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Abstract and Applied Analysis
Volume 2012, Article ID 190768, 25 pages
http://dx.doi.org/10.1155/2012/190768
Research Article

Multigrid Discretization and Iterative Algorithm for Mixed Variational Formulation of the Eigenvalue Problem of Electric Field

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

Received 9 July 2012; Revised 4 September 2012; Accepted 12 September 2012

Academic Editor: Xinan Hao

Copyright © 2012 Yidu Yang 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. D. Boffi, P. Fernandes, L. Gastaldi, and I. Perugia, “Computational models of electromagnetic resonators: analysis of edge element approximation,” SIAM Journal on Numerical Analysis, vol. 36, no. 4, pp. 1264–1290, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. A. Buffa and I. Perugia, “Discontinuous Galerkin approximation of the Maxwell eigenproblem,” SIAM Journal on Numerical Analysis, vol. 44, no. 5, pp. 2198–2226, 2006. View at Publisher · View at Google Scholar
  3. A. Buffa, P. Ciarlet Jr., and E. Jamelot, “Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements,” Numerische Mathematik, vol. 113, no. 4, pp. 497–518, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. P. Ciarlet Jr. and G. Hechme, “Computing electromagnetic eigenmodes with continuous Galerkin approximations,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 2, pp. 358–365, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. S. Caorsi, P. Fernandes, and M. Raffetto, “On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems,” SIAM Journal on Numerical Analysis, vol. 38, no. 2, pp. 580–607, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. F. Kikuchi, “Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism,” Computer Methods in Applied Mechanics and Engineering, vol. 64, pp. 509–521, 1987. View at Google Scholar
  7. Y. Yang, W. Jiang, Y. Zhang, W. Wang, and H. Bi, “A two-scale discretization scheme for mixed variational formulation of eigenvalue problems,” Abstract and Applied Analysis, vol. 2012, Article ID 812914, 29 pages, 2012. View at Publisher · View at Google Scholar
  8. Y. Yang and H. Bi, “Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 49, no. 4, pp. 1602–1624, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. H. Bi and Y. Yang, “Multi-scale discretizaiton scheme based on the Rayleigh quotient iterative method for the Steklov eigenvalue problem,” Mathematical Problems in Engineering, vol. 2012, Article ID 487207, 18 pages, 2012. View at Publisher · View at Google Scholar
  10. L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia, Pa, USA, 1997. View at Publisher · View at Google Scholar
  11. M. Costabel and M. Dauge, “Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of nodal finite elements,” Numerische Mathematik, vol. 93, no. 2, pp. 239–277, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. Yang, Finite Element Methods for Eigenvalue Problems, Science Press, Beijing, China, 2012.
  13. D. Boffi, F. Brezzi, and L. Gastaldi, “On the convergence of eigenvalues for mixed formulations,” Annali della Scuola Normale Superiore di Pisa IV, vol. 25, no. 1-2, pp. 131–154, 1997. View at Google Scholar · View at Zentralblatt MATH
  14. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, vol. 15, Springer, New York, NY, USA, 1991. View at Publisher · View at Google Scholar
  15. I. Babuška and J. Osborn, “Eigenvalue problems,” in Finite Element Methods(Part 1), Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, Eds., vol. 2, pp. 641–787, Elsevier Science Publishers, North-Holand, 1991. View at Google Scholar · View at Zentralblatt MATH
  16. B. Mercier, J. Osborn, J. Rappaz, and P.-A. Raviart, “Eigenvalue approximation by mixed and hybrid methods,” Mathematics of Computation, vol. 36, no. 154, pp. 427–453, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. F. Chatelin, Spectral Approximation of Linear Operators, Academic Press, New York, NY, USA, 1983.
  18. H. Chen, S. Jia, and H. Xie, “Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems,” Applications of Mathematics, vol. 54, no. 3, pp. 237–250, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. P. G. Ciarlet, “Basic error estimates for elliptic problems,” in Finite Element Methods (Part1), Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, Eds., vol. 2, pp. 21–343, Elsevier Science Publishers, North-Holand, 1991. View at Google Scholar · View at Zentralblatt MATH
  20. X. Dai, J. Xu, and A. Zhou, “Convergence and optimal complexity of adaptive finite element eigenvalue computations,” Numerische Mathematik, vol. 110, no. 3, pp. 313–355, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. V. Heuveline and R. Rannacher, “A posteriori error control for finite approximations of elliptic eigenvalue problems,” Advances in Computational Mathematics, vol. 15, no. 1–4, pp. 107–138, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. D. Mao, L. Shen, and A. Zhou, “Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates,” Advances in Computational Mathematics, vol. 25, no. 1–3, pp. 135–160, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, “Vector potentials in three-dimensional non-smooth domains,” Mathematical Methods in the Applied Sciences, vol. 21, no. 9, pp. 823–864, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. M. Costabel, “A coercive bilinear form for Maxwell's equations,” Journal of Mathematical Analysis and Applications, vol. 157, no. 2, pp. 527–541, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. P. Ciarlet Jr., “Augmented formulations for solving Maxwell equations,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 2–5, pp. 559–586, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. P. Ciarlet Jr. and V. Girault, “inf-sup condition for the 3D, P2-iso-P1, Taylor-Hood finite element application to Maxwell equations,” Comptes Rendus Mathématique, vol. 335, no. 10, pp. 827–832, 2002. View at Publisher · View at Google Scholar