# Analysis of approximate solutions of coupled dynamical thermoelasticity and related problems

Jozef Kačur; Alexander Ženíšek

Aplikace matematiky (1986)

- Volume: 31, Issue: 3, page 190-223
- ISSN: 0862-7940

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topKačur, Jozef, and Ženíšek, Alexander. "Analysis of approximate solutions of coupled dynamical thermoelasticity and related problems." Aplikace matematiky 31.3 (1986): 190-223. <http://eudml.org/doc/15448>.

@article{Kačur1986,

abstract = {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 $C(I;W\frac\{1\}\{2\}(\Omega ))$ 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.},

author = {Kačur, Jozef, Ženíšek, Alexander},

journal = {Aplikace matematiky},

keywords = {Euler's backward formula; regularity; Rothe's method; coupled consolidation of clay; coupled dynamical thermoelasticity; convergence; semidiscrete approximate solutions; time discretization; discretization in space; finite element method; weakest assumptions; Euler's backward formula; regularity; Rothe's method; coupled consolidation of clay; coupled dynamical thermoelasticity; convergence; semidiscrete approximate solutions; time discretization; discretization in space; finite element method; weakest assumptions},

language = {eng},

number = {3},

pages = {190-223},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Analysis of approximate solutions of coupled dynamical thermoelasticity and related problems},

url = {http://eudml.org/doc/15448},

volume = {31},

year = {1986},

}

TY - JOUR

AU - Kačur, Jozef

AU - Ženíšek, Alexander

TI - Analysis of approximate solutions of coupled dynamical thermoelasticity and related problems

JO - Aplikace matematiky

PY - 1986

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 31

IS - 3

SP - 190

EP - 223

AB - 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 $C(I;W\frac{1}{2}(\Omega ))$ 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.

LA - eng

KW - Euler's backward formula; regularity; Rothe's method; coupled consolidation of clay; coupled dynamical thermoelasticity; convergence; semidiscrete approximate solutions; time discretization; discretization in space; finite element method; weakest assumptions; Euler's backward formula; regularity; Rothe's method; coupled consolidation of clay; coupled dynamical thermoelasticity; convergence; semidiscrete approximate solutions; time discretization; discretization in space; finite element method; weakest assumptions

UR - http://eudml.org/doc/15448

ER -

## References

top- 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. (1983) MR0705448
- B. A. Boley J. H. Weiner, Theory of Thermal Stresses, John Wiley and Sons, New York-London-Sydney, 1960. (1960) MR0112414
- 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. (1981) MR0636964
- P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. (1978) Zbl0383.65058MR0520174
- G. Duvaut J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal. 46 (1972), 241-279. (1972) MR0346289
- J. Kačur, Application of Rothe's method to perturbed linear hyperbolic equations and variational inequalities, Czech. Math. J. 34 (109) (1984), 92-105. (1984) Zbl0554.35086MR0731982
- J. Kačur, 10.1002/mana.19851230119, Math. Nachr. 123 (1985), 205-224. (1985) MR0809346DOI10.1002/mana.19851230119
- J. Kačur A. Wawruch, On an approximate solution for quasilinear parabolic equations, Czech. Math. J. 27 (102) (1977), 220-241. (1977) MR0605665
- A. Kufner O. John S. Fučík, Function Spaces, Academia, Prague, 1977. (1977) MR0482102
- J. L. Lions E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 2. Dunod, Paris, 1968. (1968) MR0247244
- J. Nečas, Les méthodes directes en théorie des équations elliptiques, Academia, Prague 1967. (1967) MR0227584
- M. Zlámal, 10.1137/0710022, SIAM J. Numer. Anal. 10 (1973), 229-240. (1973) MR0395263DOI10.1137/0710022
- A. Ženíšek, Curved triangular finite ${C}^{m}$-elements, Apl. Mat. 23 (1978), 346-377. (1978) MR0502072
- A. Ženíšek, Finite element methods for coupled thermoelasticity and coupled consolidation of clay, R.A.I.R.O. Numer. Anal. 18 (1984), 183-205. (1984) MR0743885
- A. Ženíšek, The existence and uniqueness theorem in Biot's consolidation theory, Apl. Mat. 29 (1984), 194-211. (1984) Zbl0557.35005MR0747212
- A. Ženíšek, Approximations of parabolic variational inequalities, Apl. Mat. 30 (1985), 11-35. (1985) MR0779330
- A. Ženíšek M. Zlámal, 10.1002/nme.1620020302, Int. J. Num. Meth. Engng. 2 (1970), 307-310. (1970) MR0284016DOI10.1002/nme.1620020302

## Citations in EuDML Documents

top- Jiří V. Horák, On solvability of one special problem of coupled thermoelasticity. I. Classical boundary conditions and steady sources
- Marián Slodička, Application of Rothe's method to evolution integrodifferential systems
- Marián Slodička, An investigation of convergence and error estimate of approximate solution for a quasilinear parabolic integrodifferential equation

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