A Posteriori Error Estimation and Mesh Adaptivity for Finite Volume and Finite Element Methods

  • Timothy J. Barth
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 41)


Error representation formulas and a posteriori error estimates for numerical solutions of hyperbolic conservation laws are considered with specialized variants given for the Godunov finite volume and discontinuous Galerkin finite element methods. The error representation formulas utilize the solution of a dual problem to capture the nonlocal error behavior present in hyperbolic problems. The error representation formulas also provide a framework for understanding superconvergence properties of functionals and fundamental differences between finite element and Godunov finite volume methods. Computable error estimates are then constructed for practical implementation in computer codes. The error representation formulas and computable error estimates also suggest a straightforward strategy for mesh adaptivity which is demonstrated on numerical hyperbolic problems of interest.

Key words

A posteriori error estimates error representation Godunov finite volume methods finite element methods unstructured meshes 


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  1. Abg94.
    R. Abgrall. An essentially non-oscillatory reconstruction procedure on finite-element type meshes. Comp. Meth. Appl. Mech. Engrg., 116:95–101, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  2. Bar98.
    T.J. Barth. Numerical methods for gasdynamic systems on unstructured meshes. In Kröner, Ohlberger, and Rohde, editors, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, volume 5 of Lecture Notes in Computational Science and Engineering, pages 195–285. Springer-Verlag, Heidelberg, 1998.Google Scholar
  3. BD02.
    T.J. Barth and H. Deconinck (eds). Error estimation and adaptive discretization methods in CFD. In Barth and Deconinck, editors, Error Estimation and Adaptive Discretization Methods in CFD, volume 25 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Heidelberg, 2002.Google Scholar
  4. BF90.
    T. J. Barth and P.O. Frederickson. Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. Technical Report 90-0013, AIAA, Reno, NV, 1990.Google Scholar
  5. BJ89.
    T. J. Barth and D. C. Jespersen. The design and application of upwind schemes on unstructured meshes. Technical Report 89-0366, AIAA, Reno, NV, 1989.Google Scholar
  6. BL02.
    T.J. Barth and M.G. Larson. A-posteriori error estimation for higher order Godunov finite volume methods on unstructured meshes. In Herbin and Kröner, editors, Finite Volumes for Complex Applications III, pages 41–63. Hermes Science Pub., London, 2002.Google Scholar
  7. BR98.
    R. Becker and R. Rannacher. Weighted a posteriori error control in FE methods. In Proc. ENUMATH-97, Heidelberg. World Scientific Pub., Singapore, 1998.Google Scholar
  8. Chi85.
    G. Chiocchia. Exact solutions to transonic and supersonic flows. Technical Report AR-211, AGARD, 1985.Google Scholar
  9. CLS89.
    B. Cockburn, S.Y. Lin, and C.W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems. J. Comp. Phys., 84:90–113, 1989.MathSciNetCrossRefGoogle Scholar
  10. CS97.
    B. Cockburn and C.W. Shu. The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. Technical Report 201737, Institite for Computer Applications in Science and Engineering (ICASE), NASA Langley R.C., 1997.Google Scholar
  11. DOE90.
    L. Durlofsky, S. Osher, and B. Engquist. Triangle based TVD schemes for hyperbolic conservation laws. Technical Report 90-10, Institite for Computer Applications in Science and Engineering (ICASE), NASA Langley R.C., 1990.Google Scholar
  12. EEHJ95.
    K. Eriksson, D. Estep, P. Hansbo, and C. Johnson. Introduction to numerical methods for differential equations. Acta Numerica, pages 105–158, 1995.Google Scholar
  13. GLLS97.
    M. Giles, M. Larson, M. Levenstam, and E. Süli. Adaptive error control for finite element approximations of the lift and drag coefficients in viscous flow. preprint NA-97/06, Comlab, Oxford University, 1997.Google Scholar
  14. God59.
    S. K. Godunov. A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb., 47:271–290, 1959.zbMATHMathSciNetGoogle Scholar
  15. GP99.
    M. Giles and N.A. Pierce. Improved lift and drag estimates using adjoint Euler equations. Technical Report 99-3293, AIAA, Reno, NV, 1999.Google Scholar
  16. Har83.
    A. Harten. High resolution schemes for hyperbolic conservation laws. J. Comp. Phys., 49:357–393, 1983.zbMATHMathSciNetCrossRefGoogle Scholar
  17. Har89.
    A. Harten. ENO schemes with subcell resolution. J. Comp. Phys., 83:148–184, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  18. HH02.
    R. Hartmann and P. Houston. Adaptive discontinuous galerkin methods for the compressible euler equations. J. Comp. Phys., 182(2):508–532, 2002.MathSciNetCrossRefGoogle Scholar
  19. HOEC87.
    A. Harten, S. Osher, B. Engquist, and S. Chakravarthy. Uniformly high-order accurate essentially nonoscillatory schemes III. J. Comp. Phys., 71(2):231–303, 1987.MathSciNetCrossRefGoogle Scholar
  20. Joh87.
    C. Johnson. Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge, 1987.Google Scholar
  21. JP86.
    C. Johnson and J. Pitkäranta. An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp., 46:1–26, 1986.MathSciNetCrossRefGoogle Scholar
  22. JRB95.
    C. Johnson, R. Rannacher, and M. Boman. Numerics and hydrodynamics stability theory: towards error control in CFD. SIAM J. Numer. Anal., 32:1058–1079, 1995.MathSciNetCrossRefGoogle Scholar
  23. LB99.
    M.G. Larson and T.J. Barth. A posteriori error estimation for adaptive discontinuous Galerkin approximations of hyperbolic systems. In Cockburn, Karniadakis, and Shu, editors, Discontinuous Galerkin Methods, volume 11 of Lecture Notes in Computational Science and Engineering. Springer-Verlag, Heidelberg, 1999.Google Scholar
  24. OP99.
    J. T. Oden and S. Prudhomme. Goal-oriented error estimation and adaptivity for the finite element method. Technical Report 99-015, TICAM, U. Texas, Austin,TX, 1999.Google Scholar
  25. PO99.
    S. Prudhomme and J.T. Oden. On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comp. Meth. Appl. Mech. and Eng., pages 313–331, 1999.Google Scholar
  26. RH73.
    W. H. Reed and T. R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, New Mexico, 1973.Google Scholar
  27. S\(\ddot 9\)8._E. Süli. A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems. In Kröner, Ohlberger, and Rohde, editors, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, volume 5 of Lecture Notes in Computational Science and Engineering, pages 122–194. Springer-Verlag, Heidelberg, 1998.Google Scholar
  28. Van93.
    P. Vankeirsblick. Algorithmic Developments for the Solution of Hyperbolic Conservation Laws on Adaptive Unstructured Grids. PhD thesis, Katholieke Universiteit Leuven, Belgium, 1993.Google Scholar
  29. vL79.
    B. van Leer. Towards the ultimate conservative difference schemes V. A second order sequel to Godunov’s method. J. Comp. Phys., 32:101–136, 1979.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Timothy J. Barth
    • 1
  1. 1.NASA Ames Research CenterMoffett FieldUSA

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