Skip to main content
SpringerLink
Log in

Local a priori/a posteriori error estimates of conforming finite elements approximation for Steklov eigenvalue problems

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Based on the work of Xu and Zhou (2000), this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems, and proves a local a priori error estimate and a new local a posteriori error estimate in \(\left\| \cdot \right\|_{1,\Omega _0 }\) norm for conforming elements eigenfunction, which has not been studied in existing literatures.

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. Alonso A, Russo A D. Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods. J Comput Appl Math, 2009, 223: 177–197

    Article  MATH  MathSciNet  Google Scholar 

  2. Andreev A B, Todorov T D. Isoparametric finite element approximation of a Steklov eigenvalue problem. IMA J Numer Anal, 2004, 24: 309–322

    Article  MATH  MathSciNet  Google Scholar 

  3. Armentano M G. The effect of reduced integration in the Steklov eigenvalue problem. Math Model Numer Anal, 2004, 38: 27–36

    Article  MATH  MathSciNet  Google Scholar 

  4. Armentano M G, Padra C. A posteriori error estimates for the Steklov eigenvalue problem. Appl Numer Math, 2008, 58: 593–601

    Article  MATH  MathSciNet  Google Scholar 

  5. Babuska I, Osborn J E. Eigenvalue problems. In: Ciarlet P G, Lions J L, eds. Finite Element Methods (Part I). Handbook of Numerical Analysis, vol. 2. North-Holand: Elsevier Science Publishers, 1991: 641–787

    Chapter  Google Scholar 

  6. Bergman S, Schiffer M. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. New York: Academic Press, 1953

    Google Scholar 

  7. Bermudez A, Rodriguez R, Santamarina D. A finite element solution of an added mass formulation for coupled fluidsolid vibrations. Numer Math, 2000, 87: 201–227

    Article  MATH  MathSciNet  Google Scholar 

  8. Bi H, Ren S X, Yang Y D. Conforming finite element approximations for a fourth-order Steklov eigenvalue problem. Math Probab Eng, 2011, 2011: 1–13, doi:10.1155/2011/873152

    Article  Google Scholar 

  9. Bi H, Yang Y D. A two-grid method of the non-conforming Crouzeix-Raviart element for the Steklov eigenvalue problem. Appl Math Comput, 2011, 217: 9669–9678

    Article  MATH  MathSciNet  Google Scholar 

  10. Bi H, Yang Y D. Multi-scale discretizaiton scheme based on the Rayleigh quotient iterative method for the Steklov eigenvalue problem. Math Probab Eng, 2012, 2012: 1–18, doi:10.1155/2012/487207

    Article  Google Scholar 

  11. Bramble J H, Osborn J E. Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators. In: Aziz A K, ed. Math Foundations of the Finite Element Method with Applications to PDE. New York: Academic Press, 1972, 387–408

    Google Scholar 

  12. Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods. New York: Springer-Verlag, 2002

    Book  MATH  Google Scholar 

  13. Bucur D, Ionescu I R. Asymptotic analysis and scaling of friction parameters. Z Angew Math Phys, 2006, 57: 1042–1056

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  15. Ciarlet P G. The Finite Element Method for Elliptic Proplems. Amsterdam: North-Holand, 1978

    Google Scholar 

  16. Conca C, Planchard J, Vanninathanm M. Fluid and Periodic Structures. New York: John Wiley & Sons, 1995

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Dai X, Zhou A H. Three-scale finite element discretizations for quantum eigenvalue problems. SIAM J Numer Anal, 2008, 46: 295–324

    Article  MATH  MathSciNet  Google Scholar 

  19. Dauge M. Elliptic boundary value problems on corner domains: Smoothness and asymptotics of solutions. In: Lecture Notes in Mathematics, vol. 1341. Berlin: Springer-Verlag, 1988

    MATH  Google Scholar 

  20. Garau E M, Morin P. Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems. IMA J Numer Anal, 2011, 31: 914–946

    Article  MATH  MathSciNet  Google Scholar 

  21. Li M X, Lin Q, Zhang S H. Extrapolation and superconvergence of the Steklov eigenvalue problems. Adv Comput Math, 2010, 33: 25–44

    Article  MATH  MathSciNet  Google Scholar 

  22. Li Q, Yang Y D. A two-grid discretization scheme for the Steklov eigenvalue problem. J Appl Math Comput, 2011, 36: 129–139

    Article  MATH  MathSciNet  Google Scholar 

  23. Luo X B, Chen Y P, Huang Y Q. A priori error estimates of finite volume element method for hyperbolic optimal control problems. Sci China Math, 2013, 56: 901–914

    Article  MATH  MathSciNet  Google Scholar 

  24. Russo A D, Alonso A E. Aposteriori error estimates for nonconforming approximations of Steklov eigenvalue problems. Comput Math Appl, 2011, 62: 4100–4117

    Article  MATH  MathSciNet  Google Scholar 

  25. Savaré G. Regularity results for elliptic equations in Lipschitz domains. J Funct Anal, 1998, 152: 176–201

    Article  MATH  MathSciNet  Google Scholar 

  26. Shi Z C, Wang M. Finite Element Method. Beijing: Science Press, 2010

    Google Scholar 

  27. Verfürth R. A Riview of A Posteriori Error Estimates and Adaptive Mesh-Refinement Techniques. New York: Wiley-Teubner, 1996

    Google Scholar 

  28. Verfürth R. A posteriori error estimates for convection-diffusion equations. Numer Math, 1998, 80: 641–663

    Article  MATH  MathSciNet  Google Scholar 

  29. Wahlbin L B. Local behavior in finite element methods. In: Ciarlet P G, Lions J L, eds. Handbook of Numerical Analysis, vol. II. Finite Element Methods, part I. North-Holand: Elsevier Science Publishers, 1991

    Google Scholar 

  30. Wahlbin L B. Superconvergence in Galerkin Finite Element Methods. Berlin: Springer-Verlag, 1995

    MATH  Google Scholar 

  31. Wang L H, Xu X J. Foundation of Mathematics in Finite Element Methods. Beijing: Science Press, 2004

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  33. Xu J C, Zhou A H. A two-grid discretization scheme for eigenvalue problems. Math Comp, 1999, 70: 17–25

    Article  MathSciNet  Google Scholar 

  34. Xu J C, Zhou A H. Local and parallel finite element algorithms for eigenvalue problems. Acta Math Appl Sin Engl Ser, 2002, 18: 185–200

    Article  MATH  MathSciNet  Google Scholar 

  35. Yang Y D, Li Q, Li S R. Nonconforming finite element approximations of the Steklov eigenvalue problem. Appl Numer Math, 2009, 59: 2388–2401

    Article  MATH  MathSciNet  Google Scholar 

  36. Yang Y D, Bi H. A two-grid discretization scheme based on shifted-inverse power method. SIAM J Numer Anal, 2011, 49: 1602–1624

    Article  MATH  MathSciNet  Google Scholar 

  37. Yang Y D, Jiang W. Upper spectral bounds and a posteriori error analysis of several mixed finite element approximations for the Stokes eigenvalue problem. Sci China Math, 2013, 56: 1313–1330

    Article  MATH  MathSciNet  Google Scholar 

  38. Zuppa C. A posteriori error estimates for the finite element approximation of Steklov eigenvalue problems. Mecá Comput, 2007, 26: 724–735

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, 550001, China

    YiDu Yang & Hai Bi

Authors
  1. YiDu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hai Bi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YiDu Yang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, Y., Bi, H. Local a priori/a posteriori error estimates of conforming finite elements approximation for Steklov eigenvalue problems. Sci. China Math. 57, 1319–1329 (2014). https://doi.org/10.1007/s11425-013-4709-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-013-4709-7

Keywords

MSC(2010)

Search

Navigation