Geodesic
Estimates of deviations from exact solutions of variational problems with linear growth functionals
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Tome 370 (2009), pp. 132-150 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this paper, we derive estimates of deviations from exact solutions of variational problems with linear growth functionals. Since original variational problem may have no minimizer in a reflexive Banach space, the estimates are presented in terms of the dual problem. We prove the consistency of these estimates and obtain their computationally convenient forms. Bibl. – 36 titles. 
@article{ZNSL_2009_370_a7,
     author = {S. I. Repin},
     title = {Estimates of deviations from exact solutions of variational problems with linear growth functionals},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {132--150},
     year = {2009},
     volume = {370},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a7/}
}
TY  - JOUR
AU  - S. I. Repin
TI  - Estimates of deviations from exact solutions of variational problems with linear growth functionals
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2009
SP  - 132
EP  - 150
VL  - 370
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a7/
LA  - en
ID  - ZNSL_2009_370_a7
ER  - 
%0 Journal Article
%A S. I. Repin
%T Estimates of deviations from exact solutions of variational problems with linear growth functionals
%J Zapiski Nauchnykh Seminarov POMI
%D 2009
%P 132-150
%V 370
%U http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a7/
%G en
%F ZNSL_2009_370_a7
S. I. Repin. Estimates of deviations from exact solutions of variational problems with linear growth functionals. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 40, Tome 370 (2009), pp. 132-150. http://geodesic.mathdoc.fr/item/ZNSL_2009_370_a7/

[1] G. Acosta, R. Duran, “An optimal Poincaré inequality in $L^1$ for convex domains”, Proceesings of the American Mathematical Society, 132 (2003), 195–202 | DOI | MR

[2] G. Anzellotti, M. Giaquinta, “Existence of the displacements field for an elastic-plastic body subjected to Hencky's law and von Mises yield condition”, Manuscripta Math., 32 (1980), 101–136 | DOI | MR | Zbl

[3] M. Bildhauer, Convex Variational Problems, Lecture Notes in Mathematics, 1818, Springer, Berlin, 2003 | DOI | MR | Zbl

[4] M. Bildhauer, M. Fuchs, S. Repin, “The elasto-plastic torsion problem: a posteriori estimates for approximate solutions”, Numer. Functional Analysis and Optimization, 30 (2009), 653–664 | DOI | MR | Zbl

[5] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, North-Holland, New York, 1976 | MR | Zbl

[6] R. Finn, Equilibrium capillary surfaces, Springer, New York, 1986 | MR | Zbl

[7] J. Freshe, J. Malek, Asymptotic error estimates for finite element approximations in elasto-perfect plasticity, Preprint, 1994

[8] M. Fuchs, S. Repin, “Functional a posteriori error estimates for variational inequalities describing the stationary flow of certain viscous incompressible fluids”, Math. Mathematical Methods in Applied Sciences (M2AS) (to appear)

[9] M. Fuchs, G. A. Seregin, Variational methods for problems from plasticity theory and for generalized Newtonian fluids, Lect. Notes in Mathematics, 1749, Springer-Verlag, Berlin, 2000 | DOI | MR | Zbl

[10] M. Giaquinta, G. Modica, J. Souček, “Functionals with linear growth in the calculus of variations”, Comm. Math. Univ. Carolinae, 20 (1979), 143–171 | MR

[11] E. Giusti, Minimal surfaces and functions of bounded variation, Birkhauser, Boston, 1984 | MR | Zbl

[12] C. Johnson, V. Thomee, “Error estimates for a finite element approximation of a minimal surface”, Math. Comput., 29 (1975), 343–349 | DOI | MR

[13] C. Jouron, “Résolution nummérique du probleme des surfaces minima”, Arch. Rat. Mech. Anal., 59 (1975), 311–341 | DOI | MR

[14] O. A. Ladyzhenskaya, N. N. Uraltseva, “Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations”, Comm. Pure. Appl. Math., 23 (1970), 677–703 | DOI | MR | Zbl

[15] S. G. Mikhlin, Variational Methods in Mathematical Physics, Pergamon, Oxford, 1964 | MR | Zbl

[16] P. Mosolov, V. Myasnikov, Mechanics of rigid plastic bodies, Nauka, M., 1981 (Russian) | MR | Zbl

[17] P. Neittaanmaki, S. Repin, V. Rivkind, “Discontinuous finite element approximations for functionals with linear growth”, East-West J. Numer. Math., 2:3 (1994), 212–228 | MR

[18] L. E. Payne, H. F. Weinberger, “An optimal Poincaré inequality for convex domains”, Arch. Rat. Mech. Anal., 5 (1960), 286–292 | DOI | MR | Zbl

[19] S. Repin, “Variational-difference method for problems of perfect plasticity using discontinuous conventional finite elements method”, Zh. Vychisl. Mat. i Mat. Fiz., 28 (1988), 449–453 (Russian) | Zbl

[20] S. Repin, “Variational-difference method for solving problems with functionals of linear growth”, Zh. Vychisl. Mat. i Mat. Fiz., 29:5 (1989), 693–708 (Russian) | MR

[21] S. Repin, “Numerical analysis of no nonsmooth variational problems of perfect plasticity”, Russ. J. Numer. Anal. Math. Modell., 9:1 (1994), 61–74 | DOI | MR | Zbl

[22] S. Repin, “A priori error estimates of variational-difference methods for Hencky plasticity problems”, Zap. Nauchn. Semin. V. A. Steklov Mathematical Institute (POMI), 221, 1995, 226–234 | MR | Zbl

[23] S. Repin, “Errors of finite element methods for perfectly elasto-plastic problems”, Math. Models Methods Appl. Sci., 6 (1996), 587–604 | DOI | MR | Zbl

[24] S. Repin, “A posteriori error estimation for nonlinear variational problems by duality theory”, Zap. Nauchn. Semin. V. A. Steklov Mathematical Institute (POMI), 243, 1997, 201–214 | MR | Zbl

[25] S. Repin, “Estimates of deviations from exact solutions of variational inequalities based upon Payne–Weinberger inequality”, J. Math. Sci. (N.Y.), 157 (2009), 874–884 | DOI | MR | Zbl

[26] St.-Petersburg Mathematical Journal, 11:4 (2000), 651–672 | MR | Zbl

[27] S. Repin, “A posteriori error estimates for variational problems with uniformly convex functionals”, Math. Comput., 69:230 (2000), 481–500 | DOI | MR | Zbl

[28] Amer. Math. Soc. Transl. Ser. 2, 209, Amer. Math. Soc., Providence, RI, 2003, 143–171 | MR | MR | Zbl

[29] St.-Petersburg Mathematical Journal, 11 (2000), 651–672 | MR | Zbl

[30] S. Repin, A posteriori estimates for partial differential equations, Walter de Gruyter, Berlin, 2008 | MR

[31] S. Repin, G. Seregin, “Error estimates for stresses in the finite element analysis of the two-dimensional elasto-plastic problems”, Internat. J. Engrg. Sci., 33 (1995), 255–268 | DOI | MR | Zbl

[32] S. Repin, G. Seregin, “Existence of a weak solutions of the minimax problem in Coulomb-Mohr plasticity”, Nonlinear evolution equations, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995, 189–220 | MR | Zbl

[33] G. Seregin, “Variational-difference scheme for problems in the mechanics of ideally elastoplastic media”, Zh. Vychisl. Mat. i Mat. Fiz., 25:2 (1985), 237–253 | MR | Zbl

[34] Sov. Phys. Dokl., 29:5 (1984), 396–398 | MR | Zbl

[35] P. Suquet, “Existence et regularite des solutions des equations de la plasticite parfaite”, C. R. Acad. Sc. Paris Serie D, 286 (1978), 1201–1204 | MR | Zbl

[36] R. Temam, Problemes mathématiques en plasticité, Bordas, Paris, 1983 | MR | Zbl