Virtual Element and Discontinuous Galerkin Methods

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


Virtual element methods (VEM) are the latest evolution of the Mimetic Finite Difference Method and can be considered to be more close to the Finite Element approach. They combine the ductility of mimetic finite differences for dealing with rather weird element geometries with the simplicity of implementation of Finite Elements. Moreover, they make it possible to construct quite easily high-order and high-regularity approximations (and in this respect they represent a significant improvement with respect to both FE and MFD methods). In the present paper we show that, on the other hand, they can also be used to construct DG-type approximations, although numerical tests should be done to compare the behavior of DG-VEM versus DG-FEM.


Discontinuous Galerkin Virtual elements Mimetic finite differences 


  1. [1]
    D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), pp. 742–760.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    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
  3. [3]
    L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D Marini, and A. Russo, Basic Principles of Virtual Element Methods, Math. Models Methods Appl. Sci. 1 (2013), pp. 199–214.CrossRefGoogle Scholar
  4. [4]
    S. C. Brenner and R. L. Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, 15. Springer-Verlag, New York, 2008.Google Scholar
  5. [5]
    F. Brezzi and L.D. Marini, A quick tutorial on DG methods, (this book).Google Scholar
  6. [6]
    P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978.Google Scholar
  7. [7]
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    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.CrossRefzbMATHGoogle Scholar
  9. [9]
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    M.F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978), pp. 152–161.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.IUSS-PaviaPaviaItaly
  2. 2.King Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Università di PaviaPaviaItaly

Personalised recommendations