Prox-regularization and solution of ill-posed elliptic variational inequalities

Alexander Kaplan; Rainer Tichatschke

Applications of Mathematics (1997)

  • Volume: 42, Issue: 2, page 111-145
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Kaplan, Alexander, and Tichatschke, Rainer. "Prox-regularization and solution of ill-posed elliptic variational inequalities." Applications of Mathematics 42.2 (1997): 111-145. <http://eudml.org/doc/32972>.

@article{Kaplan1997,
abstract = {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.},
author = {Kaplan, Alexander, Tichatschke, Rainer},
journal = {Applications of Mathematics},
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; contact problem; strong convergence; weakly coercive operators},
language = {eng},
number = {2},
pages = {111-145},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Prox-regularization and solution of ill-posed elliptic variational inequalities},
url = {http://eudml.org/doc/32972},
volume = {42},
year = {1997},
}

TY - JOUR
AU - Kaplan, Alexander
AU - Tichatschke, Rainer
TI - Prox-regularization and solution of ill-posed elliptic variational inequalities
JO - Applications of Mathematics
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 42
IS - 2
SP - 111
EP - 145
AB - 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.
LA - eng
KW - prox-regularization; ill-posed elliptic variational inequalities; finite element methods; two-body contact problem; stable numerical methods; contact problem; strong convergence; weakly coercive operators; contact problem; strong convergence; weakly coercive operators
UR - http://eudml.org/doc/32972
ER -

References

top
  1. Contribution à la résolution numérique des inclusions différentielles, Thèse de 3 cycle, Montpellier, 1985. (1985) 
  2. 10.1007/BF01586942, Math. Programming 51 (1991), 307–331. (1991) MR1130329DOI10.1007/BF01586942
  3. Regularization methods for convex programming problems, Ekonomika i Mat. Metody 11 (1975), 336–342. (Russian) (1975) 
  4. On a method for convex programs using a symmetrical modification of the Lagrange function, Ekonomika i Mat. Metody 12 (1976), 1164–1173 (in Russian). (1976) 
  5. Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman, London, 1984. (1984) MR0773850
  6. 10.1090/S0002-9947-1986-0837797-X, Transactions of the Amer. Math. Soc. 296 (1984), 33–60. (1984) MR0837797DOI10.1090/S0002-9947-1986-0837797-X
  7. 10.1007/BFb0121157, Mathem. Programming Study 30 (1987), 102–126. (1987) Zbl0616.90052MR0874134DOI10.1007/BFb0121157
  8. 10.1007/BF00939042, JOTA 55 (1987), 1–21. (1987) MR0915675DOI10.1007/BF00939042
  9. About the solution of variational inequalities, Soviet Math. Doklady 15 (1974), 1705–1710. (1974) 
  10. 10.1007/BF01442654, Appl. Math. Optim. 15 (1987), 251–277. (1987) MR0879498DOI10.1007/BF01442654
  11. 10.1007/BF02761171, Israel J. of Math. 29 (1978), 329–345. (1978) MR0491922DOI10.1007/BF02761171
  12. The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. (1978) Zbl0383.65058MR0520174
  13. Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972. (1972) MR0464857
  14. Boundary Value Problems of Elasticity with Unilateral Constraints, Springer-Verlag, Berlin 1972. 
  15. Practical Methods of Optimization, J. Wiley & Sons, Chichester-New York-Brisbane-Toronto-Singapure, 1990. (1990) MR1867781
  16. 10.1287/opre.31.1.101, Operations Research 31 (1983), 101–113. (1983) Zbl0495.90066MR0695607DOI10.1287/opre.31.1.101
  17. Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981. (1981) Zbl1169.65064MR0635927
  18. 10.1137/0329022, SIAM. J. Control and Optim. 29 (1991), 403–419. (1991) MR1092735DOI10.1137/0329022
  19. 10.1090/qam/1146620, Quarterly of Applied Mathematics 50 (1992), 11–25. (1992) Zbl0743.73025MR1146620DOI10.1090/qam/1146620
  20. 10.1137/0328029, SIAM J. on Control and Optimization 28 (1990), 503–512. (1990) Zbl0699.49037MR1047419DOI10.1137/0328029
  21. The reciprocal variational approach to the Signorini problem with friction, Approximation Results. Proc. Roy. Soc. Edinburgh 98A (1984), 365–383. (1984) MR0768357
  22. Numerical Solution of Variational Inequalities, Springer-Verlag, Berlin-Heidelberg-New York, 1988. (1988) 
  23. 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. (1970) MR0252844
  24. 10.1007/BF00253807, Computational Optimization and Application 1 (1992), 207–226. (1992) Zbl1168.47046MR1226336DOI10.1007/BF00253807
  25. Algorithm for convex programming using a smoothing for exact penalty functions, Sibirskij Mat. Journal 23 (1982), 53–64. (Russian) (1982) MR0668335
  26. 10.1080/02331939208843852, Optimization 26 (1992), 187–214. (1992) MR1236607DOI10.1080/02331939208843852
  27. 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. (1993, 341–356) MR1241232
  28. 10.1080/02331939508844083, Optimization 33 (1995), 287–319. (1995) MR1332941DOI10.1080/02331939508844083
  29. Solution of variational inequalities by mathematical programming methods, Ph. D. Thesis, Novosibirsk, 1987. (Russian) (1987) 
  30. The proximal algorithm, Internat. Series of Numerical Mathematics 87 (1989), 73–87. (1989) Zbl0692.90079MR1001168
  31. 10.1137/0322019, SIAM J. on Control and Optimization 22 (1984), 277–293. (1984) MR0732428DOI10.1137/0322019
  32. Régularisation d’inéquations variationelles par approximations successives, RIRO 4 (1970), 154–159. (1970) MR0298899
  33. 10.1016/0001-8708(69)90009-7, Advances in Mathematics 3 (1969), 510–585. (1969) Zbl0192.49101MR0298508DOI10.1016/0001-8708(69)90009-7
  34. Une méthode de pénalisation exponentielle associée à une régularisation proximale, Bull. Soc. Roy. Sc. de Liège 56 (1987), 181–192. (1987) MR0911355
  35. Mathematical Theory of Elastic and Elasto-plastic Bodies, An Introduction, Elsevier, Amsterdam, 1981. (1981) MR0600655
  36. 10.1090/S0002-9904-1967-11761-0, Bull. Amer. Math. Soc. 73 (1967), 591–597. (1967) MR0211301DOI10.1090/S0002-9904-1967-11761-0
  37. 10.1007/BF00534623, Ing. Archiv 44 (1975), 421–432. (1975) Zbl0332.73018MR0426584DOI10.1007/BF00534623
  38. Inequality Problems in Mechanics and Applications, Birkhäuser-Verlag, Boston-Basel-Stuttgart, 1985. (1985) Zbl0579.73014MR0896909
  39. Introduction to Optimization, Optimization Software, Inc. Publ. Division, New York, 1987. (1987) MR1099605
  40. 10.1137/0314056, SIAM J. Control and Optimization 15 (1976), 877–898. (1976) Zbl0358.90053MR0410483DOI10.1137/0314056
  41. 10.1287/moor.1.2.97, Math. Oper. Res. 1 (1976), 97–116. (1976) MR0418919DOI10.1287/moor.1.2.97
  42. 10.1051/m2an/1977110201971, RAIRO Anal. Numér. 11 (1977), 197–208. (1977) MR0488860DOI10.1051/m2an/1977110201971
  43. 10.1007/BF01448388, Applied Mathematics and Optimization 10 (1983), 247–265. (1983) Zbl0524.90072MR0722489DOI10.1007/BF01448388
  44. Application of the method of partial inverses to convex programming: Decomposition, Math. Programming 32 (1985), 199–223. (1985) MR0793690
  45. Algorithme du point proximal perturbé et applications, Prépublication No. 90-015, Inst. de Mathématique, Univ. de Liège, 1990. (1990) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.