Advertisement

BIT Numerical Mathematics

, Volume 55, Issue 4, pp 1243–1266 | Cite as

A type of multi-level correction scheme for eigenvalue problems by nonconforming finite element methods

  • Hehu Xie
Article

Abstract

In this paper, a type of multi-level correction scheme is proposed to solve eigenvalue problems by nonconforming finite element methods. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an eigenvalue problem on a coarse finite element space. This correction scheme can improve the efficiency of solving eigenvalue problems by the nonconforming finite element method. Furthermore, the same as the direct eigenvalue solving by the nonconforming finite element method, this multi-level correction method can also produce the lower-bound approximations of the eigenvalues.

Keywords

Eigenvalue problem Multi-level correction Multigrid  Finite element method 

Mathematics Subject Classification

65N30 65N25 65L15 65B99 

References

  1. 1.
    Adams, R.: Sobolev spaces. Academic Press, New York (1975)MATHGoogle Scholar
  2. 2.
    Andreev, A., Lazarov, R., Racheva, M.: Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems. J. Comput. Appl. Math. 182, 333–349 (2005)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Andreev, A.B., Todorov, T.D.: Isoparametric finite-element approximation of a Steklov eigenvalue problem. IMA J. Numer. Anal. 24, 309–322 (2004)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Arbogast, T., Chen, Z.: On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comput. 64, 943–972 (1995)MathSciNetMATHGoogle Scholar
  5. 5.
    Babuška, I., Osborn, J.: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comput. 52, 275–297 (1989)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Babuška, I., Osborn, J.: Eigenvalue Problems. In: Lions, C.L., Ciarlet, P.G. (eds.) Handbook of Numerical Analysis, vol. II, Finite Element Methods (Part 1), pp. 641–787. North-Holland, Amsterdam (1991)Google Scholar
  7. 7.
    Brenner, S.: Convergence of nonconforming \(V\)-cycle and \(F\)-cycle multigrid algorithms for second order elliptic boundary value problems. Math. Comput. 73(247), 1041–1066 (2003)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)CrossRefMATHGoogle Scholar
  9. 9.
    Chatelin, F.: Spectral Approximation of Linear Operators. Academic Press Inc, New York (1983)MATHGoogle Scholar
  10. 10.
    Chen, H., Jia, S., Xie, H.: Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems. Appl. Math. 54(3), 237–250 (2009)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Cheng, S., Dey, T., Shewchuk, J.: Delaunay Mesh Generation. CRC Press, Boca Raton (2012)MATHGoogle Scholar
  12. 12.
    Ciarlet, P.: The finite Element Method for Elliptic Problem. North-Holland Amsterdam (1978)Google Scholar
  13. 13.
    Crouzeix, M., Raviart, P.: Conforming and nonconforming finite element for solving the stationary Stokes equations. RAIRO Anal. Numer. 3, 33–75 (1973)MathSciNetMATHGoogle Scholar
  14. 14.
    Grisvard, P.: Singularities in Boundary Value Problems. Masson and Springer (1985)Google Scholar
  15. 15.
    Hu, X., Cheng, X.: Acceleration of a two-grid method for eigenvalue problems. Math. Comput. 80(275), 1287–1301 (2011)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Hu, J., Huang, Y., Lin, Q.: Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods. J. Sci. Comput. (2014). doi: 10.1007/s10915-014-9821-5
  17. 17.
    Li, Q., Lin, Q., Xie, H.: Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations. Appl. Math. 58(2), 129–151 (2013)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Lin, Q., Tobiska, L., Zhou, A.: Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation. IMA J. Numer. Anal. 25, 160–181 (2005)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Lin, Q., Xie, H.: A Type of Multigrid Method for Eigenvalue Problem. Res. Rep. ICMSEC 2011–06 (2011)Google Scholar
  20. 20.
    Lin, Q., Xie, H.: The asymptotic lower bounds of eigenvalue problems by nonconforming finite element methods. Math. Pract. Theory 42(11), 219–226 (2012)MathSciNetMATHGoogle Scholar
  21. 21.
    Lin, Q., Xie, H., Luo, F., Li, Y., Yang, Y.: Stokes eigenvalue approximation from below with nonconforming mixed finite element methods. Math. Pract. Theory 19, 157–168 (2010)MathSciNetGoogle Scholar
  22. 22.
    Lin, Q., Xie, H., Xu, J.: Lower bounds of the discretization for piecewise polynomials. Math. Comput. 83(285), 1–13 (2014)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Lin, Q., Yan, N.: The Construction and Analysis of High Efficiency Finite Element Methods. Hebei University Publishers (1995)Google Scholar
  24. 24.
    Luo, F., Lin, Q., Xie, H.: Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods. Sci. China Math. 55, 1069–1082 (2012)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Mercier, B., Osborn, J., Rappaz, J., Raviart, P.: Eigenvalue approximation by mixed and hybrid methods. Math. Comput. 36(154), 427–453 (1981)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Rannacher, R.: Nonconforming finite element methods for eigenvalue problems in linear plate theory. Numer. Math. 33, 23–42 (1979)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Racheva, M., Andreev, A.: Superconvergence postprocessing for Eigenvalues. Comp. Methods Appl. Math. 2(2), 171–185 (2002)MathSciNetMATHGoogle Scholar
  28. 28.
    Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Xu, J.: A new class of iterative methods for nonselfadjoint or indefinite problems. SIAM J. Numer. Anal. 29, 303–319 (1992)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Xu, J.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33(5), 1759–1777 (1996)CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70(233), 17–25 (2001)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Yang, Y., Chen, Z.: The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators. Sci. China Ser. A 51(7), 1232–1242 (2008)CrossRefMathSciNetMATHGoogle Scholar
  34. 34.
    Yang, Y., Bi, H.: Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems. SIAM J. Numer. Anal. 49(20), 1602–1624 (2011)CrossRefMathSciNetMATHGoogle Scholar
  35. 35.
    Zhou, A.: Multi-level adaptive corrections in finite dimensional approximations. J. Comput. Math. 28(1), 45–54 (2010)MathSciNetMATHGoogle Scholar
  36. 36.
    Zienkiewicz, O., Zhu, J.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 24, 337–357 (1987)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.LSEC, NCMIS, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

Personalised recommendations