Skip to main content
SpringerLink
Log in

Convergence and optimal complexity of adaptive finite element eigenvalue computations

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper, an adaptive finite element method for elliptic eigenvalue problems is studied. Both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved. The analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams R.A. (1975) Sobolev Spaces. Academic Press, New York

    MATH  Google Scholar 

  2. Arnold D.N., Mukherjee A., Pouly L. (2000) Locally adapted tetrahedral meshes using bisection. SIAM J. Sci. Comput. 22: 431–448

    Article  MATH  MathSciNet  Google Scholar 

  3. Babuska I., Osborn J.E. (1989) Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comp. 52: 275–297

    Article  MATH  MathSciNet  Google Scholar 

  4. Babuska I., Osborn J.E. (1991) Eigenvalue problems. In: Ciarlet P.G., Lions J.L.(eds) Handbook of Numerical Analysis, vol. II.. North Holland, Amsterdam, pp 641–792

    Google Scholar 

  5. Babuska I., Rheinboldt W.C. (1978) Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15: 736–754

    Article  MATH  MathSciNet  Google Scholar 

  6. Babuska I., Vogelius M. (1984) Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44: 75–102

    Article  MATH  MathSciNet  Google Scholar 

  7. Bartels S., Carstensen C. (2002) Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II. Higher order FEM. Math. Comp. 71: 971–994

    Article  MATH  MathSciNet  Google Scholar 

  8. Becker R., Rannacher R. (2001) An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10: 1–102

    Article  MATH  MathSciNet  Google Scholar 

  9. Binev P., Dahmen W., DeVore R. (2004) Adaptive finite element methods with convergence rates. Numer. Math. 97: 219–268

    Article  MATH  MathSciNet  Google Scholar 

  10. Carstensen C. (2005) A unifying theory of a posteriori finite element error control. Numer. Math. 100: 617–637

    Article  MATH  MathSciNet  Google Scholar 

  11. Carstensen C., Bartels S. (2002) Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I. Low order conforming, nonconforming, and mixed FEM. Math. Comp. 71: 945–969

    Article  MATH  MathSciNet  Google Scholar 

  12. Carstensen C., Hoppe R.H.W. (2006) Error reduction and convergence for an adaptive mixed finite element method. Math. Comp. 75: 1033–1042

    Article  MATH  MathSciNet  Google Scholar 

  13. Carstensen C., Hoppe R.H.W. (2006) Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. 103: 251–266

    Article  MATH  MathSciNet  Google Scholar 

  14. Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. preprint (2007)

  15. Chatelin F. (1983) Spectral Approximations of Linear Operators. Academic Press, New York

    Google Scholar 

  16. Chen, L., Holst, M., Xu, J.: Convergence and optimality of adaptive mixed finite element methods. Math. Comp. (to appear) (2008)

  17. Chen Z., Nochetto R.H. (2000) Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84: 527–548

    Article  MATH  MathSciNet  Google Scholar 

  18. Ciarlet, P.G., Lions, J.L. (eds.): Finite Element Methods, Volume II of Handbook of Numerical Analysis, vol. II. North Holland, Amsterdam (1991)

    Google Scholar 

  19. Dörfler W. (1996) A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33: 1106–1124

    Article  MATH  MathSciNet  Google Scholar 

  20. Dörfler W., Wilderotter O. (2000) An adaptive finite element method for a linear elliptic equation with variable coefficients. ZAMM 80: 481–491

    Article  MATH  Google Scholar 

  21. Durán R.G., Padra C., Rodríguez R. (2003) A posteriori error estimates for the finite element approximation of eigenvalue problems. Math. Mod. Meth. Appl. Sci. 13: 1219–1229

    Article  MATH  Google Scholar 

  22. Gong X., Shen L., Zhang D., Zhou A. (2008) Finite element approximations for schrödinger equations with applications to electronic structure computations. J. Comput. Math. 26: 310–323

    MathSciNet  Google Scholar 

  23. Greiner W. (1994) Quantum Mechanics: An Introduction, 3rd edn. Springer, Berlin

    MATH  Google Scholar 

  24. Heuveline V., Rannacher R. (2001) A posteriori error control for finite element approximations of ellipic eigenvalue problems. Adv. Comput. Math. 15: 107–138

    Article  MATH  MathSciNet  Google Scholar 

  25. Larson M.G. (2001) A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems. SIAM J. Numer. Anal. 38: 608–625

    Article  MathSciNet  Google Scholar 

  26. Lin Q., Xie G. (1981) Accelerating the finite element method in eigenvalue problems. Kexue Tongbao 26: 449–452 (in Chinese)

    MathSciNet  Google Scholar 

  27. Mao D., Shen L., Zhou A. (2006) Adaptive finite algorithms for eigenvalue problems based on local averaging type a posteriori error estimates. Adv. Comput. Math. 25: 135–160

    Article  MATH  MathSciNet  Google Scholar 

  28. Maubach J. (1995) Local bisection refinement for n-simplicial grids generated by reflection. SIAM J. Sci. Comput. 16: 210–227

    Article  MATH  MathSciNet  Google Scholar 

  29. Mekchay K., Nochetto R.H. (2005) Convergence of adaptive finite element methods for general second order linear elliplic PDEs. SIAM J. Numer. Anal. 43: 1803–1827

    Article  MATH  MathSciNet  Google Scholar 

  30. Morin P., Nochetto R.H., Siebert K. (2000) Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38: 466–488

    Article  MATH  MathSciNet  Google Scholar 

  31. Morin P., Nochetto R.H., Siebert K. (2002) Convergence of adaptive finite element methods. SIAM Rev. 44: 631–658

    Article  MATH  MathSciNet  Google Scholar 

  32. Nochetto, R.H.: Adaptive finite element methods for elliptic PDE. Lecture Notes of 2006 CNA Summer School. Carnegie Mellon University, Pittsburgh (2006)

  33. Schneider R., Xu Y., Zhou A. (2006) An analysis of discontinue Galerkin method for elliptic problems. Adv. Comput. Math. 5: 259–286

    Article  MathSciNet  Google Scholar 

  34. Shen L., Zhou A. (2006) A defect correction scheme for finite element eigenvalues with applications to quantum chemistry. SIAM J. Sci. Comput. 28: 321–338

    Article  MATH  MathSciNet  Google Scholar 

  35. Sloan I.H. (1976) Iterated Galerkin method for eigenvalue problems. SIAM J. Numer. Anal. 13: 753–760

    Article  MATH  MathSciNet  Google Scholar 

  36. Stevenson R. (2007) Optimality of a standard adaptive finite element method. Found. Comput. Math. 7: 245–269

    Article  MATH  MathSciNet  Google Scholar 

  37. Stevenson R. (2008) The completion of locally refined simplicial partitions created by bisection. Math. Comp. 77: 227–241

    Article  MATH  MathSciNet  Google Scholar 

  38. Traxler C.T. (1997) An algorithm for adaptive mesh refinement in n dimensions. Computing 59: 115–137

    Article  MATH  MathSciNet  Google Scholar 

  39. Veeser A. (2002) Convergent adaptive finite elements for the nonlinear Laplacian. Numer. Math. 92: 743–770

    Article  MATH  MathSciNet  Google Scholar 

  40. Verfürth R. (1996) A Riview of a Posteriori Error Estimates and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New York

    Google Scholar 

  41. Wu H., Chen Z. (2006) Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems. Sci. China Ser. A 49: 1405–1429

    Article  MATH  MathSciNet  Google Scholar 

  42. Xu J. (1992) Iterative methods by space decomposition and subspace correction. SIAM Rev. 34: 581–613

    Article  MATH  MathSciNet  Google Scholar 

  43. Xu J., Zhou A. (2000) Local and parallel finite element algorithms based on two-grid discretizations. Math. Comp. 69: 881–909

    Article  MATH  MathSciNet  Google Scholar 

  44. Xu J., Zhou A. (2001) A two-grid discretization scheme for eigenvalue problems. Math. Comp. 70: 17–25

    Article  MATH  MathSciNet  Google Scholar 

  45. Yan N., Zhou A. (2001) Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes. Comput. Methods Appl. Mech. Eng. 190: 4289–4299

    Article  MATH  MathSciNet  Google Scholar 

  46. Yserentant H. (2004) On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98: 731–759

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Xiaoying Dai & Aihui Zhou

  2. Graduate University of Chinese Academy of Sciences, Beijing, 100190, China

    Xiaoying Dai

  3. Center for Computational Mathematics and Applications, Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Jinchao Xu

Authors
  1. Xiaoying Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinchao Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aihui Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinchao Xu.

Additional information

This work was partially supported by the National Science Foundation of China under grant 10425105 and the National Basic Research Program under grant 2005CB321704.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dai, X., Xu, J. & Zhou, A. Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer. Math. 110, 313–355 (2008). https://doi.org/10.1007/s00211-008-0169-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-008-0169-3

Mathematics Subject Classification (2000)

Search

Navigation