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An oscillation-free adaptive FEM for symmetric eigenvalue problems

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Abstract

A refined a posteriori error analysis for symmetric eigenvalue problems and the convergence of the first-order adaptive finite element method (AFEM) is presented. The H 1 stability of the L 2 projection provides reliability and efficiency of the edge-contribution of standard residual-based error estimators for P 1 finite element methods. In fact, the volume contributions and even oscillations can be omitted for Courant finite element methods. This allows for a refined averaging scheme and so improves (Mao et al. in Adv Comput Math 25(1–3):135–160, 2006). The proposed AFEM monitors the edge-contributions in a bulk criterion and so enables a contraction property up to higher-order terms and global convergence. Numerical experiments exploit the remaining L 2 error contributions and confirm our theoretical findings. The averaging schemes show a high accuracy and the AFEM leads to optimal empirical convergence rates.

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References

  1. Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. In: Pure and Applied Mathematics. Wiley-Interscience, New York (2000)

  2. Alberty J., Carstensen C., Funken S.A.: Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms 20(2–3), 117–137 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alberty J., Carstensen C., Funken S.A., Klose R.: Matlab implementation of the finite element method in elasticity. Computing 69(3), 239–263 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuška, I., Osborn, J.: Eigenvalue problems. In: Handbook of Numerical Analysis, vol II, pp. 641–787. North-Holland, Amsterdam (1991)

  5. Babuška I., Osborn J.E.: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comput. 52(186), 275–297 (1989)

    MATH  Google Scholar 

  6. Bangerth W., Rannacher R.: Adaptive finite element methods for differential equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2003)

    Google Scholar 

  7. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. In: Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)

  8. Carstensen C.: An adaptive mesh-refining algorithm allowing for an H 1 stable L 2 projection onto Courant finite element spaces. Constr. Approx. 20(4), 549–564 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carstensen C.: All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable. Math. Comput. 73(247), 1153–1165 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Carstensen C.: A unifying theory of a posteriori finite element error control. Numer. Math. 100(4), 617–637 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cascon J.M., Kreuzer C., Nochetto R.H., Siebert K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chatelin F.: Spectral approximation of linear operators. Computer Science and Applied Mathematics. Academic Press, New York (1983)

    Google Scholar 

  13. Durán R.G., Padra C., Rodríguez R.: A posteriori error estimates for the finite element approximation of eigenvalue problems. Math. Models Methods Appl. Sci. 13(8), 1219–1229 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Giani S., Graham I.G.: A convergent adaptive method for elliptic eigenvalue problems. SIAM J. Numer. Anal. 47(2), 1067–1091 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Heuveline V., Rannacher R.: A posteriori error control for finite approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15(1–4), 107–138 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Knyazev A.V.: New estimates for Ritz vectors. Math. Comput. 66(219), 985–995 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Knyazev A.V., Osborn J.E.: New a priori FEM error estimates for eigenvalues. SIAM J. Numer. Anal. 43(6), 2647–2667 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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(2), 608–625 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Larsson, S., Thomée, V.: Partial differential equations with numerical methods. In: Texts in Applied Mathematics, vol. 45. Springer, Berlin (2003)

  20. Mao D., Shen L., Zhou A.: Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates. Adv. Comput. Math. 25(1–3), 135–160 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Raviart P.A., Thomas J.M.: Introduction à l’analyse numérique des équations aux dérivées partielles. Collection Mathématiques Appliquées pour la Maî trise. Masson, Paris (1983)

    Google Scholar 

  22. Sauter S.: hp-finite elements for elliptic eigenvalue problems: error estimates which are explicit with respect to λ, h, and p. SIAM J. Numer. Anal. 48(1), 95–108 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Strang G., Fix G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)

    Google Scholar 

  24. Verfürth R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley and Teubner, New York (1996)

    MATH  Google Scholar 

  25. Walsh T.F., Reese G.M., Hetmaniuk U.L.: Explicit a posteriori error estimates for eigenvalue analysis of heterogeneous elastic structures. Comput. Methods Appl. Mech. Eng. 196(37–40), 3614–3623 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Carsten Carstensen & Joscha Gedicke

  2. Department of Computational Science and Engineering, Yonsei University, Seoul, 120-749, Korea

    Carsten Carstensen

Authors
  1. Carsten Carstensen
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  2. Joscha Gedicke
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Correspondence to Carsten Carstensen.

Additional information

Supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin and the Hausdorff Institute for Mathematics in Bonn, Germany.

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Carstensen, C., Gedicke, J. An oscillation-free adaptive FEM for symmetric eigenvalue problems. Numer. Math. 118, 401–427 (2011). https://doi.org/10.1007/s00211-011-0367-2

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