Previous |  Up |  Next

Article

Keywords:
prox-regularization; ill-posed elliptic variational inequalities; finite element methods; two-body contact problem; stable numerical methods; contact problem; strong convergence; weakly coercive operators
Summary:
In this paper new methods for solving elliptic variational inequalities with weakly coercive operators are considered. The use of the iterative prox-regularization coupled with a successive discretization of the variational inequality by means of a finite element method ensures well-posedness of the auxiliary problems and strong convergence of their approximate solutions to a solution of the original problem. In particular, regularization on the kernel of the differential operator and regularization with respect to a weak norm of the space are studied. These approaches are illustrated by two nonlinear problems in elasticity theory.
References:
[1] P. Alart: Contribution à la résolution numérique des inclusions différentielles. Thèse de 3 cycle, Montpellier, 1985.
[2] P. Alart and B. Lemaire: Penalization in non-classical convex programming via variational convergence. Math. Programming 51 (1991), 307–331. DOI 10.1007/BF01586942 | MR 1130329
[3] A.S. Antipin: Regularization methods for convex programming problems. Ekonomika i Mat. Metody 11 (1975), 336–342. (Russian)
[4] A.S. Antipin: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Mat. Metody 12 (1976), 1164–1173 (in Russian).
[5] H. Attouch: Variational Convergence for Functions and Operators, Applicable Mathematics Series. Pitman, London, 1984. MR 0773850
[6] H. Attouch and R.J.B. Wets: Isometries for the Legendre-Fenchel transformation. Transactions of the Amer. Math. Soc. 296 (1984), 33–60. DOI 10.1090/S0002-9947-1986-0837797-X | MR 0837797
[7] A. Auslender: Numerical methods for nondifferentiable convex optimization. Mathem. Programming Study 30 (1987), 102–126. DOI 10.1007/BFb0121157 | MR 0874134 | Zbl 0616.90052
[8] A. Auslender; J.P. Crouzeix and P. Fedit: Penalty proximal methods in convex programming. JOTA 55 (1987), 1–21. DOI 10.1007/BF00939042 | MR 0915675
[9] A.B. Bakushinski and B.T. Polyak: About the solution of variational inequalities. Soviet Math. Doklady 15 (1974), 1705–1710.
[10] P. Boiri; F. Gastaldi and D. Kinderlehrer: Existence, uniqueness and regularity results for the two-body contact problem. Appl. Math. Optim. 15 (1987), 251–277. DOI 10.1007/BF01442654 | MR 0879498
[11] H. Brezis and P.L. Lions: Produits infinis de resolvantes. Israel J. of Math. 29 (1978), 329–345. DOI 10.1007/BF02761171 | MR 0491922
[12] Ph.G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. MR 0520174 | Zbl 0383.65058
[13] G. Duvaut and J.L. Lions: Les Inéquations en Mécanique et en Physique. Dunod, Paris, 1972. MR 0464857
[14] G. Fichera: Boundary Value Problems of Elasticity with Unilateral Constraints. Springer-Verlag, Berlin 1972.
[15] R. Fletcher: Practical Methods of Optimization. J. Wiley & Sons, Chichester-New York-Brisbane-Toronto-Singapure, 1990. MR 1867781
[16] M. Fukushima: An outer approximation algorithm for solving general convex programs. Operations Research 31 (1983), 101–113. DOI 10.1287/opre.31.1.101 | MR 0695607 | Zbl 0495.90066
[17] R. Glowinski; J.L. Lions and R. Trémolières: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam, 1981. MR 0635927 | Zbl 1169.65064
[18] O. Güler: On the convergence of the proximal point algorithm for convex minimization. SIAM. J. Control and Optim. 29 (1991), 403–419. DOI 10.1137/0329022 | MR 1092735
[19] J. Gwinner: Finite-element convergence for contact problems in plane linear elastostatics. Quarterly of Applied Mathematics 50 (1992), 11–25. MR 1146620 | Zbl 0743.73025
[20] C.D. Ha: A generalization of the proximal point algorithm. SIAM J. on Control and Optimization 28 (1990), 503–512. DOI 10.1137/0328029 | MR 1047419 | Zbl 0699.49037
[21] J. Haslinger and P.D. Panagiotopoulos: The reciprocal variational approach to the Signorini problem with friction. Approximation Results. Proc. Roy. Soc. Edinburgh 98A (1984), 365–383. MR 0768357
[22] I. Hlaváček; J. Haslinger; I. Nečas and J. Lovišek: Numerical Solution of Variational Inequalities. Springer-Verlag, Berlin-Heidelberg-New York, 1988.
[23] I. Hlaváček and I. Nečas: On inequalities of Korn’s type, I. Boundary-value problems for elliptic systems of partial differential equations. Arch. Rat. Mech. Anal. 306 (1970), 305–311. MR 0252844
[24] S. Ibaraki; M. Fukushima and T. Ibaraki: Primal-dual proximal point algorithm for linearly constrained convex programming problems. Computational Optimization and Application 1 (1992), 207–226. DOI 10.1007/BF00253807 | MR 1226336 | Zbl 1168.47046
[25] A. Kaplan: Algorithm for convex programming using a smoothing for exact penalty functions. Sibirskij Mat. Journal 23 (1982), 53–64. (Russian) MR 0668335
[26] A. Kaplan and R. Tichatschke: Variational inequalities and convex semi-infinite programming problems. Optimization 26 (1992), 187–214. DOI 10.1080/02331939208843852 | MR 1236607
[27] A. Kaplan and R. Tichatschke: Regularized penalty methods for semi-infinite programming problems. Proc. of the 3rd Intern. Conf. On Parametric Optimization., F. Deutsch. B. Brosowski and J. Guddat (eds.), Ser. Approximation and Optimization, vol. 3, P. Lang Verlag, Frankfurt/Main, 1993, 341–356. MR 1241232
[28] A. Kaplan and R. Tichatschke: Multi-step prox-regularization methods for solving convex variational problems. Optimization 33 (1995), 287–319. DOI 10.1080/02331939508844083 | MR 1332941
[29] V.I. Kustova: Solution of variational inequalities by mathematical programming methods. Ph. D. Thesis, Novosibirsk, 1987. (Russian)
[30] B. Lemaire: The proximal algorithm. Internat. Series of Numerical Mathematics 87 (1989), 73–87. MR 1001168 | Zbl 0692.90079
[31] E.J. Luque: Asymptotic convergence analysis of the proximal point algorithms. SIAM J. on Control and Optimization 22 (1984), 277–293. DOI 10.1137/0322019 | MR 0732428
[32] B. Martinet: Régularisation d’inéquations variationelles par approximations successives. RIRO 4 (1970), 154–159. MR 0298899
[33] U. Mosco: Convergence of convex sets and of solutions of variational inequalities. Advances in Mathematics 3 (1969), 510–585. DOI 10.1016/0001-8708(69)90009-7 | MR 0298508 | Zbl 0192.49101
[34] K. Mouallif and P. Tossings: Une méthode de pénalisation exponentielle associée à une régularisation proximale. Bull. Soc. Roy. Sc. de Liège 56 (1987), 181–192. MR 0911355
[35] J. Nečas and I. Hlaváček: Mathematical Theory of Elastic and Elasto-plastic Bodies, An Introduction. Elsevier, Amsterdam, 1981. MR 0600655
[36] Z. Opial: Weak convergence of the successive approximations for nonexpansive mappings in Banach spaces. Bull. Amer. Math. Soc. 73 (1967), 591–597. DOI 10.1090/S0002-9904-1967-11761-0 | MR 0211301
[37] P.D. Panagiotopoulos: A nonlinear programming approach to the unilateral contact- and friction-boundary value problem in the theory of elasticity. Ing. Archiv 44 (1975), 421–432. DOI 10.1007/BF00534623 | MR 0426584 | Zbl 0332.73018
[38] P.D. Panagiotopoulos: Inequality Problems in Mechanics and Applications. Birkhäuser-Verlag, Boston-Basel-Stuttgart, 1985. MR 0896909 | Zbl 0579.73014
[39] B.T. Polyak: Introduction to Optimization. Optimization Software, Inc. Publ. Division, New York, 1987. MR 1099605
[40] R.T. Rockafellar: Monotone operators and the proximal point algorithm. SIAM J. Control and Optimization 15 (1976), 877–898. DOI 10.1137/0314056 | MR 0410483 | Zbl 0358.90053
[41] R.T. Rockafellar: Augmented Lagrange multiplier functions and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1 (1976), 97–116. DOI 10.1287/moor.1.2.97 | MR 0418919
[42] F. Scarpini and M.A. Vivaldi: Error estimates for the approximation of some unilateral problems. RAIRO Anal. Numér. 11 (1977), 197–208. MR 0488860
[43] J.E. Spingarn: Partial inverse of a monotone operator. Applied Mathematics and Optimization 10 (1983), 247–265. DOI 10.1007/BF01448388 | MR 0722489 | Zbl 0524.90072
[44] J.E. Spingarn: Application of the method of partial inverses to convex programming: Decomposition. Math. Programming 32 (1985), 199–223. MR 0793690
[45] P. Tossings: Algorithme du point proximal perturbé et applications. Prépublication No. 90-015, Inst. de Mathématique, Univ. de Liège, 1990.
Partner of
EuDML logo