Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

An Unconditionally Stable Quadratic Finite Volume Scheme over Triangular Meshes for Elliptic Equations

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

Abstract

In this note, we present and analyze a special quadratic finite volume scheme over triangular meshes for elliptic equations. The scheme is designed with the second degree Gauss points on the edges and the barycenters of the triangle elements. With a novel from-the-trial-to-test-space mapping, the inf–sup condition of the scheme is shown to hold independently of the minimal angle of the underlying mesh. As a direct consequence, the \(H^1\) norm error of the finite volume solution is shown to converge with the optimal order.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

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. Barth, T., Ohlberger, M.: Finite volume methods: foundation and analysis. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 1, chapter 15. Wiley (2004)

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

    Article  MathSciNet  MATH  Google Scholar 

  4. 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  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Chen, Z., Xu, Y., Zhang, Y.: A construction of higher-order finite volume methods. Math. Comput. 84, 599–628 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chou, S.H., Kwak, D.Y.: Multigrid algorithms for a vertex-centered covolume method for elliptic problems. Numer. Math. 90, 441–458 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. Delanaye, M., Essers, J.A.: Finite volume scheme with quadratic reconstruction on unstructured adaptive meshes applied to turbomachinery flows. ASME paper 95-GT-234 presented at the international gas turbine and aeroengine congress and exposition, Houston, June 5–8, 1995, also in the ASME J. Eng. Power

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

  19. Lv, J., Li, Y.: Optimal biquadratic finite volume element methods on quadrilateral meshes. SIAM J. Numer. Anal. 50, 2397–2399 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  22. 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 

  23. 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 

  24. Süli, E.: Convergence of finite volume schemes for Poissons equation on nonuniform meshes. SIAM J. Numer. Anal. 28, 1419–1430 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, Z., Zou, Q.: Vertex centered finite volume schemes of any order over quadrilateral meshes for elliptic equations. Numer. Math. 130, 363–393 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is indebted to Dr. Hailong Guo of Wayne State University for the figures designed in the paper.

Author information

Authors and Affiliations

  1. School of Data and Computer Science and Guangdong Province Key Laboratory of Computational Science, Sun Yat-Sen University, Guangzhou, 510275, People’s Republic of China

    Qingsong Zou

Authors
  1. Qingsong Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qingsong Zou.

Additional information

Supported in part by the National Natural Science Foundation of China through the Grants 11571384, 11171359 and 11428103, in part by Guangdong Provincial Natural Science Foundation of China through the grant 2014A030313179, and in part by the Fundamental Research Funds for the Central Universities of China through the grant 16lgjc80.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zou, Q. An Unconditionally Stable Quadratic Finite Volume Scheme over Triangular Meshes for Elliptic Equations. J Sci Comput 70, 112–124 (2017). https://doi.org/10.1007/s10915-016-0244-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-016-0244-3

Keywords

Mathematics Subject Classification