Numerische Mathematik

, Volume 65, Issue 1, pp 23–50

A unified approach to a posteriori error estimation using element residual methods

  • Mark Ainsworth
  • J. Tinsley Oden
Article

Summary

This paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm. Moreover, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error. A key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems. The work extends and combines several existing methods to the case of fullh-p finite element approximation on possibly irregular meshes with, elements of non-uniform degree. As a special case, the analysis proves a conjecture made by Bank and Weiser [Some A Posteriori Error Estimators for Elliptic Partial Differential Equations, Math. Comput.44, 283–301 (1985)].

Mathematics Subject Classification (1991)

65N30 

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References

  1. 1.
    Adams, R.A. (1975): Sobolev spaces, Academic Press, New YorkGoogle Scholar
  2. 2.
    Ainsworth, M., Craig, A.W. (1992): A posteriori error estimators in the finite element method. Numer. Math.60, 429–463Google Scholar
  3. 3.
    Babuŝka, I., Miller, A.D. (1987): A feedback finite element method with a posteriori error estimation: Part 1. Comput. Methods Appl. Mech. Eng.61, pp. 1–40Google Scholar
  4. 4.
    Babuŝka, I., Rheinboldt, W.C. (1978): A posteriori error estimates for adaptive finite element computations. SIAM J. Numer. Anal.15, pp. 736–754Google Scholar
  5. 5.
    Bank, R.E., Weiser, A. (1985): Some a posteriori error estimators for elliptic partial differential equations, Math. Comput.44, 283–301Google Scholar
  6. 6.
    Demkowicz, L., Oden, J.T., Strouboulis, T. (1985): An adaptivep-version finite element method for transient flow problems with moving boundaries. In: R.H.Gallagher, G. Carey, J.T. Oden, O.C. Zienkiewicz, eds., Finite elements in fluids, Vol. 6. Wiley, Chichester, pp. 291–306Google Scholar
  7. 7.
    Fraejis de Veubeke, B. (1974): Variational principles and the patch test, Int. J. Numer. Methods Eng.8, 783–801Google Scholar
  8. 8.
    Girault, V., Raviart, P.A. (1986): Finite element methods for Navier Stokes equations: Theory and algorithms Springer Series in Computational Mathematics5. Springer, Berlin Heidelberg New YorkGoogle Scholar
  9. 9.
    Kelly, D.W. (1984): The self equilibration of residuals and complementary a posteriori error estimates in the finite element method. Int. J. Numer. Methods Eng.20, 1491–1506Google Scholar
  10. 10.
    Ladeveze, D., Leguillon (1983): Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal.20, 485–509Google Scholar
  11. 11.
    Oden, J.T., Demkowicz, L., Rachowicz, W., Westermann, T.A. (1989): Towards a universalh-p finite element strategy. Part II. A posteriori error estimation, Comput. Methods Appl. Mech. Eng.77, 113–180Google Scholar
  12. 12.
    Oden, J.T., Demkowicz, L., Strouboulis, T., Devloo, P. (1986): Adaptive methods for problems in solid and fluid mechanics. In: I. Babuŝka, O.C. Zienkiewicz, J. Gago, E.R. de A. Oliveira, eds., Accuracy estimates and adaptive refinements in finite element computations, Wiley Chichester, pp. 249–280.Google Scholar
  13. 13.
    Percell, P., Wheeler, M.F. (1978): A local finite element procedure for elliptic equation., SIAM J. Numer. Anal.15, pp. 705–714Google Scholar
  14. 14.
    Raviart, P.A., Thomas, J.M. (1977): Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comput.31, 391–413Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Mark Ainsworth
    • 1
  • J. Tinsley Oden
    • 1
  1. 1.Texas Institute for Computational MechanicsThe University of Texas at AustinAustinUSA
  2. 2.Mathematics DepartmentLeicester UniversityLeicesterUK

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