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A posteriori error estimator for eigenvalue problems by mixed finite element method

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Abstract

In this paper, a residual type of a posteriori error estimator for the general second order elliptic eigenpair approximation by the mixed finite element method is derived and analyzed, based on a type of superconvergence result of the eigenfunction approximation. Its efficiency and reliability are proved by both theoretical analysis and numerical experiments.

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References

  1. Ainsworth M, Oden J. A Posteriori Error Estimation in Finite Element Analysis. New York: Wiley-Interscience, 2000

    Book  MATH  Google Scholar 

  2. Babuška I, Osborn J. Eigenvalue problems. In: Lions P G, Ciarlet P G, Eds. Handbook of Numerical Analysis, vol. II. Finite Element Methods (Part 1). Amsterdam: North-Holland, 1991, 641–787

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuška I, Rheinboldt W C. A posteriori error estimators in the finite element method. Int J Numer Meth Eng, 1978, 12: 1597–1615

    Article  MATH  Google Scholar 

  5. Bacuta C, Bramble J H. Regularity estimates for solutions of the equations of linear elasticity in convex plane polygonal domains. Z Angew Math Phys, 2003, 54: 874–878

    Article  MathSciNet  MATH  Google Scholar 

  6. Bank R E, Sherman A H, Weiser A. Refinement algorithms and data structures for regular local mesh refinement. In: Scientific Computing. Amsterdam: North-Holland Publishing, 1983, 3–17

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Boffi D. Finite element approximation of eigenvalue problems. Acta Numer, 2010, 19: 1–120

    Article  MathSciNet  MATH  Google Scholar 

  9. Boffi D, Brezzi F, Gastaldi L. On the convergenc of eigenvalues for mixed formulations. Ann Sc Norm Sup Pisa Cl Sci, 1997, 25: 131–154

    MathSciNet  MATH  Google Scholar 

  10. Boffi D, Brezzi F, Gastaldi L. On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math Comp, 2000, 69: 121–140

    Article  MathSciNet  MATH  Google Scholar 

  11. Boffi D, Brezzi F, Gastaldi L. Some remarks on eigenvalue approximation by finite elements. In: Fronitiers in Numerical Analysis. New York: Springer-Verlag, 2010, 1–77

    Google Scholar 

  12. Brenner S, Scott L. The Mathematical Theory of Finite Element Methods. New York: Springer-Verlag, 1994

    Book  MATH  Google Scholar 

  13. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. New York: Springer-Verlag, 1991

    Book  MATH  Google Scholar 

  14. Chatelin F. Spectral Approximation of Linear Operators. New York: Academic Press, 1983

    MATH  Google Scholar 

  15. Chen H, Jia S, Xie H. Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem. Appl Numer Math, 2011, 61: 615–629

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen Z. A posteriori error analysis and adaptive methods for partial differential equations. In: International Congress of Mathematicians, vol. III. Zürich: Europ Math Soc, 2006, 1163–1180

    Google Scholar 

  17. Ciarlet P G. The finite Element Method for Elliptic Problem. Amsterdam: North-holland, 1978

    Google Scholar 

  18. Clément P. Approximation by finite elementfunctions using local regularization. RARIO Anal Numer, 1975, 2: 77–84

    Google Scholar 

  19. Dai X, Xu J, Zhou A. Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer Math, 2008, 110: 313–355

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Durán R, Gastaldi L, Padra C. A posteriori error estimators for mixed approximations of eigenvalue problems. Math Models Methods Appl Sci, 1999, 9: 1165–1178

    Article  MathSciNet  MATH  Google Scholar 

  22. Durán R G, Padra C, Rodriguez R. A posteriori estimates for the finite element approximation of eigenvalue problems. Math Models Methods Appl Sci, 2003, 13: 1219–1229

    Article  MathSciNet  MATH  Google Scholar 

  23. Garau E, Morin P, Zuppa C. Convergence of adaptive finite element methods for eigenvalue problems. Math Models Methods Appl Sci, 2009, 19: 721–747

    Article  MathSciNet  MATH  Google Scholar 

  24. Gardini F. A posteriori error estimates for an eigenvalue problem arising from fluid-structure interaction. Istituto Lombardo Rend Sc, 2004, 138: 17–34

    MathSciNet  Google Scholar 

  25. Gardini F. Mixed approximation of eigenvalue problems: a superconvergence result. ESAIM Math Model Numer Anal, 2009, 43: 853–865

    Article  MathSciNet  MATH  Google Scholar 

  26. Giani S, Graham I G. A convergent adaptive method for elliptic eigenvalue problems. SIAM J Numer Anal, 2009, 47: 1067–1091

    Article  MathSciNet  MATH  Google Scholar 

  27. Girault V, Raviart P. Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, 1986

    Book  MATH  Google Scholar 

  28. Grisvard P. Singularities in Boundary Value Problems. Paris-Berlin: Masson-Springer-Verlag, 1992

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Huang J G, Xu Y F. Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation. Sci China Math, 2012, 55: 1083–1098

    Article  MathSciNet  MATH  Google Scholar 

  31. Larson M G. A posteriori and a priori error analysis for finite element approximations of selfadjoint elliptic eigenvalue problems. SIAM J Numer Anal, 2000, 38: 608–625

    Article  MathSciNet  MATH  Google Scholar 

  32. Lin Q, Xie H. A Superconvergence result for mixed finite element approximations of the eigenvalue problem. ESAIM Math Model Numer Anal, 2012, 46: 797–812

    Article  MathSciNet  Google Scholar 

  33. Lovadina C, Stenberg R. Energy norm a posteriori error estimates for mixed finite element methods. Math Comp, 2006, 75: 1659–1674

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Mercier B, Osborn J, Rappaz J, et al. Eigenvalue approximation by mixed and hybrid methods. Math Comp, 1981, 36: 427–453

    Article  MathSciNet  MATH  Google Scholar 

  36. Mitchell W F. A comparison of adaptive refinement techniques for elliptic problems. ACM Trans Math Softw, 1989, 15: 326–347

    Article  MATH  Google Scholar 

  37. Rivara M C. Design and data structure for fully adaptive, multigrid finite element software. ACM Trans Math Softw, 1984, 10: 242–264

    Article  MathSciNet  MATH  Google Scholar 

  38. Rivara M C. Mesh refinement processes based on the generalized bisection of simplices. SIAM J Numer Anal, 1984, 21: 604–613

    Article  MathSciNet  MATH  Google Scholar 

  39. Scott L, Zhang S. Finite element interpolation of nonsmooth functions staisfying boundary conditions. Math Comp, 1990, 54: 483–493

    Article  MathSciNet  MATH  Google Scholar 

  40. Sewell E G. Automatic generation of triangulations for piecewise polynomial approximation. PhD dissertation, Purdue University, 1972

    Google Scholar 

  41. Verfürth R. A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques. Chichester-New York: Wiley & Teubner, 1996

    MATH  Google Scholar 

  42. Xie X P, Xu J C. New mixed finite elements for plane elasticity and Stokes equations. Sci China Math, 2011, 54: 1499–1519

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Applied Mathematics, Central University of Finance and Economics, Beijing, 100081, China

    ShangHui Jia

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    HongTao Chen

  3. LSEC, NCMIS, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    HeHu Xie

Authors
  1. ShangHui Jia
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  2. HongTao Chen
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  3. HeHu Xie
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Correspondence to HongTao Chen.

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Jia, S., Chen, H. & Xie, H. A posteriori error estimator for eigenvalue problems by mixed finite element method. Sci. China Math. 56, 887–900 (2013). https://doi.org/10.1007/s11425-013-4614-0

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  • DOI: https://doi.org/10.1007/s11425-013-4614-0

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