Error estimates and mesh adaption for a cell vertex finite volume scheme

  • J. A. Mackenzie
  • D. F. Mayers
  • A. J. Mayfield
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NNFM, volume 44)


In this paper we investigate a posteriori error estimates for the cell vertex finite volume method and their applicability to adaptive mesh generation. We start from the cell vertex scheme reformulated as a Petrov-Galerkin finite element method and base the error estimate on the calculation of the finite element residual of the approximate solution. An analogy can be made between the residual in finite element methods and the truncation error of finite difference schemes. Numerical investigation of the performance of the error indicator for a model problem of two dimensional advection demonstrate its reliability and efficiency. The error indicator is then applied to a transonic Euler calculation where it is contrasted with an alternative procedure based on first and second differences of physical flow quantities which can be related to local interpolation error. Finally, we discuss the effective use of error estimates in adaptive mesh generation.


Truncation Error Posteriori Error Estimate Error Indicator Local Truncation Error Posteriori Error Estimator 
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  1. [1]
    M.J. Berger and J. Oliger. Adaptive mesh refinement for hyperbolic partial differential equations. Journal of Computational Physics, 53: 484–512, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    A. Brandt. Multi-level adaptive solutions to boundary value problems. Mathematics of Computation, 31: 333–390, 1977.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    H. Cartan. Calcul Différentiel. Herrmann, 1967.Google Scholar
  4. [4]
    I. Christie, D. F. Griffiths, A. R. Mitchell, and J. M. Sanz-Serna. Product approximation for non-linear problems in the finite element method. IMA. J. Numer. Anal., 1: 253–266, 1981.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    P.I. Crumpton, J.A. Mackenzie, and K.W. Morton. Cell vertex algorithms for the compressible Navier-Stokes equations. Technical Report NA91/12, Oxford University Computing Laboratory, 11 Keble Road, Oxford, OX1 3QD, 1991. Submitted for publication.Google Scholar
  6. R.G. Jones. Error estimators for adaptive methods. Technical Report NA90/17, Oxford University Computing Laboratory, 11 Keble Road, Oxford, OX1 3QD, 1990.Google Scholar
  7. [7]
    A.J. Mayfield. Discussion and presentation of a new method for mesh movement. Technical Report NA92/11, Oxford University Computing Laboratory, 11 Keble Road, Oxford, OX1 3QD, 1992.Google Scholar
  8. [8]
    A.J. Mayfield. Design and implementation of an adaptive multigrid algorithm. D.Phil thesis, Oxford University, 1992.Google Scholar
  9. A.J. Mayfield. A new efficient data structure for mesh enrichment algorithms. Technical Report NA92/12, Oxford University Computing Laboratory, 11 Keble Road, Oxford, OX1 3QD, 1992.Google Scholar
  10. [10]
    T.A. Manteuffel and A.B. White Jr. The numerical solution of second-order boundary value problems on nonuniform meshes. Mathematics of Computation, 47 (176): 511–535, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    K.W. Morton and M.F. Paisley. A finite volume scheme with shock fitting for the steady Euler equations. Journal of Computational Physics, 80: 168–203, 1989.zbMATHCrossRefGoogle Scholar
  12. [12]
    H. Schlichting. Boundary-Layer Theory. McGraw-Hill, 1955.Google Scholar
  13. [13]
    E. Sùli. Finite volume methods on distorted meshes: stability, accuracy, adaptivity. Technical Report NA89/6, Oxford University Computing Laboratory, 11 Keble Road, Oxford, OX1 3QD, 1989.Google Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 1993

Authors and Affiliations

  • J. A. Mackenzie
    • 1
  • D. F. Mayers
    • 1
  • A. J. Mayfield
    • 1
  1. 1.Oxford University Computing LaboratoryOxfordEngland

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