Skip to main content
SpringerLink
Log in

A Family of Finite Volume Schemes of Arbitrary Order on Rectangular Meshes

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we analyze vertex-centered finite volume method (FVM) of any order for elliptic equations on rectangular meshes. The novelty is a unified proof of the inf-sup condition, based on which, we show that the FVM approximation converges to the exact solution with the optimal rate in the energy norm. Furthermore, we discuss superconvergence property of the FVM solution. With the help of this superconvergence result, we find that the FVM solution also converges to the exact solution with the optimal rate in the \(L^2\)-norm. Finally, we validate our theory with numerical experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Arnold, D., Brezzi, F., Cockburn, B., Marini, L.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2003)

    Article  MathSciNet  Google Scholar 

  2. Aavatsmark, I., Barkve, T., Boe, O., Mannseth, T.: Discretization on non-orthogonal quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127, 2–14 (1996)

    Article  MATH  Google Scholar 

  3. Bank, R.E., Rose, D.J.: Some error estimates for the box scheme. SIAM J. Numer. Anal. 24, 777–787 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barth, T., Ohlberger, M., Finite volume methods: foundation and analysis. In: Encyclopedia of Computational Mechanics, vol. 1, Chapter 15. Wiley, London (2004)

  5. Boyer, F., Hubert, F.: Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 46, 3032–3070 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43, 1872–1896 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cai, Z.: On the finite volume element method. Numer. Math. 58, 713–735 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cai, Z., Douglas, J., Park, M.: Development and analysis of higher order finite volume methods over rectangles for elliptic equations. Adv. Comput. Math. 19, 3–33 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cao, W., Zhang, Z., Zou, Q.: Superconvergence finite volume schemes for 1D general elliptic equations. J. Sci. Comput. (2013). doi:10.1007/s10915-013-9691-2

  10. Chen, L.: A new class of high order finite volume methods for second order elliptic equations. SIAM J. Numer. Anal. 47, 4021–4043 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, Z., Wu, J., Xu, Y.: Higher-order finite volume methods for elliptic boundary value problems. Adv. Comput. Math. 37, 191–253 (2012)

    Google Scholar 

  12. Chou, S.H., Kwak, D.Y., Li, Q.: \(L^p\) error estimates and superconvergence for covolume or finite volume element methods. Numer. Methods Partial Differ. Equ. 19, 463–486 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cui, M., Ye, X.: Unified analysis of finite volume methods for the Stokes equations. SIAM J. Numer. Anal. 48, 824–839 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, Boston (1984)

    MATH  Google Scholar 

  15. Domelevo, K., Omnes, P.: A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. M2AN Math. Model. Numer. Anal. 39, 1203–1249 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Douglas, J., Dupont, T.: Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces. Numer. Math. 22, 99–109 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  17. Droniou, J., Eymard, R.: A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105, 35–71 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Emonot, Ph.: Methods de volums elements finis: applications aux equations de Navier–Stokes et resultats de convergence. Dissertation Lyon (1992)

  19. Ewing, R., Lin, T., Lin, Y.: On the accuracy of the finite volume element based on piecewise linear polynomials. SIAM J. Numer. Anal. 39, 1865–1888 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Eymard, R., Gallouet, T., Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes, SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30, 1009–1043 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Eymard, R., Guichard, C., Herbin, R.: Small-stencil 3D schemes for diffusive flows in porous media. M2AN Math. Model. Numer. Anal. 46, 265–290 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Eymard, R., Gallouet, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Analysis VII, pp. 713–1020. North-Holland, Amsterdam (2000)

  23. Hackbusch, W.: On first and second order box methods. Computing 41, 277–296 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hermeline, F.: Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Eng. 192(16–18), 1939–1959 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hyman, J.M., Knapp, R., Scovel, J.C.: High order finite volume approximations of differential operators on nonuniform grids. Physica D 60, 112–138 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lazarov, R., Michev, I., Vassilevski, P.: Finite volume methods for convection-diffusion problems. SIAM J. Numer. Anal. 33, 31–55 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  28. Li, R., Chen, Z., Wu, W.: The Generalized Difference Methods for Partial differential Equations. Marcel Dikker, New York (2000)

    Google Scholar 

  29. Liebau, F.: The finite volume element method with quadratic basis function. Computing 57, 281–299 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  30. Mattiussi, C.: An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology. J. Comput. Phys. 133, 289–309 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  31. Nicolaides, R.A., Porsching, T.A., Hall, C.A.: Covolume methods in computational fluid dynamics. In: Hafez, M., Oshima, K. (eds.) Computational Fluid Dynamics Review, pp. 279–299. Wiley, New York (1995)

  32. Ollivier-Gooch, C., Altena, M.: A high-order-accurate unconstructed mesh finite-volume scheme for the advection-diffusion equation. J. Comput. Phys. 181, 729–752 (2002)

    Article  MATH  Google Scholar 

  33. Patanker, S.V.: Numerical Heat Transfer and Fluid Flow, Ser. Comput. Methods Mech. Thermal Sci. McGraw Hill, New York (1980)

    Google Scholar 

  34. Plexousakis, M., Zouraris, G.: On the construction and analysis of high order locally conservative finite volume type methods for one dimensional elliptic problems. SIAM J. Numer. Anal. 42, 1226–1260 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. Shu, C.W.: High order finite difference and finite volume WENO schemes and discontinous Galerkin methods for CFD. J. Comput. Fluid Dyn. 17, 107–118 (2003)

    Article  MATH  Google Scholar 

  36. Tian, M., Chen, Z.: Quadratical element generalized differential methods for elliptic equations. Numer. Math. J. Chin. Univ. 13, 99–113 (1991)

    MATH  MathSciNet  Google Scholar 

  37. Wang, T., Gu, Y.: Superconvergence biquadratic finite volume element method for two dimensional Poisson’s equation. Comput. Appl. Math. 234, 447–460 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Xu, J., Zikatanov, L.: Some observations on Babuska–Brezzi conditions. Numer. Math. 94, 195–202 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Xu, J., Zou, Q.: Analysis of linear and quadratic simplitical finite volume methods for elliptic equations. Numer. Math. 111, 469–492 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zhang, Z., Zou, Q.: Finite volume methods of any order on quadrilateral meshes for general elliptic equations (2013) (submitted)

  41. Zhang, Z.: Finite element superconvergent approximation for one-dimensional singularly perturbed problems. Numer. Methods Partial Differ. Equ. 18, 374–395 (2002)

    Article  MATH  Google Scholar 

  42. Zhang, Z.: Superconvergence of spectral collocation and p-version methods in one dimensional problems. Math. Comput. 74, 1621–1636 (2005)

    Article  MATH  Google Scholar 

  43. Zhu, Q., Lin, Q.: Superconvergence Theory of the Finite Element Method (in Chinese). Hunan Science Press, Hunan (1989)

    Google Scholar 

Download references

Acknowledgments

The authors would like to thank a Ph.D. student, Waixiang Cao for her assistance in the numerical experiments.

Author information

Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100084, People’s Republic of China

    Zhimin Zhang

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

  3. College of Mathematics and Scientific Computing, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

    Qingsong Zou

  4. Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

    Qingsong Zou

Authors
  1. Zhimin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qingsong Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qingsong Zou.

Additional information

Zhimin Zhang was supported in part by the US National Science Foundation through grant DMS-1115530, the Ministry of Education of China through the Changjiang Scholars program, and Guangdong Provincial Government of China through the “Computational Science Innovative Research Team” program.

Qingsong Zou was supported in part by the National Natural Science Foundation of China under the grant 11171359 and in part by the Fundamental Research Funds for the Central Universities of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Z., Zou, Q. A Family of Finite Volume Schemes of Arbitrary Order on Rectangular Meshes. J Sci Comput 58, 308–330 (2014). https://doi.org/10.1007/s10915-013-9737-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-013-9737-5

Keywords

Mathematics Subject Classification

Search

Navigation