Geodesic
The method of successive approximations for solving quasi-variational Signorini inequality
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2017), pp. 44-52.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the method of successive approximations for solving the semicoercive quasi-variational Signorini inequality corresponding to the contact problem of elasticity theory with friction. Each outer step of the iterative process involves the Signorini problem with given friction, which is solved by the Uzawa method based on an iterative proximal regularization of a modified Lagrangian functional. We investigate stabilization sequence of auxiliary finite element solutions on the outer steps of the method of successive approximations and present the results of numerical calculation.
Keywords: contact problem of the elasticity theory, Lagrangian functional, saddle point, Uzawa method, proximal regularization, finite element method. 
@article{IVM_2017_1_a4,
     author = {R. V. Namm and G. I. Tsoi},
     title = {The method of successive approximations for solving quasi-variational {Signorini} inequality},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {44--52},
     publisher = {mathdoc},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2017_1_a4/}
}
TY  - JOUR
AU  - R. V. Namm
AU  - G. I. Tsoi
TI  - The method of successive approximations for solving quasi-variational Signorini inequality
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2017
SP  - 44
EP  - 52
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2017_1_a4/
LA  - ru
ID  - IVM_2017_1_a4
ER  - 
%0 Journal Article
%A R. V. Namm
%A G. I. Tsoi
%T The method of successive approximations for solving quasi-variational Signorini inequality
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2017
%P 44-52
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2017_1_a4/
%G ru
%F IVM_2017_1_a4
R. V. Namm; G. I. Tsoi. The method of successive approximations for solving quasi-variational Signorini inequality. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2017), pp. 44-52. http://geodesic.mathdoc.fr/item/IVM_2017_1_a4/

[1] Glavachek I., Gaslinger Ya., Nechas I., Lovishek Ya., Reshenie variatsionnykh neravenstv v mekhanike, Mir, M., 1986

[2] Kikuchi N., Oden T., Contact problem in elasticity: a study of variational inequalities and finite element methods, SIAM, Philadelphia, 1988 | MR

[3] Haslinger J., Hlavacek I., Necas J., “Numerical methods for unilateral problems in solid mechanics”, Handbook of Numerical Analysis, v. IV, eds. P. G. Ciarlet, J. L. Lions, North Holland, 1996, 313–486 | MR

[4] Vikhtenko E. M., Namm R. V., “Skhema dvoistvennosti dlya resheniya polukoertsitivnoi zadachi Sinorini s treniem”, Zhurn. vychisl. matem. i matem. fiz., 47:12 (2007), 2023–2036 | MR

[5] Vikhtenko E. M., Namm R. V., “Iterativnaya proksimalnaya regulyarizatsiya modifitsirovannogo funktsionala Lagranzha dlya resheniya polukoertsitivnogo kvazivariatsionnogo neravenstva Sinorini”, Zhurn. vychisl. matem. i matem. fiz., 48:9 (2008), 1–9 | MR

[6] Kravchuk A. S., Variatsionnye i kvazivariatsionnye neravenstva v mekhanike, MGAPI, M., 1997

[7] Antipin A. S., Gradientnyi i ekstragradientnyi podkhody v bilineinom ravnovesnom programmirovanii, VTs RAN, M., 2002

[8] Antipin A. S., Golikov A. I., Khoroshilova E. V., “Funktsiya chuvstvitelnosti, ee svoistva i prilozheniya”, Zhurn. vychisl. matem. i matem. fiz., 51:12 (2011), 1–17 | MR

[9] Glovinski R., Lions Zh. L., Tremoler R., Chislennoe issledovanie variatsionnykh neravenstv, Mir, M., 1979

[10] Glowinski R., Numerical methods for nonlinear variational problems, Springer, New York, 1984 | MR | Zbl

[11] Vu G., Namm R. V., Sachkov S. A., “Iteratsionnyi metod poiska sedlovoi tochki dlya polukoertsitivnoi zadachi Sinorini, osnovannyi na modifitsirovannom funktsionale Lagranzha”, Zhurn. vychisl. matem. i matem. fiz., 46:1 (2006), 26–36 | MR | Zbl

[12] Vu G., Kim S., Namm R. V., Sachkov S. A., “Metod iterativnoi proksimalnoi regulyarizatsii dlya poiska sedlovoi tochki v polukoertsitivnoi zadache Sinorini”, Zhurn. vychisl. matem. i matem. fiz., 46:11 (2006), 2024–2031 | MR

[13] Vikhtenko E. M., Vu G., Namm R. V., “Metody resheniya polukoertsitivnykh variatsionnykh neravenstv mekhaniki na osnove modifitsirovannykh funktsionalov Lagranzha”, Dalnevostochn. matem. zhurn., 14:1 (2014), 6–17 | MR | Zbl

[14] Vikhtenko E. M., Vu G., Namm R. V., “Funktsionaly chuvstvitelnosti v kontaktnykh zadachakh teorii uprugosti”, Zhurn. vychisl. matem. i matem. fiz., 54:7 (2014), 1218–1228 | DOI | MR | Zbl

[15] Konnov I., Gwinner J., “A strongly convergent combined relaxation method in Hilbert spaces”, Numerical Funct. Anal. Optim., 35:7–9 (2014), 1066–1077 | DOI | MR | Zbl

[16] Marchuk G. I., Agoshkov V. I., Vvedenie v proektsionno-setochnye metody, Fizmatlit, M., 1981

[17] Zenkevich O., Morgan K., Konechnye elementy i approksimatsiya, Mir, M., 1986

[18] Namm R. V., Sachkov S. A., “Reshenie kvazivariatsionnogo neravenstva Sinorini metodom posledovatelnykh priblizhenii”, Zhurn. vychisl. matem. i matem. fiz., 49:5 (2009), 805–814 | MR | Zbl