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 "Analysis and Optimization of Differential Systems" (eds. V. Barbu, I. Lasiecka, D. Tiba and C. Varsan) (Constanta, 2002), Kluwer Academic Publishers, Boston, MA, (2003), 31-42. Available from: http://imar.ro/~lbadea/pub.html.  Google Scholar

[2]

L. Badea, Convergence rate of a Schwarz multilevel method for the constrained minimization of non-quadratic functionals, SIAM J. Numer. Anal., 44 (2006), 449-477. doi: 10.1137/S003614290342995X.  Google Scholar

[3]

L. Badea, Schwarz methods for inequalities with contraction operators, J. Comp. Appl. Math., 215 (2008), 196-219. doi: 10.1016/j.cam.2007.04.004.  Google Scholar

[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), 573-599. doi: 10.1007/s00211-011-0423-y.  Google Scholar

[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, No. 0804, Institute for Numerical Simulation, University of Bonn, June, 2008. Google Scholar

[6]

A. Capatina and M. Cocou, Internal approximation of quasi-variational inequalities, Numer. Math., 59 (1991), 385-398. doi: 10.1007/BF01385787.  Google Scholar

[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), 1804-1827. doi: 10.1002/mma.1112.  Google Scholar

[8]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.  Google Scholar

[9]

M. Cocou, E. Pratt and M. Raous, Formulation and approximation of quasistatic frictional contact, Int. J. Engrg. Sci., 34 (1996), 783-798. doi: 10.1016/0020-7225(95)00121-2.  Google Scholar

[10]

R. Glowinski, "Numerical Methods for Nonlinear Variational Problems," Springer Series in Computational Physics, Springer-Verlag, New York, 1984.  Google Scholar

[11]

R. Glowinski, J.-L. Lions and R. Trémolières, "Analyse Numérique des Inéquations Variationnelles," Dunod, Paris, 1976. Google Scholar

[12]

R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities. I, Numer. Math., 69 (1994), 167-184.  Google Scholar

[13]

R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities. II, Numer. Math., 72 (1996), 481-499. doi: 10.1007/s002110050178.  Google Scholar

[14]

R. Kornhuber, "Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems," Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1997.  Google Scholar

[15]

J. Mandel, A multilevel iterative method for symmetric, positive definite linear complementarity problems, Appl. Math. Opt., 11 (1984), 77-95. doi: 10.1007/BF01442171.  Google Scholar

[16]

J. Mandel, Étude algébrique d'une méthode multigrille pour quelques problèmes de frontière libre, C. R. Acad. Sci. Série I Math., 298 (1984), 469-472.  Google Scholar

[17]

M. Raous, L. Cangémi and M. Cocu, A consistent model coupling adhesion, friction, and unilateral contact, Comput. Meth. Appl. Mech. Engrg., 177 (1999), 383-399. doi: 10.1016/S0045-7825(98)00389-2.  Google Scholar

[18]

A. Toselli and O. Widlund, "Domains Decomposition Methods - Algorithms and Theory," Springer Series in Computational Mathematics, 34, Springer-Verlag, Berlin, 2005.  Google Scholar

show all references

References:
[1]

L. Badea, Convergence rate of a multiplicative Schwarz method for strongly nonlinear inequalities, in "Analysis and Optimization of Differential Systems" (eds. V. Barbu, I. Lasiecka, D. Tiba and C. Varsan) (Constanta, 2002), Kluwer Academic Publishers, Boston, MA, (2003), 31-42. Available from: http://imar.ro/~lbadea/pub.html.  Google Scholar

[2]

L. Badea, Convergence rate of a Schwarz multilevel method for the constrained minimization of non-quadratic functionals, SIAM J. Numer. Anal., 44 (2006), 449-477. doi: 10.1137/S003614290342995X.  Google Scholar

[3]

L. Badea, Schwarz methods for inequalities with contraction operators, J. Comp. Appl. Math., 215 (2008), 196-219. doi: 10.1016/j.cam.2007.04.004.  Google Scholar

[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), 573-599. doi: 10.1007/s00211-011-0423-y.  Google Scholar

[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, No. 0804, Institute for Numerical Simulation, University of Bonn, June, 2008. Google Scholar

[6]

A. Capatina and M. Cocou, Internal approximation of quasi-variational inequalities, Numer. Math., 59 (1991), 385-398. doi: 10.1007/BF01385787.  Google Scholar

[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), 1804-1827. doi: 10.1002/mma.1112.  Google Scholar

[8]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems," Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978.  Google Scholar

[9]

M. Cocou, E. Pratt and M. Raous, Formulation and approximation of quasistatic frictional contact, Int. J. Engrg. Sci., 34 (1996), 783-798. doi: 10.1016/0020-7225(95)00121-2.  Google Scholar

[10]

R. Glowinski, "Numerical Methods for Nonlinear Variational Problems," Springer Series in Computational Physics, Springer-Verlag, New York, 1984.  Google Scholar

[11]

R. Glowinski, J.-L. Lions and R. Trémolières, "Analyse Numérique des Inéquations Variationnelles," Dunod, Paris, 1976. Google Scholar

[12]

R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities. I, Numer. Math., 69 (1994), 167-184.  Google Scholar

[13]

R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities. II, Numer. Math., 72 (1996), 481-499. doi: 10.1007/s002110050178.  Google Scholar

[14]

R. Kornhuber, "Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems," Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1997.  Google Scholar

[15]

J. Mandel, A multilevel iterative method for symmetric, positive definite linear complementarity problems, Appl. Math. Opt., 11 (1984), 77-95. doi: 10.1007/BF01442171.  Google Scholar

[16]

J. Mandel, Étude algébrique d'une méthode multigrille pour quelques problèmes de frontière libre, C. R. Acad. Sci. Série I Math., 298 (1984), 469-472.  Google Scholar

[17]

M. Raous, L. Cangémi and M. Cocu, A consistent model coupling adhesion, friction, and unilateral contact, Comput. Meth. Appl. Mech. Engrg., 177 (1999), 383-399. doi: 10.1016/S0045-7825(98)00389-2.  Google Scholar

[18]

A. Toselli and O. Widlund, "Domains Decomposition Methods - Algorithms and Theory," Springer Series in Computational Mathematics, 34, Springer-Verlag, Berlin, 2005.  Google Scholar

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