About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 156095, 12 pages
http://dx.doi.org/10.1155/2012/156095
Research Article

Superconvergence Analysis of Finite Element Method for a Second-Type Variational Inequality

1Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China
2Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China
3Department of Mathematics, Tongji University, Shanghai 200092, China

Received 10 May 2012; Accepted 14 October 2012

Academic Editor: Song Cen

Copyright © 2012 Dongyang Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. P. Hartman and G. Stampacchia, “On some non-linear elliptic differential-functional equations,” Acta Mathematica, vol. 115, pp. 271–310, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. G. Strang, “The finite element method—linear and nonlinear applications,” in Proceedings of the International Congress of Mathematicians, pp. 429–435, Vancouver, Canada, 1974.
  3. F. Brezzi and G. Sacchi, “A finite element approximation of a variational inequality related to hydraulics,” Calcolo, vol. 13, no. 3, pp. 257–273, 1976. View at Zentralblatt MATH
  4. F. Brezzi, W. W. Hager, and P.-A. Raviart, “Error estimates for the finite element solution of variational inequalities,” Numerische Mathematik, vol. 28, no. 4, pp. 431–443, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. L. Wang, “On the quadratic finite element approximation to the obstacle problem,” Numerische Mathematik, vol. 92, no. 4, pp. 771–778, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. L. Wang, “On the error estimate of nonconforming finite element approximation to the obstacle problem,” Journal of Computational Mathematics, vol. 21, no. 4, pp. 481–490, 2003. View at Zentralblatt MATH
  7. D. Y. Shi and C. X. Wang, “Anisotropic nonconforming finite element approximation to variational inequality problems with displacement obstacle,” Chinese Journal of Engineering Mathematics, vol. 23, no. 3, pp. 399–406, 2006. View at Zentralblatt MATH
  8. D. Shi and H. Guan, “A class of Crouzeix-Raviart type nonconforming finite element methods for parabolic variational inequality problem with moving grid on anisotropic meshes,” Hokkaido Mathematical Journal, vol. 36, no. 4, pp. 687–709, 2007. View at Zentralblatt MATH
  9. F. Ben Belgacem, “Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods,” SIAM Journal on Numerical Analysis, vol. 37, no. 4, pp. 1198–1216, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Z. Belhachmi and F. B. Belgacem, “Quadratic finite element approximation of the Signorini problem,” Mathematics of Computation, vol. 72, no. 241, pp. 83–104, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. F. Ben Belgacem and Y. Renard, “Hybrid finite element methods for the Signorini problem,” Mathematics of Computation, vol. 72, no. 243, pp. 1117–1145, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. D. Hua and L. Wang, “The nonconforming finite element method for Signorini problem,” Journal of Computational Mathematics, vol. 25, no. 1, pp. 67–80, 2007. View at Zentralblatt MATH
  13. D. Y. Shi, S. P. Mao, and S. C. Chen, “A class of anisotropic Crouzeix-Raviart type finite element approximations to the Signorini variational inequality problem,” Chinese Journal of Numerical Mathematics and Applications, vol. 27, no. 1, pp. 69–78, 2005.
  14. M. Li, Q. Lin, and S. Zhang, “Superconvergence of finite element method for the Signorini problem,” Journal of Computational and Applied Mathematics, vol. 222, no. 2, pp. 284–292, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. D. Shi, J. Ren, and W. Gong, “Convergence and superconvergence analysis of a nonconforming finite element method for solving the Signorini problem,” Nonlinear Analysis A, vol. 75, no. 8, pp. 3493–3502, 2012. View at Publisher · View at Google Scholar
  16. D. Hage, N. Klein, and F. T. Suttmeier, “Adaptive finite elements for a certain class of variational inequalities of second kind,” Calcolo, vol. 48, no. 4, pp. 293–305, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. R. An and K. T. Li, “Mixed finite element approximation for the plate contact problem,” Acta Mathematica Scientia, vol. 30, no. 3, pp. 666–676, 2010. View at Zentralblatt MATH
  18. S. Zhou, Variational Inequalities and Its FEM, Hunan University press, Changsha, China, 1994.
  19. N. Kikuchi and J. T. Oden, Contact Problem in Elasticity, vol. 8 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1988.
  20. R. Glowinski, J.-L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, vol. 8 of Studies in Mathematics and its Applications, North-Holland Publishing, Amsterdam, The Netherlands, 1981.
  21. L. H. Wang, “The finite element approximation to a second type variational inequality,” Mathematica Numerica Sinica, vol. 22, no. 3, pp. 339–344, 2000.
  22. T. Zhang and C. J. Li, “Finite element approximation to the second type variational inequality,” Mathematica Numerica Sinica, vol. 25, no. 3, pp. 257–264, 2003.
  23. Q. Lin and J. Lin, Finite Element Methods: Accuracy and Improvement, Science Press, Beijing, China, 2006.
  24. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15, Springer, Berlin, Germany, 1994.
  25. D. Y. Shi and H. B. Guan, “A kind of full-discrete nonconforming finite element method for the parabolic variational inequality,” Acta Mathematicae Applicatae Sinica, vol. 31, no. 1, pp. 90–96, 2008.
  26. D. Shi, S. Mao, and S. Chen, “An anisotropic nonconforming finite element with some superconvergence results,” Journal of Computational Mathematics, vol. 23, no. 3, pp. 261–274, 2005. View at Zentralblatt MATH