Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

POSTPROCESSING FOR THE RAVIART-THOMAS MIXED FINITE ELEMENT APPROXIMATION OF THE EIGENVALUE PROBLEM

  • Received : 2018.05.10
  • Accepted : 2018.09.07
  • Published : 2018.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we present a postprocessing scheme for the Raviart-Thomas mixed finite element approximation of the second order elliptic eigenvalue problem. This scheme is carried out by solving a primal source problem on a higher order space, and thereby can improve the convergence rate of the eigenfunction and eigenvalue approximations. It is also used to compute a posteriori error estimates which are asymptotically exact for the $L^2$ errors of the eigenfunctions. Some numerical results are provided to confirm the theoretical results.

Keywords

References

  1. A. Alonso., A. D. Russo, and V. Vampa, A posteriori error estimates in finite element acoustic analysis, J. Comput. Appl. Math. 117 (2000), 105-119. https://doi.org/10.1016/S0377-0427(99)00335-0
  2. I. Babuska and J. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis II, Finite Element Methods (Part 1), edited by P.G. Lions and P.G. Ciarlet, North-Holland, Amsterdam, 1991, 641-787.
  3. D. Boffi, Finite element approximation of eigenvalue problems, Acta Numer. 19 (2010), 1-120. https://doi.org/10.1017/S0962492910000012
  4. D. Boffi, F. Brezzi, and L. Gastaldi, On the convergence of eigenvalues for mixed formulations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997), 131-154.
  5. D. Boffi, F. Brezzi, and L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Math. Comp. 69 (2000), 121-140.
  6. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, New York, 1991.
  7. H. Chen, S. Jia, and H. Xie, Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem, Appl. Numer. Math. 61 (2011), 615-629. https://doi.org/10.1016/j.apnum.2010.12.007
  8. R. Duran, L. Gastaldi, and C. Padra, A posteriori error estimators for mixed approximations of eigenvalue problems, Math. Models Methods Appl. Sci. 9 (1999), 1165-1178. https://doi.org/10.1142/S021820259900052X
  9. F. Gardini, Mixed approximation of eigenvalue problems: a superconvergence result, ESAIM: M2AN 43 (2009), 853-865. https://doi.org/10.1051/m2an/2009005
  10. P. Grisvard, Elliptic Problems in Non-Smooth Domains, Monographs and Studies in Mathematics 24, Pitman, Boston, 1985.
  11. S. Jia, H. Chen, and H. Xie, A posteriori error estimator for eigenvalue problems by mixed finite element method, Sci. China Math. 56 (2013), 887-900. https://doi.org/10.1007/s11425-013-4614-0
  12. J. Douglas Jr. and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), 39-52. https://doi.org/10.1090/S0025-5718-1985-0771029-9
  13. Q. Lin and H. Xie, A superconvergence result for mixed finite element approximations of the eigenvalue problem, ESAIM: M2AN 46 (2012), 797-812. https://doi.org/10.1051/m2an/2011065
  14. B. Mercier, J. Osborn, J. Rappaz, and P. A. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comp. 36 (1981), 427-453. https://doi.org/10.1090/S0025-5718-1981-0606505-9
  15. A. Naga and Z. Zhang, Function value recovery and its application in eigenvalue problems, SIAM J. Numer. Anal. 50 (2012), 272-286. https://doi.org/10.1137/100797709
  16. M. R. Racheva and A. B. Andreev, Superconvergence postprocessing for eigenvalues, Comp. Methods Appl. Math. 2 (2002), 171-185.
  17. J. Xu and A. Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comp. 70 (2001), 17-25.