m$-elements.Apl. Mat. 23 (1978), 346-377. MR 0502072 |
| Reference:
|
[14] A. Ženíšek: Finite element methods for coupled thermoelasticity and coupled consolidation of clay.R.A.I.R.O. Numer. Anal. 18 (1984), 183-205. MR 0743885 |
| Reference:
|
[15] A. Ženíšek: The existence and uniqueness theorem in Biot's consolidation theory.Apl. Mat. 29 (1984), 194-211. Zbl 0557.35005, MR 0747212 |
| Reference:
|
[16] A. Ženíšek: Approximations of parabolic variational inequalities.Apl. Mat. 30 (1985), 11-35. MR 0779330 |
| Reference:
|
[17] A. Ženíšek M. Zlámal: Convergence of a finite element procedure for solving boundary value problems of the fourth order.Int. J. Num. Meth. Engng. 2 (1970), 307-310. MR 0284016, 10.1002/nme.1620020302 |
| . |
