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Science China Mathematics

, Volume 61, Issue 6, pp 973–992 | Cite as

Residual-based a posteriori error estimates for symmetric conforming mixed finite elements for linear elasticity problems

  • Long Chen
  • Jun Hu
  • Xuehai Huang
  • Hongying Man
Articles Progress of Projects Supported by NSFC
  • 68 Downloads

Abstract

A posteriori error estimators for the symmetric mixed finite element methods for linear elasticity problems with Dirichlet and mixed boundary conditions are proposed. Reliability and efficiency of the estimators are proved. Finally, numerical examples are presented to verify the theoretical results.

Keywords

symmetric mixed finite element linear elasticity problems a posteriori error estimator adaptive method 

MSC(2010)

65N30 73C02 

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Notes

Acknowledgements

This work was supported by National Science Foundation of USA (Grant No. DMS- 1418934), the Sea Poly Project of Beijing Overseas Talents, National Natural Science Foundation of China (Grant Nos. 11625101, 91430213, 11421101, 11771338, 11671304 and 11401026), Zhejiang Provincial Natural Science Foundation of China Projects (Grant Nos. LY17A010010, LY15A010015 and LY15A010016) and Wen- zhou Science and Technology Plan Project (Grant No. G20160019). The last author thanks the support of the China Scholarship Council and the University of California, Irvine during her visit to UC Irvine from 2014 to 2015.

References

  1. 1.
    Adams S, Cockburn B. A mixed finite element method for elasticity in three dimensions. J Sci Comput, 2001, 25: 515–521MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alonso A. Error estimators for a mixed method. Numer Math, 1996, 74: 385–395MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arnold D N. Differential complexes and numerical stability. In: Proceedings of the International Congress of Mathematicians, vol 1. Beijing: Higher Education Press, 2002, 137–157Google Scholar
  4. 4.
    Arnold D N, Awanou G. Rectangular mixed finite elements for elasticity. Math Models Methods Appl Sci, 2005, 15: 1417–1429MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Arnold D N, Awanou G, Winther R. Finite elements for symmetric tensors in three dimensions. Math Comp, 2008, 77: 1229–1251MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Arnold D N, Brezzi F, Douglas J. Peers: A new mixed finite element for plane elasticity. Jpn J Appl Math, 1984, 1: 347–367MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Arnold D N, Falk R S, Winther R. Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math Comp, 2007, 76: 1699–1724MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Arnold D N, Winther R. Mixed finite elements for elasticity. Numer Math, 2002, 92: 401–419MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Arnold D N, Winther R. Nonconforming mixed elements for elasticity. Math Models Methods Appl Sci, 2003, 13: 295–307MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Boff D, Brezzi F, Fortin M. Reduced symmetry elements in linear elasticity. Commun Pure Appl Anal, 2009, 8: 95–121MathSciNetMATHGoogle Scholar
  11. 11.
    Boff D, Brezzi F, Fortin M. Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Heidelberg: Springer, 2013Google Scholar
  12. 12.
    Bramble J H, Xu J. A local post-processing technique for improving the accuracy in mixed finite-element approximations. SIAM J Numer Anal, 1989, 26: 1267–1275MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Carstensen C. A posteriori error estimate for the mixed finite element method. Math Comp, 1997, 66: 465–477MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Carstensen C, Dolzmann G. A posteriori error estimates for mixed FEM in elasticity. Numer Math, 1998, 81: 187–209MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Carstensen C, Eigel M, Gedicke J. Computational competition of symmetric mixed FEM in linear elasticity. Comput Methods Appl Mech Engrg, 2011, 200: 2903–2915MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Carstensen C, Gedicke J. Robust residual-based a posteriori Arnold-Winther mixed finite element analysis in elasticity. Comput Methods Appl Mech Engrg, 2016, 300: 245–264MathSciNetCrossRefGoogle Scholar
  17. 17.
    Carstensen C, Günther D, Reininghaus J, et al. The Arnold-Winther mixed FEM in linear elasticity. Part I: Implementation and numerical verification. Comput Methods Appl Mech Engrg, 2008, 197: 3014–3023MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Carstensen C, Hu J. A unifying theory of a posteriori error control for nonconforming finite element methods. Numer Math, 2007, 107: 473–502MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Chen L, Holst M, Xu J. Convergence and optimality of adaptive mixed finite element methods. Math Comp, 2009, 78: 35–35MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Chen L, Hu J, Huang X. Fast auxiliary space preconditioner for linear elasticity in mixed form. Math Comp, 2017, https://doi.org/10.1090/mcom/3285Google Scholar
  21. 21.
    Cockburn B, Gopalakrishnan J, Guzmán J. A new elasticity element made for enforcing weak stress symmetry. Math Comp, 2010, 79: 1331–1349MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Gatica G N, Maischak M. A posteriori error estimates for the mixed finite element method with Lagrange multipliers. Numer Methods Partial Differential Equations, 2005, 21: 421–450MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Girault V, Scott L R. Hermite interpolation of nonsmooth functions preserving boundary conditions. Math Comp, 2002, 71: 1043–1074MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Grisvard P. Singularities in Boundary Value Problems. Research in Applied Mathematics, vol. 22. Paris: Masson, 1992Google Scholar
  25. 25.
    Guzmán J. A unified analysis of several mixed methods for elasticity with weak stress symmetry. J Sci Comput, 2010, 44: 156–169MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Hoppe R H W,Wohlmuth B. Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems. SIAM J Numer Anal, 1997, 34: 1658–1681MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Hu J. Finite element approximations of symmetric tensors on simplicial grids in Rn: The higher order case. J Comput Math, 2015, 33: 283–296MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Hu J. A new family of effcient conforming mixed finite elements on both rectangular and cuboid meshes for linear elasticity in the symmetric formulation. SIAM J Numer Anal, 2015, 53: 1438–1463MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Hu J, Zhang S. A family of conforming mixed finite elements for linear elasticity on triangular grids. ArXiv:1406.7457, 2014Google Scholar
  30. 30.
    Hu J, Zhang S. A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids. Sci China Math, 2015, 58: 297–307MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Hu J, Zhang S. Finite element approximations of symmetric tensors on simplicial grids in Rn: The lower order case. Math Models Methods Appl Sci, 2016, 26: 1649–1669MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kim K Y. A posteriori error estimator for linear elasticity based on nonsymmetric stress tensor approximation. J Korean Soc Ind Appl Math, 2012, 16: 1–13MathSciNetMATHGoogle Scholar
  33. 33.
    Larson M G, Målqvist A. A posteriori error estimates for mixed finite element approximations of elliptic problems. Numer Math, 2008, 108: 487–500MathSciNetCrossRefGoogle Scholar
  34. 34.
    Lonsing M, Verfürth R. A posteriori error estimators for mixed finite element methods in linear elasticity. Numer Math, 2004, 97: 757–778MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Lovadina C, Stenberg R. Energy norm a posteriori error estimates for mixed finite element methods. Math Comp, 2006, 75: 1659–1674MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Morgan J, Scott R. A nodal basis for C 1 piecewise polynomials of degree n ≥ 5. Math Comp, 1975, 29: 736–740MATHGoogle Scholar
  37. 37.
    Shi Z C, Wang M. Finite Element Methods. Beijing: Science Press, 2013Google Scholar
  38. 38.
    Stenberg R. A family of mixed finite elements for the elasticity problem. Numer Math, 1988, 53: 513–538MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at IrvineIrvineUSA
  2. 2.LMAM and School of Mathematical SciencesPeking UniversityBeijingChina
  3. 3.College of Mathematics and Information ScienceWenzhou UniversityWenzhouChina
  4. 4.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina

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