Journal of Scientific Computing

, Volume 58, Issue 2, pp 308–330

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

Article

## 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.

## Keywords

High order Finite volume method Inf-sup condition Superconvergence

## Mathematics Subject Classification

Primary 65N30 Secondary 45N08

## Notes

### Acknowledgments

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

## References

1. 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)
2. 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)
3. 3.
Bank, R.E., Rose, D.J.: Some error estimates for the box scheme. SIAM J. Numer. Anal. 24, 777–787 (1987)
4. 4.
Barth, T., Ohlberger, M., Finite volume methods: foundation and analysis. In: Encyclopedia of Computational Mechanics, vol. 1, Chapter 15. Wiley, London (2004)Google Scholar
5. 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)
6. 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)
7. 7.
Cai, Z.: On the finite volume element method. Numer. Math. 58, 713–735 (1991)
8. 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)
9. 9.
Cao, W., Zhang, Z., Zou, Q.: Superconvergence finite volume schemes for 1D general elliptic equations. J. Sci. Comput. (2013). doi:
10. 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)
11. 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. 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)
13. 13.
Cui, M., Ye, X.: Unified analysis of finite volume methods for the Stokes equations. SIAM J. Numer. Anal. 48, 824–839 (2010)
14. 14.
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, Boston (1984)
15. 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)
16. 16.
Douglas, J., Dupont, T.: Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces. Numer. Math. 22, 99–109 (1974)
17. 17.
Droniou, J., Eymard, R.: A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105, 35–71 (2006)
18. 18.
Emonot, Ph.: Methods de volums elements finis: applications aux equations de Navier–Stokes et resultats de convergence. Dissertation Lyon (1992)Google Scholar
19. 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)
20. 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)
21. 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)
22. 22.
Eymard, R., Gallouet, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Analysis VII, pp. 713–1020. North-Holland, Amsterdam (2000)Google Scholar
23. 23.
Hackbusch, W.: On first and second order box methods. Computing 41, 277–296 (1989)
24. 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)
25. 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)
26. 26.
Lazarov, R., Michev, I., Vassilevski, P.: Finite volume methods for convection-diffusion problems. SIAM J. Numer. Anal. 33, 31–55 (1996)
27. 27.
LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)
28. 28.
Li, R., Chen, Z., Wu, W.: The Generalized Difference Methods for Partial differential Equations. Marcel Dikker, New York (2000)Google Scholar
29. 29.
Liebau, F.: The finite volume element method with quadratic basis function. Computing 57, 281–299 (1996)
30. 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)
31. 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)Google Scholar
32. 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)
33. 33.
Patanker, S.V.: Numerical Heat Transfer and Fluid Flow, Ser. Comput. Methods Mech. Thermal Sci. McGraw Hill, New York (1980)Google Scholar
34. 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)
35. 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)
36. 36.
Tian, M., Chen, Z.: Quadratical element generalized differential methods for elliptic equations. Numer. Math. J. Chin. Univ. 13, 99–113 (1991)
37. 37.
Wang, T., Gu, Y.: Superconvergence biquadratic finite volume element method for two dimensional Poisson’s equation. Comput. Appl. Math. 234, 447–460 (2010)
38. 38.
Xu, J., Zikatanov, L.: Some observations on Babuska–Brezzi conditions. Numer. Math. 94, 195–202 (2003)
39. 39.
Xu, J., Zou, Q.: Analysis of linear and quadratic simplitical finite volume methods for elliptic equations. Numer. Math. 111, 469–492 (2009)
40. 40.
Zhang, Z., Zou, Q.: Finite volume methods of any order on quadrilateral meshes for general elliptic equations (2013) (submitted)Google Scholar
41. 41.
Zhang, Z.: Finite element superconvergent approximation for one-dimensional singularly perturbed problems. Numer. Methods Partial Differ. Equ. 18, 374–395 (2002)
42. 42.
Zhang, Z.: Superconvergence of spectral collocation and p-version methods in one dimensional problems. Math. Comput. 74, 1621–1636 (2005)
43. 43.
Zhu, Q., Lin, Q.: Superconvergence Theory of the Finite Element Method (in Chinese). Hunan Science Press, Hunan (1989)Google Scholar