Numerische Mathematik

, Volume 123, Issue 1, pp 97–119 | Cite as

New locally conservative finite element methods on a rectangular mesh

Article

Abstract

A new family of locally conservative, finite element methods for a rectangular mesh is introduced to solve second-order elliptic equations. Our approach is composed of generating PDE-adapted local basis and solving a global matrix system arising from a flux continuity equation. Quadratic and cubic elements are analyzed and optimal order error estimates measured in the energy norm are provided for elliptic equations. Next, this approach is exploited to approximate Stokes equations. Numerical results are presented for various examples including the lid driven-cavity problem.

Mathematics Subject Classification

65N12 65N30 

References

  1. 1.
    Antonietti, P.F., Brezzi, F., Marini, L.D.: Stabilizations of the Baumann–Oden DG formulation: the 3D case. Boll. Unione Mat. Ital. (9) 1(3), 629–643 (2008)MathSciNetMATHGoogle Scholar
  2. 2.
    Arbogast, T., Pencheva, G., Wheeler, M.F., Yotov, I.: A multiscale mortar mixed finite element method. Multiscale Model. Simul. 6(1), 319–346 (2007)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Arnold, D.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Arnold, D., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. Modél. Math. Anal. Numér. 19, 7–32 (1985)MathSciNetMATHGoogle Scholar
  5. 5.
    Arnold, D., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1740–1779 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Baumann, C.E., Oden, J.T.: A discontinuous \(hp\) finite element method for convection–diffusion problems. Comput. Methods Appl. Mech. Eng 175(3–4), 311–341 (1999)Google Scholar
  7. 7.
    Brezzi, F., Marini, L.D.: Bubble stabilization of discontinuous Galerkin methods. In: Fitzgibbon, W., Hoppe, R., Periaux, J., Pironneau, O., Vassilevski, Y. (eds.) Advances in Numerical Mathematics. Proceedings of the International Conference on the Occasion of the 60th Birthday of Y.A. Kuznetsov. Institute of Numerical Mathematics of the Russian Academy of Sciences, Moscow, pp. 25–36 (2006)Google Scholar
  8. 8.
    Cockburn, B., Gopalakrishnan, J.: Chracerization of hybrid mixed methods for second order elliptic problems. SIAM J. Numer. Anal. 42, 283–301 (2004)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Güzey, S., Cockburn, B., Stolarski, H.K.: The embedded discontinuous Galerkin method: application to linear shell problems. Int. J. Numer. Methods Eng. 70(7), 757–790 (2007)MATHCrossRefGoogle Scholar
  11. 11.
    Hesthaven, J.S., Warburton, T.: Nodal discontinuous Galerkin methods: algorithms, analysis, and applications. Springer Texts in Applied Mathematics, vol. 54. Springer, New York (2008)Google Scholar
  12. 12.
    Jeon, Y.: A multiscale cell boundary element method for elliptic problems. Appl. Numer. Math 59(11), 2801–2813 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Jeon, Y., Park, E.-J.: Nonconforming cell boundary element methods for elliptic problems on triangular mesh. Appl. Numer. Math. 58, 800–814 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Jeon, Y., Park, E.-J.: A hybrid discontinuous Galerkin method for elliptic problems. SINUM 48, 1968–1983 (2010)MathSciNetMATHGoogle Scholar
  15. 15.
    Jeon, Y., Park, E.-J., Sheen, D.: A cell boundary element method for elliptic problems. Numer. Methods Partial Differ. Equ. 21, 496–511 (2005)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Jeon, Y., Sheen, D.: Analysis of a cell boundary element method. Adv. Comput. Math. 22(3), 201–222 (2005)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Oden, J.T., Babuska, I., Baumann, C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146, 491–519 (1998)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Tong, P.: New displacement hybrid finite element models for solid continua. Int. J. Numer. Methods Eng. 2, 73–83 (1970)MATHCrossRefGoogle Scholar
  19. 19.
    Riviére, B., Wheeler, M.F., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems I. Comput. Geosci. 3, 337–360 (1999)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Rivière, B., Wheeler, M.F., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SINUM 39(3), 902–931 (2001)MATHGoogle Scholar
  21. 21.
    Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SINUM 15, 152–161 (1978)MATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsAjou UniversitySuwonKorea
  2. 2.Department of Computational Science and EngineeringYonsei UniversitySeoulKorea

Personalised recommendations