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Analysis of approximate solutions of coupled dynamical thermoelasticity and related problems. (English) Zbl 0627.73002
The paper deals with existence and uniqueness of solutions of various variational formulations of coupled dynamical thermoelasticity and with the convergence of approximate solutions. The first part deals with semidiscrete approximate solutions which are obtained by time discretization of the original variational problem. In the second part the authors consider in addition discretization in space by the finite element method. In the third part the weakest assumptions possible are imposed which require a different definition of the variational solution.
Reviewer: R.Schmidt

MSC:
74F05 Thermal effects in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
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References:
[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.
[2] B. A. Boley J. H. Weiner: Theory of Thermal Stresses. John Wiley and Sons, New York-London-Sydney, 1960. · Zbl 0095.18407
[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. · Zbl 0494.73071
[4] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. · Zbl 0383.65058
[5] G. Duvaut J. L. Lions: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Rational Mech. Anal. 46 (1972), 241-279. · Zbl 0264.73027
[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
[7] J. Kačur: On an approximate solution of variational inequalities. Math. Nachr. 123 (1985), 205-224.
[8] J. Kačur A. Wawruch: On an approximate solution for quasilinear parabolic equations. Czech. Math. J. 27 (102) (1977), 220-241. · Zbl 0377.35036
[9] A. Kufner O. John S. Fučík: Function Spaces. Academia, Prague, 1977.
[10] J. L. Lions E. Magenes: Problèmes aux limites non homogènes et applications. Vol. 2. Dunod, Paris, 1968. · Zbl 0212.43801
[11] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague 1967. · Zbl 1225.35003
[12] M. Zlámal: Curved elements in the finite element method. I. SIAM J. Numer. Anal. 10 (1973), 229-240. · Zbl 0285.65067
[13] A. Ženíšek: Curved triangular finite \(C^m\)-elements. Apl. Mat. 23 (1978), 346-377. · Zbl 0404.35041
[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. · Zbl 0539.73005
[15] A. Ženíšek: The existence and uniqueness theorem in Biot’s consolidation theory. Apl. Mat. 29 (1984), 194-211. · Zbl 0557.35005
[16] A. Ženíšek: Approximations of parabolic variational inequalities. Apl. Mat. 30 (1985), 11-35. · Zbl 0574.65066
[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. · Zbl 0256.65055
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