Advertisement

A Quick Tutorial on DG Methods for Elliptic Problems

  • F. Brezzi
  • L. D. Marini
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 157)

Abstract

In this paper we recall a few basic definitions and results concerning the use of DG methods for elliptic problems. As examples we consider the Poisson problem and the linear elasticity problem. A hint on the nearly incompressible case is given, just to show one of the possible advantages of DG methods over continuous ones. At the end of the paper we recall some physical principles for linear elasticity problems, just to open the door towards possible new developments.

Keywords

Discontinuous Galerkin Elliptic problems Linear elasticity 

References

  1. [1]
    P.F. Antonietti, F. Brezzi and L.D. Marini Bubble stabilization of discontinuous Galerkin methods, Comput. Methods Appl. Mech. Engrg. 198 (2009), pp. 1651–1659.Google Scholar
  2. [2]
    P.F. Antonietti, F. Brezzi and L.D. Marini Stabilizations of the Baumann-Oden DG formulation: the 3D case, Boll. Unione Mat. Ital. 9 (2008), pp. 629–643.Google Scholar
  3. [3]
    D.N. Arnold An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), pp. 742–760.Google Scholar
  4. [4]
    D.N. Arnold and F. Brezzi Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), pp. 7–32.Google Scholar
  5. [5]
    D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), pp. 1749–1779.Google Scholar
  6. [6]
    D.N. Arnold, R. Falk, and R. Winther Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comput 76 (2007), pp. 1699–1723.Google Scholar
  7. [7]
    D.N. Arnold and R. Winther Nonconforming mixed elements for elasticity, Math. Models Methods Appl. Sci. 13 (2003), pp. 295–307.Google Scholar
  8. [8]
    B. Ayuso de Dios, F. Brezzi, O. Havle, and L.D. Marini L2-estimates for the DG IIPG-0 scheme, Numer. Methods Partial Differential Equations 28 (2012), pp. 1440–1465.Google Scholar
  9. [9]
    B. Ayuso de Dios and L. Zikatanov Uniformly convergent iterative methods for discontinuous Galerkin discretizations, J. Sci. Comput. 40 (2009), pp. 4–36.Google Scholar
  10. [10]
    C.E. Baumann and J.T. Oden A discontinuous hp finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 175 (1999), pp. 311–341.Google Scholar
  11. [11]
    D. Boffi, F. Brezzi, M. Fortin Reduced symmetry elements in linear elasticity, Commun. Pure Appl. Anal. 8 (2009), pp. 95–121.Google Scholar
  12. [12]
    S.C. Brenner Korn’s inequalities for piecewise H 1 vector fields, Math. Comp. 73 (2004), pp. 1067–1087.Google Scholar
  13. [13]
    S.C. Brenner, L.R. Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, 15. Springer-Verlag, New York, 1994.Google Scholar
  14. [14]
    F. Brezzi, J. Douglas, Jr., and L.D. Marini Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47(1985), pp. 217–235.Google Scholar
  15. [15]
    F. Brezzi and L.D. Marini Bubble stabilization of discontinuous Galerkin methods, in Advances in numerical mathematics, (W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, Y. Vassilevski, Eds) Institute of Numerical Mathematics of the Russian Academy of Sciences, Moscow (2006), pp. 25–36.Google Scholar
  16. [16]
    E. Burman, P. Hansbo A stabilized non-conforming finite element method for incompressible flow, Comput. Methods Appl. Mech. Engrg. 195 (2006), pp. 2881–2899.Google Scholar
  17. [17]
    E. Burman and B. Stamm Symmetric and non-symmetric discontinuous Galerkin methods stabilized using bubble enrichment, C.R. Math. Acad. Sci. Paris 346 (2008), pp. 103–106.Google Scholar
  18. [18]
    P.G. Ciarlet, The finite element method for elliptic problems, North-Holland, 1978.Google Scholar
  19. [19]
    B. Cockburn Discontinuous Galerkin methods,ZAMM Z. Angew. Math. Mech. 83 (2003), 731754.Google Scholar
  20. [20]
    B. Cockburn and J. Gopalakrishnan A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal. 42 (2004), pp. 283–301.Google Scholar
  21. [21]
    B. Cockburn, J. Gopalakrishnan and R. Lazarov Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), pp. 1319–1365.Google Scholar
  22. [22]
    B. Cockburn, J. Gopalakrishnan and J. Guzmán A new elasticity element made for enforcing weak stress symmetry, Math. Comput. 79 (2010), pp. 1331–1349.Google Scholar
  23. [23]
    B. Cockburn, G. Kanschat and D. Shötzau A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, J. Sci. Comput. 31 (2007), pp. 61–73.Google Scholar
  24. [24]
    C. Dawson, S. Sun, and M.F. Wheeler Compatible algorithms for coupled flow and transport, Comput. Methods Appl. Mech. Engrg. 193 (2004), pp. 2565–2580.Google Scholar
  25. [25]
    X. Feng and O.A. Karakashian A. Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems, SIAM J. Numer. Anal. 39 (2001), pp. 1343–1365.Google Scholar
  26. [26]
    B.X. Fraeijis de Veubeke Displacement and equilibrium models in the finite element method, in Stress Analysis, O.C. Zienkiewicz and G. Holister Eds., Wiley (1965).Google Scholar
  27. [27]
    R. Falk Finite elements for linear elasticity, in Mixed Finite Elements: Compatibility Conditions Stress Analysis, Lecture Notes in Math. v. 1939, D. Boffi and L. Gastaldi Eds., Springer, Heidelberg (2008).Google Scholar
  28. [28]
    J. Gopalakrishnan and J. Guzmán Symmetric nonconforming mixed finite elements for linear elasticity, SIAM J. Numer. Anal. 49 (2011), pp. 1504–1520.Google Scholar
  29. [29]
    J. Guzmán A unified analysis of several mixed methods for elasticity with weak stress symmetry, J. Sci. Comput. 44 (2010) 156–169.Google Scholar
  30. [30]
    O.A. Karakashian and F. Pascal A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), pp. 2374–2399.Google Scholar
  31. [31]
    O.A. Karakashian and F. Pascal Convergence of adaptive discontinuous Galerkin approximations of second-order elliptic problems, SIAM J. Numer. Anal. 45 (2007), pp. 641–665.Google Scholar
  32. [32]
    C. Lovadina and L.D. Marini A-Posteriori Error Estimates for Discontinuous Galerkin Approximations of Second Order Elliptic Problems, J. Sci. Comput. 40 (2009), pp. 340–359.Google Scholar
  33. [33]
    L.D. Marini An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method, SIAM J. Numer. Anal. 22 (1985), pp. 493–496.Google Scholar
  34. [34]
    B. Rivière, M.F. Wheeler, and V. Girault A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems,SIAM J. Numer. Anal. 39 (2001), pp. 902–931.Google Scholar
  35. [35]
    C.W. Shu Discontinuous Galerkin methods: general approach and stability, Numerical solutions of partial differential equations, 149201, Adv. Courses Math. CRM Barcelona, Birkhuser, Basel, 2009Google Scholar
  36. [36]
    C.W. Shu Paper in this book Google Scholar
  37. [37]
    S. Sun and M.F. Wheeler Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media, SIAM J. Numer. Anal. 43 (2005), pp. 195–219.Google Scholar
  38. [38]
    M.F. Wheeler An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978), pp. 152–161.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Istituto Universitario di Studi Superiori (IUSS)PaviaItaly
  2. 2.Department of MathematicsKing Abulaziz UniversityJeddahSaudi Arabia
  3. 3.Dipartimento di MatematicaUniversità di PaviaPaviaItaly

Personalised recommendations