Skip to main content
SpringerLink
Log in

A Weak Galerkin Mixed Finite Element Method for Biharmonic Equations

  • Conference paper
  • First Online:
Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 45))

Abstract

This article introduces and analyzes a weak Galerkin mixed finite element method for solving the biharmonic equation. The weak Galerkin method, first introduced by two of the authors (J. Wang and X. Ye) in (Wang et al., Comput. Appl. Math. 241:103–115, 2013) for second-order elliptic problems, is based on the concept of discrete weak gradients. The method uses completely discrete finite element functions, and, using certain discrete spaces and with stabilization, it works on partitions of arbitrary polygon or polyhedron. In this article, the weak Galerkin method is applied to discretize the Ciarlet–Raviart mixed formulation for the biharmonic equation. In particular, an a priori error estimation is given for the corresponding finite element approximations. The error analysis essentially follows the framework of Babus̆ka, Osborn, and Pitkäranta (Math. Comp. 35:1039–1062, 1980) and uses specially designed mesh-dependent norms. The proof is technically tedious due to the discontinuous nature of the weak Galerkin finite element functions. Some computational results are presented to demonstrate the efficiency of the method.

AMS subject classifications. Primary, 65N15, 65N30

The research of Wang was supported by the NSF IR/D program, while working at the Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

This research was supported in part by National Science Foundation Grant DMS-1115097.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Adams, R., Fournier, J.: Sobolev Spaces. Academic press, New York (2003)

    MATH  Google Scholar 

  2. Adini, A., Glough, R.W.: Analysis of plate bending by the finite element method. NSF report G, 7337 (1961)

    Google Scholar 

  3. Arnold, D.N., Scott, L.R., Vogelius, M.: Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15(4), 169–192 (1988)

    MathSciNet  MATH  Google Scholar 

  4. Babus̆ka, I.: The finite element method with Lagrange multipliers. Numer. Math. 20, 179–192 (1973)

    Google Scholar 

  5. Babus̆ka, I., Osborn, J., Pitk aranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comp. 35, 1039–1062 (1980)

    Google Scholar 

  6. Behrens, E.M., Guzmán, J.: A mixed method for the biharmonic problem based on a system of first-order equations. SIAM J. Numer. Anal. 49, 789–817 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Blum, H., Rannacher, R., Leis, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2, 556–581 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brenner, S.C., Sung, L.-Y.: C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23, 83–118 (2005)

    Google Scholar 

  9. Brezzi, F.: On the existence, uniqueness, and approximation of saddle point problems arising from Lagrange multipliers. RAIRO 8, 129–151 (1974)

    MathSciNet  MATH  Google Scholar 

  10. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Elements. Springer, New York (1991)

    Book  Google Scholar 

  11. Ciarlet, P., Raviart, P.: A mixed finite element for the biharmonic equation. In: de Boor, C. (ed.) Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 125–143. Academic Press, New York (1974)

    Google Scholar 

  12. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, New York (1978)

    MATH  Google Scholar 

  13. Cockburn, B., Dong, B., Guzmán, J.: A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems. J. Sci. Comput. 40, 141–187 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Engel, G., Garikipati, K., Hughes, T.J.R., Larson, M.G., Mazzei, L., Taylor, R.L.: Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Eng. 191, 3669–3750 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Falk, R.S., Osborn, J.E.: Error estimates for mixed methods. RAIRO. Numer. Anal. 14, 249–277 (1980)

    MathSciNet  MATH  Google Scholar 

  16. Fraeijs de Veubeke, B.X.: Displacement and equilibrium models in the finite element method. In: Zienkiewicz, O.C., Holister, G. (eds.) “Stress Analysis”. Wiley, New York (1965)

    Google Scholar 

  17. Fraeijs de Veubeke, B.X.: Stress function approach. In: International Congress on the Finite Element Methods in Structural Mechanics, Bournemouth, 1975

    Google Scholar 

  18. Gastaldi, L., Nochetto, R.H.: Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations. M2AN 23, 103–128 (1989)

    Google Scholar 

  19. Glowinski, R., Pironneau, O.: Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Rev. 21, 167–212 (1979)

    Google Scholar 

  20. Gudi, T., Nataraj, N., Pani, A.K.: Mixed discontinuous Galerkin finite element method for the biharmonic equation. J. Sci. Comput. 37, 139–161 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Herrmann, L.: A bending analysis for plates. In: Matrix Methods in Structural Mechanics, On Proceedings of the Conference held at Wright-Patterson Air Force Base, Ohio, AFFDL technical report No. AFFDL-TR-66-88 pp. 577–604 (1965)

    Google Scholar 

  22. Herrmann, L.: Finite element bending analysis for plates. J. Eng. Mech. Div. ASCE EM5 93, 49–83 (1967)

    Google Scholar 

  23. Johnson, C.: On the convergence of a mixed finite element method for plate bending problems. Numer. Math. 21, 43–62 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  24. Johnson, C., Pitkäranta, J.: Analysis of some mixed finite element methods related to reduced integration. Math. Comp. 38, 375–400 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lascaux, P., Lesaint, P.: Some nonconforming finite elements for the plate bending problem. RAIRO Anal. Numer. R-1, 9–53 (1985)

    Google Scholar 

  26. Malkus, D.S., Hughes, T.J.R.: Mixed finite element methods-reduced and selective integration techniques: A unification of concepts. Comput. Methods Appl. Mech. Eng. 15, 63–81 (1978)

    Article  MATH  Google Scholar 

  27. Miyoshi, T.: A finite element method for the solution of fourth order partial differential equations. Kunamoto J. Sci. (Math.) 9, 87–116 (1973)

    Google Scholar 

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

    Google Scholar 

  29. Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods on polytopal meshes. arXiv:1204.3655v2

    Google Scholar 

  30. Mu, L., Wang, J., Wang, Y., Ye, X.: A computational study of the weak Galerkin method for second order elliptic equations. Numer. Algorithm (2012). arXiv:1111.0618v1, DOI:10.1007/s11075-012-9651-1

    Google Scholar 

  31. Raviart, P., Thomas, J.: A mixed finite element method for second order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of the Finite Element Method. Lectures Notes in Math. vol. 606. Springer, New York (1977)

    Google Scholar 

  32. Scholz, R.: A mixed method for 4th order problems using linear finite elements. R.A.I.R.O. Numer. Anal. 12, 85–90 (1978)

    Google Scholar 

  33. Scholz, R.: Interior error estimates for a mixed finite element method. Numer. Funct. Anal. Optim. 1, 415–429 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, J.: Asymptotic expansions and maximum norm error estimates for mixed finite element methods for second order elliptic problems. Numer. Math. 55, 401–430 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems, arXiv:1104.2897v1 [math.NA]. J. Comput. Appl. Math. 241, 103-115 (2013)

    Google Scholar 

  36. Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second-order elliptic problems, arXiv:1202.3655v1, submitted to Math Comp.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lancing, MI, 48824, USA

    Lin Mu

  2. Division of Mathematical Sciences, National Science Foundation, Arlington, VA, 22230, USA

    Junping Wang

  3. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74075, USA

    Yanqiu Wang

  4. Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA

    Xiu Ye

Authors
  1. Lin Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junping Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yanqiu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiu Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junping Wang .

Editor information

Editors and Affiliations

  1. , Dept. of Flow and Material Simulation, Fraunhofer Institute for Industrial Math, Fraunhofer-Platz 1, Kaiserslautern, 67663, Germany

    Oleg P. Iliev

  2. Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. 25A, Sofia, 1113, Bulgaria

    Svetozar D. Margenov

  3. , Dept. of Mathematical and Statistical Sc, University of Alberta, 677 Central Academic Building, Edmonton, T6G 2G1, Alberta, Canada

    Peter D Minev

  4. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, East Avenue 7000, Livermore, 94550, California, USA

    Panayot S. Vassilevski

  5. , Department of Mathematics, Pennsylvania State University, 310 McAllister Building, University Park, 16802, Pennsylvania, USA

    Ludmil T Zikatanov

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this paper

Cite this paper

Mu, L., Wang, J., Wang, Y., Ye, X. (2013). A Weak Galerkin Mixed Finite Element Method for Biharmonic Equations. In: Iliev, O., Margenov, S., Minev, P., Vassilevski, P., Zikatanov, L. (eds) Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications. Springer Proceedings in Mathematics & Statistics, vol 45. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7172-1_13

Download citation

Publish with us

Policies and ethics

Search

Navigation