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

Abstract

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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 ( Δ t 1 / 2 ) of Rothe’s functions in the spaces C ( I ; W 1 2 ( Ω ) ) and C ( I ; L 2 ( Ω ) ) 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.

How to cite

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Kačur, 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

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  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. (1981) Zbl0494.73071MR0636964
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  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. (1984) Zbl0539.73005MR0743885
  15. A. Ženíšek, The existence and uniqueness theorem in Biot's consolidation theory, Apl. Mat. 29 (1984), 194-211. (1984) Zbl0557.35005MR0747212
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