Journal of Scientific Computing

, Volume 70, Issue 1, pp 125–148 | Cite as

Superconvergent Two-Grid Methods for Elliptic Eigenvalue Problems

  • Hailong Guo
  • Zhimin ZhangEmail author
  • Ren Zhao


Some numerical algorithms for elliptic eigenvalue problems are proposed, analyzed, and numerically tested. The methods combine advantages of the two-grid algorithm (Xu and Zhou in Math Comput 70(233):17–25, 2001), the two-space method (Racheva and Andreev in Comput Methods Appl Math 2:171–185, 2002), the shifted inverse power method (Hu and Cheng in Math Comput 80:1287–1301, 2011; Yang and Bi in SIAM J Numer Anal 49:1602–1624, 2011), and the polynomial preserving recovery enhancing technique (Naga et al. in SIAM J Sci Comput 28:1289–1300, 2006). Our new algorithms compare favorably with some existing methods and enjoy superconvergence property.


Eigenvalue problems Two-grid method Gradient recovery Superconvergence Polynomial preserving Adaptive 

Mathematics Subject Classification

65N15 65N25 65N30 


  1. 1.
    Andreev, A.B., Lazarov, R.D., Racheva, M.R.: Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems. J. Comput. Appl. Math. 182, 333–349 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ainsworth, M., Oden, J.: A Posteriori Error Estimation in Finite Element Analysis, Pure and Applied Mathematics. Wiley, New York (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Armentano, M., Durán, R.: Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. Electron. Trans. Numer. Anal. 17, 93–101 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Babuška, I., Strouboulis, T.: The Finite Element Method and its Reliability. Oxford University Press, London (2001)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Babuška, I., Osborn, J.: Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, pp. 641–787, Handbook of Numerical Analysis, North-Holland, Amsterdan (1991)Google Scholar
  7. 7.
    Betcke, T., Trefethen, L.N.: Reviving the method of particular solutions. SIAM Rev. 47, 469–491 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brenner, S., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chatelin, F.: Spectral Approximation of Linear Operators. Academic Press, New York (1983)zbMATHGoogle Scholar
  10. 10.
    Chen, L.: iFEM: An Integrated Finite Element Package in MATLAB, Technical Report, University of California at Irvine (2009)Google Scholar
  11. 11.
    Chien, C.-S., Jeng, B.-W.: A two-grid discretization scheme for semilinear elliptic eigenvalue problems. SIAM J. Sci. Comput. 27, 1287–1304 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  13. 13.
    Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Durán, R., Padra, G., Rodríguez, R.: A posteriori error estimates for the finite element approximation of eigenvalue problems. Math. Models Methods Appl. Sci. 13, 1219–1229 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fang, J., Gao, X., Zhou, A.: A finite element recovery approach approach to eigenvalue approximations with applications to electronic structure calculations. J. Sci. Comput. 55, 432–454 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gao, X., Liu, F., Zhou, A.: Three-scale finite element eigenvalue discretizations. BIT 48, 533–562 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Grebenkov, D.S., Nguyen, B.-T.: Geometric structure of Laplacian eigenfunctions. SIAM Rev. 55–4, 601–667 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Greiner, W.: Quantum Mechanics: An Introduction, 3rd edn. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  19. 19.
    Giani, S., Grubišić, L., Ovall, J.: Benchmark results for testing adaptive finite element eigenvalue procedures. Appl. Numer. Math. 62, 121–140 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Giani, S., Graham, I.G.: A convergent adaptive method for elliptic eigenvalue problems. SIAM J. Numer. Anal. 47, 1067–1091 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hoppe, R.H.W., Wu, H., Zhang, Z.: Adaptive finite element methods for the Laplace eigenvalue problem. J. Numer. Math. 18, 281–302 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hu, X., Cheng, X.: Acceleration of a two-grid method for eigenvalue problems. Math. Comput. 80, 1287–1301 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kolman, K.: A two-level method for nonsymmetric eigenvalue problems. Acta Math. Appl. Sin. 21, 1–12 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Li, H., Yang, Y.: The adaptive finite element method based on multi-scale discretizations for eigenvalue problems. Comput. Math. Appl. 65, 1086–1102 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lin, Q., Xie, H.: A multi-level correction scheme for eigenvalue problems. Math. Comput. 84(291), 71–88 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lin, Q., Xie, H., Xu, J.: Lower bounds of the discretization error for piecewise polynomials. Math. Comput. 83, 1–13 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Liu, H., Yan, N.: Enhancing finite element approximation for eigenvalue problems by projection method. Comput. Methods Appl. Mech. Eng. 233–236, 81–91 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Larson, M.G.: A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems. SIAM J. Numer. Anal. 38, 608–625 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mao, M., Shen, L., Zhou, A.: Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimators. Adv. Comput. Math. 25, 135–160 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mehrmann, V., Miedlar, A.: Adaptive computation of smallest eigenvalues of self-adjoint elliptic partial differential equations. Numer. Linear Algebra Appl. 18, 387–409 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Meng, L., Zhang, Z.: The ultraconvergence of eigenvalues for bi-quadrature finite elements. J. Comput. Math. 30, 555–564 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Naga, A., Zhang, Z.: A posteriori error estimates based on the polynomial preserving recovery. SIAM J. Numer. Anal. 42, 1780–1800 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Naga, A., Zhang, Z.: The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete Contin. Dyn. Syst. Ser. B 5, 769–798 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Naga, A., Zhang, Z., Zhou, A.: Enhancing eigenvalue approximation by gradient recovery. SIAM J. Sci. Comput. 28, 1289–1300 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Niceno, B.: EasyMesh Version 1.4: A Two-Dimensional Quality Mesh Generator,
  36. 36.
    Racheva, M.R., Andreev, A.B.: Superconvergence postprocessing for eigenvalues. Comput. Methods Appl. Math. 2, 171–185 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Shen, L., Zhou, A.: A defect correction scheme for finite element eigenvalues with applications to quantum chemistry. SIAM J. Sci. Comput. 28, 321–338 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Strang, G., Fix, G.: An Analysis of the Finite Element Method. Prentice-Hall Series in Automatic Computation. Prentice-Hall, Englewood Cliffs (1973)Google Scholar
  39. 39.
    Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-refinement Techniques. Wiley and Teubner, New York (1996)zbMATHGoogle Scholar
  40. 40.
    Wu, H., Zhang, Z.: Enhancing eigenvalue approximation by gradient recovery on adaptive meshes. IMA J. Numer. Anal. 29, 1008–1022 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Xie, H.: A Multilevel Correction Type of Adaptive Finite Element Method for Eigenvalue Problems, arXiv:1201.2308v1 [math.NA]
  42. 42.
    Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70(233), 17–25 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    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, 1602–1624 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Yang, Y., Zhang, Z., Lin, F.: Eigenvalue approximation from below using non-conforming finite elements. Sci. China Math. 53, 137–150 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Zhang, Z., Naga, A.: A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput. 26, 1192–1213 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Zhang, Z., Yang, Y., Chen, Z.: Eigenvalue approximation from below by Wilson’s element(Chinese). Math. Numer. Sin. 29, 319–321 (2007)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique. Int. J. Numer. Methods Eng. 33, 1331–1364 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Beijing Computational Science Research CenterBeijingChina
  2. 2.Department of MathematicsWayne State UniversityDetroitUSA
  3. 3.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

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