Skip to main content
SpringerLink
Log in

Higher-order finite volume methods for elliptic boundary value problems

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

This paper studies higher-order finite volume methods for solving elliptic boundary value problems. We develop a general framework for construction and analysis of higher-order finite volume methods. Specifically, we establish the boundedness and uniform ellipticity of the bilinear forms for the methods, and show that they lead to an optimal error estimate of the methods. We prove that the uniform local-ellipticity of the family of the bilinear forms ensures its uniform ellipticity. We then establish necessary and sufficient conditions for the uniform local-ellipticity in terms of geometric requirements on the meshes of the domain of the differential equation, and provide a general way to investigate the mesh geometric requirements for arbitrary higher-order schemes. Several useful examples of higher-order finite volume methods are presented to illustrate the mesh geometric requirements.

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

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai, Z., Mandel, J., McCormick, S.: The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. 28, 392–402 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Chatzipantelidis, P.: A finite volume method based on the Crouzeix-Raviart element for elliptic PDE’s in two dimensions. Numer. Math. 82, 409–432 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chatzipantelidis, P., Lazarov, R.D.: Error estimates for a finite volume element method for elliptic PDE’s in nonconvex polygonal domains. SIAM J. Numer. Anal. 42, 1932–1958 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Z.: The error estimate of generalized difference methods of 3rd-order Hermite type for elliptic partial differential equations. Northeast. Math. 8, 127–135 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Chen, Z.: L 2-estimates for linear element generalized difference methods. Acta Sci. Natur. Univ. Sunyatseni 33(4), 22–28 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Chen, Z.: Superconvergence of generalized difference methods for elliptic boundary value problem. Numer. Math. J. Chin. Univ. (English Ser.) 3, 163–171 (1994)

    MATH  Google Scholar 

  11. 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  MathSciNet  MATH  Google Scholar 

  12. Chen, Z., Xu, Y.: The Petrov–Galerkin and iterated Petrov–Galerkin methods for second-kind integral equations. SIAM J. Numer. Anal. 35, 406–434 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, Z., Li, R., Zhou, A.: A note on the optimal L 2-estimate of the finite volume element method. Adv. Comput. Math. 16, 291–303 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chou, S.-H., Li, Q.: Error estimates in L 2, H 1 and L  ∞  in covolume methods for elliptic and parabolic problems: a unified approach. Math. Comput. 69, 103–120 (2000)

    MathSciNet  MATH  Google Scholar 

  15. Chou, S.-H., Ye, X.: Unified analysis of finite volume methods for second order elliptic problems. SIAM J. Numer. Anal. 45, 1639–1653 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  17. Emonot, P.: Methodes de volumes elements finis: applications aux equations de Navier–Stokes et resultats de convergence. Dissertation, Lyon (1992)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Eymard, R., Gallouet, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. VII, pp. 713–1020. North-Holland, Amsterdam (2000)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Heinrich, B.: Difference Methods on Irregular Networks. Birkhauser, Boston (1987). ISNM 82

    Book  Google Scholar 

  22. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, World Publishing Corp (1985)

  23. Huang, J., Xi, S.: On the finite volume element method for general self-adjoint elliptic problems. SIAM J. Numer. Anal. 35, 1762–1774 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, R.: On generalized difference methods for elliptic and parabolic differential equations. In: Feng, K., Lions, J.L. (eds.) Proceedings of the Symposium on the Finite Element Method Between China and France, Beijing, China, pp. 323–360 (1982)

  25. Li, R.: Generalized difference methods for two point boundary value problems (in Chinese). Acta Sci. Natur. Univ. Jilin. 1, 26–40 (1982)

    Google Scholar 

  26. Li, R.: Generalized difference methods for a nonlinear Dirichlet problem. SIAM J. Numer. Anal. 24, 77–88 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, R., Chen, Z.: The Generalized Difference Method for Differential Equations (in Chinese). Jilin University Press, Changchun (1994)

    Google Scholar 

  28. Li, Y., Li, R.: Generalized difference methods on arbitrary quadrilateral networks. J. Comput. Math. 17, 653–672 (1999)

    MathSciNet  MATH  Google Scholar 

  29. Li, R., Zhu, P.: Generalized difference methods for second order elliptic partial differential equations (I)—triangle grids. Numer. Math. J. Chin. Univ. 2, 140–152 (1982)

    Google Scholar 

  30. Li, R., Chen, Z., Wu, W.: A survey on generalized difference methods and their analysis. In: Chen, Z., Li, Y., Micchelli, C.A., Xu, Y. (eds.) Advances in Computational Mathematics, Lecture Notes in Pure and Applied Mathematics, vol. 202, pp. 321–337 (1999)

  31. Li, R., Chen, Z., Wu, W.: Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Lv, J., Li, R.: L 2 error estimate of the finite volume element methods on quadrilateral meshes. Adv. Comput. Math. 33, 129–148 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. MacNeal, R.H.: An asymmetrical finite difference network. Q. Appl. Math. 11, 295–310 (1953)

    MathSciNet  MATH  Google Scholar 

  35. Petrila, T., Trif, D.: Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics. Springer, Berlin (2005)

    MATH  Google Scholar 

  36. 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  MathSciNet  MATH  Google Scholar 

  37. Schmidt, T.: Box schemes on quadrilateral meshes. Computing 51, 271–292 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  38. Winslow, A.M.: Numerical solution of quasi-linear Poinsson equation in a nonuniform triangle mesh. J. Comput. Phys. 1, 149–172 (1967)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  40. Tikhonov, A.N., Samarskii, A.A.: Homogeneous difference schemes of a high order of accuracy on non-uniform nets (in Russian). Ž. Vyčisl. Mat. i Mat. Fiz. 1, 425–440 (1961)

    Google Scholar 

  41. Tikhonov, A.N., Samarskii, A.A.: Homogeneous difference schemes on irregular meshes (in Russian). Ž. Vyčisl. Mat. i Mat. Fiz. 2, 812–832 (1962)

    Google Scholar 

  42. Versteeg, H., Malalasekera, W.: An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Prentice Hall, Englewood Cliffs (2007)

    Google Scholar 

  43. Wu, W., Li, R.: A generalized difference method for solving one-dimensional second-order elliptic and parabolic differential equations (in Chinese). An English summary appears in Chin. Ann. Math. Ser. B 5(3), 392 (1984); Chin. Ann. Math. Ser. A 5(3), 303–312 (1984)

    Google Scholar 

  44. Wu, H., Li, R.: Error estimates for finite volume element methods for general second-order elliptic problems. Numer. Methods Partial Differ. Equ. 19, 693–708 (2003)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhu, P., Li, R.: Generalized difference methods for second-order elliptic partial differential equations. II. Quadrilateral subdivision. Numer. Math. J. Chin. Univ. 4, 360–375 (1982)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Scientific Computing and Computer Applications, Sun Yat-Sen University, Guangzhou, 510275, People’s Republic of China

    Zhongying Chen, Junfeng Wu & Yuesheng Xu

  2. Department of Mathematics, Syracuse University, Syracuse, NY, 13244, USA

    Yuesheng Xu

Authors
  1. Zhongying Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junfeng Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuesheng Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuesheng Xu.

Additional information

Communicated by Aihui Zhou.

This paper was supported in part by Guangdong Provincial Government of China through the “Computational Science Innovative Research Team” program, and Guangdong Province Key Lab of Computational Science.

Z. Chen was supported in part by the Natural Science Foundation of China under grants 10771224 and 11071264, and the Science and Technology Section of SINOPEC.

J. Wu was supported in part by the US National Science Foundation under grant CCF-0833152.

Y. Xu was supported in part by US Air Force Office of Scientific Research under grant FA9550-09-1-0511, by the US National Science Foundation under grants DMS-0712827 and CCF-0833152, by the Natural Science Foundation of China under grant 11071286.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, Z., Wu, J. & Xu, Y. Higher-order finite volume methods for elliptic boundary value problems. Adv Comput Math 37, 191–253 (2012). https://doi.org/10.1007/s10444-011-9201-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-011-9201-8

Keywords

AMS 2000 Subject Classifications

Search

Navigation