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Convergence and optimality of the adaptive Morley element method

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Abstract

This paper is devoted to the convergence and optimality analysis of the adaptive Morley element method for the fourth order elliptic problem. A new technique is developed to establish a quasi-orthogonality which is crucial for the convergence analysis of the adaptive nonconforming method. By introducing a new parameter-dependent error estimator and further establishing a discrete reliability property, sharp convergence and optimality estimates are then fully proved for the fourth order elliptic problem.

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References

  1. Ainsworth M., Rankin R.: Robust a posteriori error estimation for the nonconforming Fortin-Soulie finite-element approximation. Math. Comput. 77, 1917–1939 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. da Veiga L.B., Niiranen J., Stenberg R.: A posteriori error estimates for the Morley plate bending element. Numer. Math. 106, 165–179 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Binev P., Dahmen W., DeVore R., Petrushev P.: Approximation classes for adaptive methods. Serdica Math. J. 28, 391–416 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Binev P., DeVore R.: Fast computation in adaptive tree approximation. Numer. Math. 97, 193–217 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Brenner S., Scott L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)

    MATH  Google Scholar 

  8. Brenner S.C., Sung L.-Y.: C 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22(23), 83–118 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Carstensen C., Bartels S., Jansche S.: A posteriori error estimates for nonconforming finite-element methods. Numer. Math. 92, 233–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carstensen C., Hu J.: A unifying theory of a posteriori error control for nonconforming finite-element methods. Numer. Math. 107, 473–502 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Carstensen C., Hu J., Orlando A.: Framework for the a posteriori error analysis of nonconforming finite elements. SIAM. J. Numer. Anal. 45, 68–82 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen L., Holst M., Xu J.: Convergence and optimality of adaptive mixed finite-element methods. Math. Comput. 78, 35–53 (2008)

    MathSciNet  Google Scholar 

  16. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North–Holland, 1978; reprinted as SIAM Classics in Applied Mathematics (2002)

  17. Dari E., Duran R., Padra C.: Error estimators for nonconforming finite-element approximations of the Stokes problem. Math. Comput. 64, 1017–1033 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dari E., Duran R., Padra C., Vampa V.: A posteriori error estimators for nonconforming finite- element methods. Math. Model. Numer. Anal. 30, 385–400 (1996)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Gudi T.: A new error analysis for discontinous finite element methods for linear problems. Math. Comput. 79, 2169–2189 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hu J., Shi Z.C.: A new a posteriori error estimate for the Morley element. Numer. Math. 112, 25–40 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hu, J., Shi, Z.C., Xu, J.C.: Convergence and optimality of adaptive nonconforming methods for high-order differential equations, Research Report 19. School of Mathematical Sciences and Institute of Mathematics, Peking University (2009)

  23. Hu, J., Xu, J.C.: Convergence of Adaptive Conforming and Nonconforming Finite Element Methods for the Perturbed Stokes Equation, Research Report 73. School of Mathematical Sciences and Institute of Mathematics, Peking University. http://www.math.pku.edu.cn:8000/var/preprint/7297.pdf (2007)

  24. Marini L.: An inexpensive method for the evaluation of the solution of the lowest order Raviart–Thomas mixed method. SIAM J. Numer. Anal. 22, 493–496 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  25. Morley L.S.D.: The triangular equilibrium element in the solutions of plate bending problem. Aero. Q. 19, 149–169 (1968)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Shi Z.C.: Error estimates for the Morley element. Chin. J. Numer. Math. Appl. 12, 102–108 (1990)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Stevenson R.: The completion of locally refined simplicial partitions created by bisection. Math.Comput. 77, 227–241 (2007)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  32. Wang, M., Zhang, S.: Local a priori and a posteriori error estimates of finite-elements for biharmonic equation. Research Report 13. School of Mathematical Sciences and Institute of Mathematics, Peking University (2006)

  33. Wang, M., Xu, J.C.: Minimal finite-element spaces for 2m-th order partial differential equations in R n. Research Report 29. School of Mathematical Sciences and Institute of Mathematics, Peking University (submitted to Mathematics of Computation) (2006)

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Jun Hu

  2. Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China

    Zhongci Shi

  3. The School of Mathematical Sciences, Peking University, Beijing, People’s Republic of China

    Jinchao Xu

  4. Department of Mathematics, Pennsylvania State University, University Park, PA, 16801, USA

    Jinchao Xu

Authors
  1. Jun Hu
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  2. Zhongci Shi
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  3. Jinchao Xu
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Corresponding author

Correspondence to Jun Hu.

Additional information

The first author was supported in part by the NSFC Key Project 11031006 under Grant 10971005, and Foundation for the Author of National Excellent Doctoral Dissertation of PR China 200718, and partially supported by the Chinesisch-Deutsches Zentrum project GZ578. The third author was supported in part by NSF 0915153 and NSFC-10528102.

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Hu, J., Shi, Z. & Xu, J. Convergence and optimality of the adaptive Morley element method. Numer. Math. 121, 731–752 (2012). https://doi.org/10.1007/s00211-012-0445-0

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  • DOI: https://doi.org/10.1007/s00211-012-0445-0

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