Numerische Mathematik

, Volume 111, Issue 3, pp 469–492 | Cite as

Analysis of linear and quadratic simplicial finite volume methods for elliptic equations

Article

Abstract

This paper is devoted to analysis of some convergent properties of both linear and quadratic simplicial finite volume methods (FVMs) for elliptic equations. For linear FVM on domains in any dimensions, the inf-sup condition is established in a simple fashion. It is also proved that the solution of a linear FVM is super-close to that of a relevant finite element method (FEM). As a result, some a posterior error estimates and also algebraic solvers for FEM are extended to FVM. For quadratic FVM on domains in two dimensions, the inf-sup condition is established under some weak condition on the grid.

Mathematics Subject Classification (2000)

65N30 65N12 65N06 

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References

  1. 1.
    Arnold D.N., Brezzi F., Cockburn B., Marini L.D.: Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems. SIAM J. on Numer. Anal. 39, 1749–1779 (2003)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Babuška, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the finite element method, Univ. Maryland, College Park, Washington DC, Techinical Note BN-748 (1972)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)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bank R.E., Xu J.: Asymptotic exact a posteriori error estimators, Part I: Grids with superconvergence. SIAM J. Numer. Anal. 41, 2294–2312 (2003)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bank R.E., Xu J.: Asymptotic exact a posteriori error estimators, Part II: General Unstructured Grids. SIAM J. Numer. Anal. 41, 2313–2332 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bramble J.H., Pasciak J.E., Xu J.: Parallel multilevel preconditioners. Math. Comput. 55, 1–22 (1990)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer-Verlag (1991)Google Scholar
  8. 8.
    Cai Z.: On the finite volume element method. Numer. Math. 58, 713–735 (1991)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    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)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Carstensen C., Lazarov R., Tomov S.: Explicit and averaging a posteriori error estimates for adaptive finite volume methods. SIAM J. Numer. Anal. 42, 2496–2521 (2005)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    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)MATHMathSciNetGoogle Scholar
  12. 12.
    Chou S.H., Tang S.: Conservative P1 conforming and nonconforming Galerkin FEMs: effective flux evaluation via a nonmixed method approach. SIAM J. Numer. Anal 38, 660–668 (2000)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATHCrossRefGoogle Scholar
  14. 14.
    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, also in the ASME Journal of Engineering for Power (1995)Google Scholar
  15. 15.
    Emonot, Ph.: Methodes de volumes elements finis: applications aux equations de Navier-Stokes et resultats de convergence, Dissertation, Lyon (1992)Google Scholar
  16. 16.
    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)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Eymard, R., Gallouet, T., Herbin, R.: Finite Volume Methods, in Handbook of Numerical Analysis VII, North-Holland, Amsterdam, pp. 713–1020 (2000)Google Scholar
  18. 28.
    Hackbusch W.: On first and second order box methods. Computing 41, 277–296 (1989)MATHCrossRefMathSciNetGoogle Scholar
  19. 29.
    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)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Liebau F.: The finite volume element method with quadratic basis Function. Computing 57, 281–299 (1996)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    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)Google Scholar
  22. 22.
    Li, Y., Shu, S., Xu, Y., Zou, Q.: Multilevel Preconditioning for Finite Volume Element Methods (in Preparation)Google Scholar
  23. 23.
    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)MATHCrossRefGoogle Scholar
  24. 24.
    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)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Patanker S.V.: Numerical Heat Transfer and Fluid Flow. Ser. Comput. Methods Mech. Thermal Sci. McGraw-Hill, New York (1980)Google Scholar
  26. 26.
    Rogiest, P., Geuzaine, Ph., Essers, J.A., Delanaye, M.: Implicit High-Order Finite-Volume Euler Solver Using Multi-Block Structured Grids. presented at the 12th AIAA CFD Conf., San Diego, June (1995)Google Scholar
  27. 27.
    Rogiest, P., Essers, J.A., Leonard, O.: Application of High-Order Upwind Finite-Volume Scheme to 2D Cascade Flows Using a Multi-Block Approach, International Conf. on Air Breathing Engines, September 1995, ISABE Paper 95-7057 (1995)Google Scholar
  28. 28.
    Shu C.: High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD. J. Comput. Fluid Dyn. 17, 107–118 (2003)MATHCrossRefGoogle Scholar
  29. 29.
    Sun, P., Xue, G., Wang, C., Xu, J.: A combined finite element-upwind finite volume- Newton’s method for liquid-feed direct methanol fuel cell simulations. Proceedings of Sixth International Fuel Cell Science, Engineering and Technology Conference 8, Denver, Colorado, USA (2008)Google Scholar
  30. 30.
    Tian M., Chen Z.: Quadratic element generalized differential methods for elliptic equations. Numer. Math. J. Chinese Univ. 13, 99–113 (1991)MATHMathSciNetGoogle Scholar
  31. 31.
    Wu H.J., Li R.H.: Error Estimates for fnite volume element methods for general second order elliptic problem. Numer. Meth. PDEs 19, 693–708 (2003)MATHGoogle Scholar
  32. 32.
    Xu J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34, 581–613 (1992)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Xu J., Zhang Z.M.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput. 73, 1139–1152 (2004)MATHGoogle Scholar
  34. 34.
    Xu J., Zikatanov L.: Some observations on Babuška and Brezzi theories. Numer. Math. 94, 195–202 (2003)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Zienkiewicz, O.C., Zhu, J.Z.: The superconvergence patch recovery and a posteriori error estimates, Part I: the recovery technique. Int. J. Numer. Methods Eng. 33, 1331–1364. MR 93c:73098 (1992)Google Scholar
  36. 36.
    Zienkiewicz, O.C., Zhu, J.Z.: The superconvergence patch recovery and a posteriori error estimates, Part II: The recovery technique. Int. J. Numer. Methods Eng. 33, 1365–1382. MR 93c:73099 (1992)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania UniversityUniversity ParkUSA
  2. 2.Department of Scientific Computation and Computer ApplicationsZhongshan (Sun Yat-sen) UniversityGuangzhouPeople’s Republic of China

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