Computing and Visualization in Science

, Volume 13, Issue 5, pp 221–228 | Cite as

A generalization of the vertex-centered finite volume scheme to arbitrary high order

  • Andreas Vogel
  • Jinchao Xu
  • Gabriel Wittum


A higher order finite volume method for elliptic problems is proposed for arbitrary order \({p \in \mathbb{N}}\) . Piecewise polynomial basis functions are used as trial functions while the control volumes are constructed by a vertex-centered technique. The discretization is tested on numerical examples utilizing triangles and quadrilaterals in 2D. In these tests the optimal error is achieved in the H 1-norm. The error in the L 2-norm is one order below optimal for even polynomial degrees and optimal for odd degrees.


Finite volume method Vertex-centered Higher order 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bank R.E., Rose D.: Some error estimates for the box method. SIAM J. Numer. Anal. 24(4), 777–787 (1987)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cai Z.: On the finite volume element method. Numerische Mathematik 58(1), 713–735 (1990)CrossRefGoogle Scholar
  3. 3.
    Ciarlet P.: The Finite Element Method for Elliptic Problems, vol. 58. North-Holland, Amsterdam (1978)Google Scholar
  4. 4.
    Ewing R.E., Lin T., Lin Y.: On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39(6), 1865–1888 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Eymard R., Gallouet T., Herbin R.: Finite Volume Methods, vol. 7. North Holland, Amsterdam (2000)Google Scholar
  6. 6.
    Hackbusch W.: On first and second order box schemes. Computing 41, 277–296 (1989)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen. Springer, Berlin; Heidelberg [u.a.] (2000)Google Scholar
  8. 8.
    Li R., Chen Z., Wu W.: Generalized Difference Methods for Differential Equations, vol. 226. Dekker, NY (2000)Google Scholar
  9. 9.
    Liebau F.: The finite volume element method with quadratic basis functions. Computing 57(4), 281–299 (1996)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Mishev I.: Finite volume element methods for non-definite problems. Numerische Mathematik 83(1), 161–175 (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Süli, E.: The accuracy of cell vertex finite volume methods on quadrilateral meshes. Math. Comput. 359–382 (1992)Google Scholar
  12. 12.
    Vogel, A.: Ein Finite-Volumen-Verfahren höherer Ordnung mit Anwendung in der Biophysik. Master’s thesis, University of Heidelberg (2008)Google Scholar
  13. 13.
    Xu, J., Zou, Q.: Analysis of linear and quadratic finite volume methods for elliptic equations. Preprint AM298, Math. Dept., Penn State, (2005)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Goethe Center for Scientific Computing (G-CSC)Goethe-University Frankfurt am MainFrankfurt am MainGermany
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

Personalised recommendations