Numerische Mathematik

, Volume 110, Issue 3, pp 313–355

Convergence and optimal complexity of adaptive finite element eigenvalue computations

Article

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.

Mathematics Subject Classification (2000)

65F15 65N15 65N25 65N30 65N50 

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References

  1. 1.
    Adams R.A. (1975) Sobolev Spaces. Academic Press, New YorkMATHGoogle Scholar
  2. 2.
    Arnold D.N., Mukherjee A., Pouly L. (2000) Locally adapted tetrahedral meshes using bisection. SIAM J. Sci. Comput. 22: 431–448MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Babuska I., Osborn J.E. (1989) Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comp. 52: 275–297MATHCrossRefMathSciNetGoogle Scholar
  4. 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 641–792Google Scholar
  5. 5.
    Babuska I., Rheinboldt W.C. (1978) Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15: 736–754MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Babuska I., Vogelius M. (1984) Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44: 75–102MATHCrossRefMathSciNetGoogle Scholar
  7. 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: 971–994MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Becker R., Rannacher R. (2001) An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10: 1–102MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Binev P., Dahmen W., DeVore R. (2004) Adaptive finite element methods with convergence rates. Numer. Math. 97: 219–268MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Carstensen C. (2005) A unifying theory of a posteriori finite element error control. Numer. Math. 100: 617–637MATHCrossRefMathSciNetGoogle Scholar
  11. 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: 945–969MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Carstensen C., Hoppe R.H.W. (2006) Error reduction and convergence for an adaptive mixed finite element method. Math. Comp. 75: 1033–1042MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Carstensen C., Hoppe R.H.W. (2006) Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. 103: 251–266MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. preprint (2007)Google Scholar
  15. 15.
    Chatelin F. (1983) Spectral Approximations of Linear Operators. Academic Press, New YorkGoogle Scholar
  16. 16.
    Chen, L., Holst, M., Xu, J.: Convergence and optimality of adaptive mixed finite element methods. Math. Comp. (to appear) (2008)Google Scholar
  17. 17.
    Chen Z., Nochetto R.H. (2000) Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84: 527–548MATHCrossRefMathSciNetGoogle Scholar
  18. 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. 19.
    Dörfler W. (1996) A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33: 1106–1124MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Dörfler W., Wilderotter O. (2000) An adaptive finite element method for a linear elliptic equation with variable coefficients. ZAMM 80: 481–491MATHCrossRefGoogle Scholar
  21. 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: 1219–1229MATHCrossRefGoogle Scholar
  22. 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: 310–323MathSciNetGoogle Scholar
  23. 23.
    Greiner W. (1994) Quantum Mechanics: An Introduction, 3rd edn. Springer, BerlinMATHGoogle Scholar
  24. 24.
    Heuveline V., Rannacher R. (2001) A posteriori error control for finite element approximations of ellipic eigenvalue problems. Adv. Comput. Math. 15: 107–138MATHCrossRefMathSciNetGoogle Scholar
  25. 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: 608–625CrossRefMathSciNetGoogle Scholar
  26. 26.
    Lin Q., Xie G. (1981) Accelerating the finite element method in eigenvalue problems. Kexue Tongbao 26: 449–452 (in Chinese)MathSciNetGoogle Scholar
  27. 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: 135–160MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Maubach J. (1995) Local bisection refinement for n-simplicial grids generated by reflection. SIAM J. Sci. Comput. 16: 210–227MATHCrossRefMathSciNetGoogle Scholar
  29. 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: 1803–1827MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Morin P., Nochetto R.H., Siebert K. (2000) Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38: 466–488MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Morin P., Nochetto R.H., Siebert K. (2002) Convergence of adaptive finite element methods. SIAM Rev. 44: 631–658MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Nochetto, R.H.: Adaptive finite element methods for elliptic PDE. Lecture Notes of 2006 CNA Summer School. Carnegie Mellon University, Pittsburgh (2006)Google Scholar
  33. 33.
    Schneider R., Xu Y., Zhou A. (2006) An analysis of discontinue Galerkin method for elliptic problems. Adv. Comput. Math. 5: 259–286CrossRefMathSciNetGoogle Scholar
  34. 34.
    Shen L., Zhou A. (2006) A defect correction scheme for finite element eigenvalues with applications to quantum chemistry. SIAM J. Sci. Comput. 28: 321–338MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Sloan I.H. (1976) Iterated Galerkin method for eigenvalue problems. SIAM J. Numer. Anal. 13: 753–760MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Stevenson R. (2007) Optimality of a standard adaptive finite element method. Found. Comput. Math. 7: 245–269MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Stevenson R. (2008) The completion of locally refined simplicial partitions created by bisection. Math. Comp. 77: 227–241MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Traxler C.T. (1997) An algorithm for adaptive mesh refinement in n dimensions. Computing 59: 115–137MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Veeser A. (2002) Convergent adaptive finite elements for the nonlinear Laplacian. Numer. Math. 92: 743–770MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Verfürth R. (1996) A Riview of a Posteriori Error Estimates and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New YorkGoogle Scholar
  41. 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: 1405–1429MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Xu J. (1992) Iterative methods by space decomposition and subspace correction. SIAM Rev. 34: 581–613MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Xu J., Zhou A. (2000) Local and parallel finite element algorithms based on two-grid discretizations. Math. Comp. 69: 881–909MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Xu J., Zhou A. (2001) A two-grid discretization scheme for eigenvalue problems. Math. Comp. 70: 17–25MATHCrossRefMathSciNetGoogle Scholar
  45. 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: 4289–4299MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Yserentant H. (2004) On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98: 731–759MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.Graduate University of Chinese Academy of SciencesBeijingChina
  3. 3.Center for Computational Mathematics and Applications, Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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