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Mathematical Problems in Engineering
Volume 2011, Article ID 873152, 13 pages
http://dx.doi.org/10.1155/2011/873152
Research Article

Conforming Finite Element Approximations for a Fourth-Order Steklov Eigenvalue Problem

School of Mathematics and Computer Science, Guizhou Normal University, GuiYang 550001, China

Received 30 May 2011; Revised 5 August 2011; Accepted 13 September 2011

Academic Editor: Bin Liu

Copyright © 2011 Hai Bi 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. S. Bergman and M. Schiffer, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, New York, NY, USA, 1953.
  2. C. Conca, J. Planchard, and M. Vanninathan, Fluids and Periodic Structures, John Wiley & Sons, New York, NY, USA, 1995.
  3. A. Bermudez, R. Rodriguez, and D. Santamarina, “A finite element solution of an added mass formulation for coupled fluid-solid vibrations,” Numerische Mathematik, vol. 87, no. 2, pp. 201–227, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. F. Gazzola and G. Sweers, “On positivity for the biharmonic operator under Steklov boundary conditions,” Archive for Rational Mechanics and Analysis, vol. 188, no. 3, pp. 399–427, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. A. B. Andreev and T. D. Todorov, “Isoparametric finite-element approximation of a Steklov eigenvalue problem,” IMA Journal of Numerical Analysis, vol. 24, no. 2, pp. 309–322, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. M. G. Armentano, “The effect of reduced integration in the Steklov eigenvalue problem,” Mathematical Modelling and Numerical Analysis, vol. 38, no. 1, pp. 27–36, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. M. G. Armentano and C. Padra, “A posteriori error estimates for the Steklov eigenvalue problem,” Applied Numerical Mathematics, vol. 58, no. 5, pp. 593–601, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. H. Bi and Y. D. Yang, “A two-grid method of the non-conforming Crouzeix-Raviart element for the Steklov eigenvalue problem,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9669–9678, 2011. View at Publisher · View at Google Scholar
  9. J. H. Bramble and J. E. Osborn, “Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators,” in Mathematical Foundation of the Finite Element Method with Applications to PDE, A. K. Aziz, Ed., pp. 387–408, Academic Press, New York, NY, USA, 1972. View at Google Scholar · View at Zentralblatt MATH
  10. M. X. Li, Q. Lin, and S. H. Zhang, “Extrapolation and superconvergence of the Steklov eigenvalue problem,” Advances in Computational Mathematics, vol. 33, no. 1, pp. 25–44, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. Q. Li and Y. Yang, “A two-grid discretization scheme for the Steklov eigenvalue problem,” Journal of Applied Mathematics and Computing, vol. 36, no. 1-2, pp. 129–139, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. D. Yang, Q. Li, and S. R. Li, “Nonconforming finite element approximations of the Steklov eigenvalue problem,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2388–2401, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. J. R. Kuttler, “Remarks on a Stekloff eigenvalue problem,” SIAM Journal on Numerical Analysis, vol. 9, pp. 1–5, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J. R. Kuttler, “Bounds for Stekloff eigenvalues,” SIAM Journal on Numerical Analysis, vol. 19, no. 1, pp. 121–125, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. D. Bucur, A. Ferrero, and F. Gazzola, “On the first eigenvalue of a fourth order Steklov eigenvalue problem,” Calculus of Variations and Partial Differential Equations, vol. 35, no. 1, pp. 103–131, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. A. Ferrero, F. Gazzola, and T. Weth, “On a fourth order Steklov eigenvalue problem,” Analysis, vol. 25, no. 4, pp. 315–332, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. Q. Wang and C. Xia, “Sharp bounds for the first non-zero Stekloff eigenvalues,” Journal of Functional Analysis, vol. 257, no. 8, pp. 2635–2644, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. D. Bucur and F. Gazzola, “The first biharmonic Steklov eigenvalue: positivity preserving and shape optimization,” Milan Journal of Mathematics, vol. 79, no. 1, pp. 247–258, 2011. View at Publisher · View at Google Scholar
  19. G. Liu, “The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues with dirichlet boundary condition on Riemannian manifolds,” Advances in Mathematics, vol. 1, 42 pages, 2009. View at Google Scholar
  20. I. Babuska and J. E. Osborn, “Eigenvalue problems,” in Finite Element Methods(Part 1), Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, Eds., vol. 2, pp. 640–787, Elsevier Science Publishers, Amsterdam, The Netherlands, 1991. View at Google Scholar · View at Zentralblatt MATH
  21. F. Chatelin, Spectral Approximation of Linear Operators, Academic Press, New York, NY, USA, 1983.
  22. Z. C. Shi and M. Wang, Finite Element Methods, Science Publishers, Beijing, China, 2010.
  23. J. L. Lions and E. Magenes, Non-Homogrneous Boundary Value Problems and Applications, Springer-Verlag, New York, NY, USA, 1972.
  24. L. H. Wang and X. J. Xu, The Mathematical Basic of Finite Element Methods, Science Publishers, Beijing, China, 2004.
  25. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, New York, NY, USA, 1977.
  26. G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, USA, 1973.
  27. Q. Lin and Q. Zhu, The Preprocessing and Postprocessing for the Finite Element Method, Shanghai Scientific & Thchnical Publishers, Shanghai, China, 1994.
  28. P. Grisvard, Elliptic problems in nonsmooth domains, Pitman Publishing, London, UK, 1985.
  29. Y. D. Yang, Q. Lin, H. Bi, and Q. Li, “Eigenvalue approximations from below using Morley elements,” Advances in Computational Mathematics. In press. View at Publisher · View at Google Scholar
  30. H. Bi and Y. D. Yang, “Matlab experiments on morley element for lower spectral bound of plate vibration poblem,” in Proceedings of the 3rd International Conference on Computer and Network Technology, vol. 7, pp. 64–66, Shanxi, China, 2011.