Numerische Mathematik

, Volume 58, Issue 1, pp 713–735 | Cite as

On the finite volume element method

  • Zhiqiang Cai


The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH1 semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h2). Results on the effects of numerical integration are also included.

Subject classifications

AMS(MOS): 65N10 65N30 CR: G1.8 


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  1. 1.
    Adams, R.A.: Sobolev spaces. Ed. New York: Academic Press 1975Google Scholar
  2. 2.
    Baliga, B.R., Patankar, S.V.: A new finite-element formulation for convection-diffusion problems. Numer. Heat Transfer3, 393–409 (1980)Google Scholar
  3. 3.
    Bank, R.E., Rose, D.J.: Some error estimates for the box method. SIAM J. Numer. Anal.24, 777–787 (1987)Google Scholar
  4. 4.
    Bramble, J.H., Hilbert, S.R.: Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal.7, 112–124 (1970)Google Scholar
  5. 5.
    Cai, Z.: A theoretical foundation of the finite volume element method. Ph. D. Thesis, University of Colorado at Denver, May 1990Google Scholar
  6. 6.
    Cai, Z., Mandel, J., McCormick, S.: The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. (to appear)Google Scholar
  7. 7.
    Cai, Z., McCormick, S.: On the accuracy of the finite volume element method for diffusion equations on composite grids. SIAM J. Numer. Anal.27, 636–655 (1990)Google Scholar
  8. 8.
    Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam. North-Holland 1978Google Scholar
  9. 9.
    Ewing, R.E., Lazarov, R.D., Vassilevski, P.S.: Local refinement techniques for elliptic problems on cell-centered grids. Univ. Wyoming E.O.R.I. rep. no 1888-16Google Scholar
  10. 10.
    Hachbusch, W.: On first and second order box schemes. Computing41, 277–296 (1989)Google Scholar
  11. 11.
    Heinrich, B.: Finite difference methods on irregular networks. Basel: Birkhäuser 1987Google Scholar
  12. 12.
    Kadlec, J.: On the regularity of the solution of the Poisson equation on a domain with boundary locally similar to the boundary of a convex domain. Czechoslovak Math. J.14, 386–393 (1964)Google Scholar
  13. 13.
    Kreiss, H.O., Manteuffel, T.A., Swartz, B., Wendroff, B., White, A.B.: Supraconvergent schemes on irregular grids. Math. Comput.47, 537–554 (1986)Google Scholar
  14. 14.
    Manteuffel, T.A., White, A.B.: The numerical solution of second-order boundary value problems on nonuniform meshes. Math. Comput.47, 511–535 (1986)Google Scholar
  15. 15.
    McCormick, S., Thomas, J.: The fast adaptive composite grid method (FAC) for elliptic boundary value problems. Math. Comput.6, 439–456 (1986)Google Scholar
  16. 16.
    Nečas, J.: Les méthodes directes en théorie des equations elliptiques. Paris: Masson 1967Google Scholar
  17. 17.
    Oganesjan, A., Ruchovec, L.A.: Variational methods of solving elliptic equations (in Russian). Erevan, Izd. AN Arm. SSR, 1979Google Scholar
  18. 18.
    Zlámal, M.: On the finite element method. Numer. Math.12, 394–409 (1968)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Zhiqiang Cai
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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