December  2013, 6(6): 1507-1524. doi: 10.3934/dcdss.2013.6.1507

Approximation results and subspace correction algorithms for implicit variational inequalities

1. 

Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest

2. 

LMA, Aix-Marseille University, CNRS, UPR 7051, Centrale Marseille, F-13402 Marseille Cedex 20, France

Received  June 2012 Revised  September 2012 Published  April 2013

This paper deals with the mathematical analysis and the subspace approximation of a system of variational inequalities representing a unified approach to several quasistatic contact problems in elasticity. Using an implicit time discretization scheme and some estimates, convergence properties of the incremental solutions and existence results are presented for a class of abstract implicit evolution variational inequalities involving a nonlinear operator. To solve the corresponding semi-discrete and the fully discrete problems, some general subspace correction algorithms are proposed, for which global convergence is analyzed and error estimates are established.
Citation: Lori Badea, Marius Cocou. Approximation results and subspace correction algorithms for implicit variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1507-1524. doi: 10.3934/dcdss.2013.6.1507
References:
[1]

L. Badea, Convergence rate of a multiplicative Schwarz method for strongly nonlinear inequalities,, in, (2003), 31.

[2]

L. Badea, Convergence rate of a Schwarz multilevel method for the constrained minimization of non-quadratic functionals,, SIAM J. Numer. Anal., 44 (2006), 449. doi: 10.1137/S003614290342995X.

[3]

L. Badea, Schwarz methods for inequalities with contraction operators,, J. Comp. Appl. Math., 215 (2008), 196. doi: 10.1016/j.cam.2007.04.004.

[4]

L. Badea and R. Krause, One- and two-level Schwarz methods for variational inequalities of the second kind and their application to frictional contact,, Numer. Math., 120 (2012), 573. doi: 10.1007/s00211-011-0423-y.

[5]

L. Badea and R. Krause, One- and two-level multiplicative Schwarz methods for variational and quasi-variational inequalities of the second kind. Part I - general convergence results,, INS Preprint, (0804).

[6]

A. Capatina and M. Cocou, Internal approximation of quasi-variational inequalities,, Numer. Math., 59 (1991), 385. doi: 10.1007/BF01385787.

[7]

A. Capatina, M. Cocou and M. Raous, A class of implicit variational inequalities and applications to frictional contact,, Math. Meth. Appl. Sci., 32 (2009), 1804. doi: 10.1002/mma.1112.

[8]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", Studies in Mathematics and its Applications, (1978).

[9]

M. Cocou, E. Pratt and M. Raous, Formulation and approximation of quasistatic frictional contact,, Int. J. Engrg. Sci., 34 (1996), 783. doi: 10.1016/0020-7225(95)00121-2.

[10]

R. Glowinski, "Numerical Methods for Nonlinear Variational Problems,", Springer Series in Computational Physics, (1984).

[11]

R. Glowinski, J.-L. Lions and R. Trémolières, "Analyse Numérique des Inéquations Variationnelles,", Dunod, (1976).

[12]

R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities. I,, Numer. Math., 69 (1994), 167.

[13]

R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities. II,, Numer. Math., 72 (1996), 481. doi: 10.1007/s002110050178.

[14]

R. Kornhuber, "Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems,", Advances in Numerical Mathematics, (1997).

[15]

J. Mandel, A multilevel iterative method for symmetric, positive definite linear complementarity problems,, Appl. Math. Opt., 11 (1984), 77. doi: 10.1007/BF01442171.

[16]

J. Mandel, Étude algébrique d'une méthode multigrille pour quelques problèmes de frontière libre,, C. R. Acad. Sci. Série I Math., 298 (1984), 469.

[17]

M. Raous, L. Cangémi and M. Cocu, A consistent model coupling adhesion, friction, and unilateral contact,, Comput. Meth. Appl. Mech. Engrg., 177 (1999), 383. doi: 10.1016/S0045-7825(98)00389-2.

[18]

A. Toselli and O. Widlund, "Domains Decomposition Methods - Algorithms and Theory,", Springer Series in Computational Mathematics, 34 (2005).

show all references

References:
[1]

L. Badea, Convergence rate of a multiplicative Schwarz method for strongly nonlinear inequalities,, in, (2003), 31.

[2]

L. Badea, Convergence rate of a Schwarz multilevel method for the constrained minimization of non-quadratic functionals,, SIAM J. Numer. Anal., 44 (2006), 449. doi: 10.1137/S003614290342995X.

[3]

L. Badea, Schwarz methods for inequalities with contraction operators,, J. Comp. Appl. Math., 215 (2008), 196. doi: 10.1016/j.cam.2007.04.004.

[4]

L. Badea and R. Krause, One- and two-level Schwarz methods for variational inequalities of the second kind and their application to frictional contact,, Numer. Math., 120 (2012), 573. doi: 10.1007/s00211-011-0423-y.

[5]

L. Badea and R. Krause, One- and two-level multiplicative Schwarz methods for variational and quasi-variational inequalities of the second kind. Part I - general convergence results,, INS Preprint, (0804).

[6]

A. Capatina and M. Cocou, Internal approximation of quasi-variational inequalities,, Numer. Math., 59 (1991), 385. doi: 10.1007/BF01385787.

[7]

A. Capatina, M. Cocou and M. Raous, A class of implicit variational inequalities and applications to frictional contact,, Math. Meth. Appl. Sci., 32 (2009), 1804. doi: 10.1002/mma.1112.

[8]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,", Studies in Mathematics and its Applications, (1978).

[9]

M. Cocou, E. Pratt and M. Raous, Formulation and approximation of quasistatic frictional contact,, Int. J. Engrg. Sci., 34 (1996), 783. doi: 10.1016/0020-7225(95)00121-2.

[10]

R. Glowinski, "Numerical Methods for Nonlinear Variational Problems,", Springer Series in Computational Physics, (1984).

[11]

R. Glowinski, J.-L. Lions and R. Trémolières, "Analyse Numérique des Inéquations Variationnelles,", Dunod, (1976).

[12]

R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities. I,, Numer. Math., 69 (1994), 167.

[13]

R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities. II,, Numer. Math., 72 (1996), 481. doi: 10.1007/s002110050178.

[14]

R. Kornhuber, "Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems,", Advances in Numerical Mathematics, (1997).

[15]

J. Mandel, A multilevel iterative method for symmetric, positive definite linear complementarity problems,, Appl. Math. Opt., 11 (1984), 77. doi: 10.1007/BF01442171.

[16]

J. Mandel, Étude algébrique d'une méthode multigrille pour quelques problèmes de frontière libre,, C. R. Acad. Sci. Série I Math., 298 (1984), 469.

[17]

M. Raous, L. Cangémi and M. Cocu, A consistent model coupling adhesion, friction, and unilateral contact,, Comput. Meth. Appl. Mech. Engrg., 177 (1999), 383. doi: 10.1016/S0045-7825(98)00389-2.

[18]

A. Toselli and O. Widlund, "Domains Decomposition Methods - Algorithms and Theory,", Springer Series in Computational Mathematics, 34 (2005).

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