On the Unilateral Contact Between Membranes. Part 1: Finite Element Discretization and Mixed Reformulation

F. Ben Belgacem; C. Bernardi; A. Blouza; M. Vohralík

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 1, page 21-43
  • ISSN: 0973-5348

Abstract

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The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. We first perform the analysis of this system. We then propose a discretization, where the displacements are approximated by standard finite elements and the action by a local postprocessing. Such a discretization admits an equivalent mixed reformulation. We prove the well-posedness of the discrete problem and establish optimal a priori error estimates.

How to cite

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Ben Belgacem, F., et al. "On the Unilateral Contact Between Membranes. Part 1: Finite Element Discretization and Mixed Reformulation." Mathematical Modelling of Natural Phenomena 4.1 (2009): 21-43. <http://eudml.org/doc/222317>.

@article{BenBelgacem2009,
abstract = { The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. We first perform the analysis of this system. We then propose a discretization, where the displacements are approximated by standard finite elements and the action by a local postprocessing. Such a discretization admits an equivalent mixed reformulation. We prove the well-posedness of the discrete problem and establish optimal a priori error estimates.},
author = {Ben Belgacem, F., Bernardi, C., Blouza, A., Vohralík, M.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {unilateral contact; elastic membranes; variational inequalities; variational inequality; local postprocessing; well-posedness; a priori error estimates},
language = {eng},
month = {1},
number = {1},
pages = {21-43},
publisher = {EDP Sciences},
title = {On the Unilateral Contact Between Membranes. Part 1: Finite Element Discretization and Mixed Reformulation},
url = {http://eudml.org/doc/222317},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Ben Belgacem, F.
AU - Bernardi, C.
AU - Blouza, A.
AU - Vohralík, M.
TI - On the Unilateral Contact Between Membranes. Part 1: Finite Element Discretization and Mixed Reformulation
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/1//
PB - EDP Sciences
VL - 4
IS - 1
SP - 21
EP - 43
AB - The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. We first perform the analysis of this system. We then propose a discretization, where the displacements are approximated by standard finite elements and the action by a local postprocessing. Such a discretization admits an equivalent mixed reformulation. We prove the well-posedness of the discrete problem and establish optimal a priori error estimates.
LA - eng
KW - unilateral contact; elastic membranes; variational inequalities; variational inequality; local postprocessing; well-posedness; a priori error estimates
UR - http://eudml.org/doc/222317
ER -

References

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  11. J.-L. Lions, G. Stampacchia. Variational inequalities. Comm. Pure and Appl. Math., 20 (1967), 493–519.  
  12. P.-A. Raviart, J.-M. Thomas. A mixed finite element method for second order elliptic problems. In: Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, 606, Springer, 1977, pp. 292–315.  
  13. L. Slimane, A. Bendali, P. Laborde. Mixed formulations for a class of variational inequalities. Math. Model. Numer. Anal., 38 (2004), 177–201.  
  14. M. Vohralík. A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization. C. R. Math. Acad. Sci. Paris, 346 (2008), 687–690.  

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