Skip to main content
SpringerLink
Log in

Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

A hybridization technique is applied to the weak Galerkin finite element method (WGFEM) for solving the linear elasticity problem in mixed form. An auxiliary function, the Lagrange multiplier defined on the boundary of elements, is introduced in this method. Consequently, the computational costs are much lower than the standard WGFEM. Optimal order error estimates are presented for the approximation scheme. Numerical results are provided to verify the theoretical results.

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. Adams S, Cockburn B. A mixed finite element method for elasticity in three dimensions. J Sci Comput, 2005, 25: 515–521

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold D N, Brezzi F. Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél Math Anal Numér, 1985, 19: 7–32

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold D N, Brezzi F, Douglas J. PEERS: a new mixed finite element for plane elasticity. Japan J Appl Math, 1984, 1: 347–367

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold D N, Douglas J, Gupta C P. A family of higher order mixed finite element methods for plane elasticity. Numer Math, 1984, 45: 1–22

    Article  MathSciNet  MATH  Google Scholar 

  5. Arnold D N, Falk R S, Winther R. Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math Comp, 2007, 76: 1699–1723

    Article  MathSciNet  MATH  Google Scholar 

  6. Arnold D N, Winther R. Mixed finite elements for elasticity. Numer Math, 2002, 92: 401–419

    Article  MathSciNet  MATH  Google Scholar 

  7. Baiocchi C, Brezzi F, Marini L D. Stabilization of Galerkin methods and applications to domain decomposition. In: Lecture Notes in Comput Sci, Vol 653. Berlin: Springer, 1992, 345–355

    Google Scholar 

  8. Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev Francaise Automat Informat Recherche Opérationnelle Sér, 1974, 8: 129–151

    MathSciNet  MATH  Google Scholar 

  9. Brezzi F, Douglas J, Marini L D. Two families of mixed finite elements for second order elliptic problems. Numer Math, 1985, 47: 217–235

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen G, Feng M, Xie X. Robust globally divergence-free weak Galerkin methods for Stokes equations. J Comput Math, 2016, 34: 549–572

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen G, Feng M, Xie X. A class of robust WG/HDG finite element method for convection-diffusion-reaction equations. J Comput Appl Math, 2017, 315: 107–125

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen G, Xie X. A robust weak Galerkin finite element method for linear elasticity with strong symmetric stresses. Comput Methods Appl Math, 2016, 16: 389–408

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen L, Wang J, Ye X. A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems. J Sci Comput, 2014, 59: 496–511

    Article  MathSciNet  MATH  Google Scholar 

  14. Johnson C, Mercier B. Some equilibrium finite element methods for two-dimensional elasticity problems. Numer Math, 1978, 30: 103–116

    Article  MathSciNet  MATH  Google Scholar 

  15. Li B, Xie X. A two-level algorithm for the weak Galerkin discretization of diffusion problems. J Comput Appl Math, 2015, 287: 179–195

    Article  MathSciNet  MATH  Google Scholar 

  16. Li J, Wang X, Zhang K. Multi-level Monte Carlo weak Galerkin method for elliptic equations with stochastic jump coefficients. Appl Math Comput, 2016, 275: 181–194

    MathSciNet  Google Scholar 

  17. Mu L, Wang J, Ye X. A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods. J Comput Phys, 2014, 273: 327–342

    Article  MathSciNet  MATH  Google Scholar 

  18. Mu L, Wang J, Ye X. Weak Galerkin finite element methods on polytopal meshes. Int J Numer Anal Model, 2015, 12: 31–53

    MathSciNet  MATH  Google Scholar 

  19. Mu L, Wang J, Ye X. A hybridized formulation for the weak Galerkin mixed finite element method. J Comput Appl Math, 2016, 307: 335–345

    Article  MathSciNet  MATH  Google Scholar 

  20. Mu L, Wang J, Ye X, Zhang S. A C0-weak Galerkin finite element method for the biharmonic equation. J Sci Comput, 2014, 59: 473–495

    Article  MathSciNet  MATH  Google Scholar 

  21. Qiu W, Demkowicz L. Mixed hp-finite element method for linear elasticity with weakly imposed symmetry. Comput Methods Appl Mech Engrg, 2009, 198: 3682–3701

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang C, Wang J, Wang R, Zhang R. A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation. J Comput Appl Math, 2016, 307: 346–366

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang J, Ye X. A weak Galerkin finite element method for second-order elliptic problems. J Comput Appl Math, 2013, 241: 103–115

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang J, Ye X. A weak Galerkin mixed finite element method for second order elliptic problems. Math Comp, 2014, 83: 2101–2126

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang J, Ye X. A weak Galerkin finite element method for the Stokes equations. Adv Comput Math, 2016, 42: 155–174

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang R, Wang X, Zhai Q, Zhang R. A weak Galerkin finite element scheme for solving the stationary Stokes equations. J Comput Appl Math, 2016, 302: 171–185

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang R, Zhang R. A weak Galerkin finite element method for the linear elasticity problem in mixed form. J Comput Math, 2018, 36: 469–491

    Article  MathSciNet  Google Scholar 

  28. Watwood V B Jr, Hartz B J. An equilibrium stress field model for finite element solutions of two-dimensional elastostatic problems. Internat J Solids Structures, 1968, 4: 857–873

    Article  MATH  Google Scholar 

  29. Zhai Q, Zhang R, Mu L. A new weak Galerkin finite element scheme for the Brinkman model. Commun Comput Phys, 2016, 19: 1409–1434

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang R, Zhai Q. A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order. J Sci Comput, 2015, 64: 559–585

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11771179, 11726102, 91630201, U1530116, 11471141, 11871245), the Program for Cheung Kong Scholars of Ministry of Education of China, and the Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University (No. 93K172018Z01).

Author information

Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, China

    Ruishu Wang, Kai Zhang & Qian Zhou

  2. Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA

    Xiaoshen Wang

Authors
  1. Ruishu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoshen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qian Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoshen Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, R., Wang, X., Zhang, K. et al. Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form. Front. Math. China 13, 1121–1140 (2018). https://doi.org/10.1007/s11464-018-0730-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-018-0730-z

Keywords

MSC

Search

Navigation