Computing and Visualization in Science

, Volume 11, Issue 4–6, pp 351–362

A posteriori estimators for obstacle problems by the hypercircle method

  • Dietrich Braess
  • Ronald H. W. Hoppe
  • Joachim Schöberl
Regular article


A posteriori error estimates for the obstacle problem are established in the framework of the hypercircle method. To this end, we provide a general theorem of Prager–Synge type. There is now no generic constant in the main term of the estimate. Moreover, the role of edge terms is elucidated, and the analysis also applies to other types of a posteriori error estimators for obstacle problems.


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  1. 1.
    Ainsworth M. and Oden T.J. (2000). A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester MATHGoogle Scholar
  2. 2.
    Bartels S. and Carstensen C. (2004). Averaging techniques yield reliable a posteriori finite element error control for obstacle problems. Numer. Math. 99: 225–249 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Braess D. (2007). Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, London MATHGoogle Scholar
  4. 4.
    Braess D. (2005). A posteriori error estimators for obstacle problems—another look. Numer. Math. 101: 415–421 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Braess D., Carstensen C. and Hoppe R.H.W. (2007). Convergence analysis of a conforming adaptive finite element method for an obstacle problem. Numer. Math. 107: 455–471 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Braess, D., Carstensen, C., Hoppe, R.H.W.: Error reduction in adaptive finite element approximation of elliptic obstacle problems (in preparation)Google Scholar
  7. 7.
    Braess D. and Schöberl J. (2008). Equilibrated residual error estimator for edge elements. Math. Comp. 77: 651–672 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Luce R. and Wohlmuth B. (2004). A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42: 1394–1414 MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Morin P., Nochetto R.H. and Siebert K.G. (2000). Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38: 466–488 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Neittaanmäki, P., Repin, S.: Reliable Methods for Computer Simulation. Error Control and A Posteriori Estimates. Elsevier, Amsterdam (2004)Google Scholar
  11. 11.
    Prager W. and Synge J.L. (1947). Approximations in elasticity based on the concept of function spaces. Q. Appl. Math. 5: 241–269 MATHMathSciNetGoogle Scholar
  12. 12.
    Repin S.I. (2003). Estimates of deviations from exact solutions of elliptic variational inequalities. J. Math. Sci. 115: 2811–2819 CrossRefMathSciNetGoogle Scholar
  13. 13.
    Siebert K. and Veeser A. (2007). A unilaterally constrained quadratic minimization with adaptive finite elements. SIAM J. Optim. 18: 260–289 MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Veeser A. (2001). Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39: 146–167 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Weiss, A., Wohlmuth, B.: A posteriori error estimator and error control for contact problems. IANS Report 12/2007, University of StuttgartGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Dietrich Braess
    • 1
  • Ronald H. W. Hoppe
    • 2
    • 3
  • Joachim Schöberl
    • 4
  1. 1.Institute of MathematicsRuhr-University of BochumBochumGermany
  2. 2.Department of MathematicsUniversity of HoustonHoustonUSA
  3. 3.Institute of MathematicsUniversity of AugsburgAugsburgGermany
  4. 4.Department of Mathematics and Center for Computational Engineering ScienceRWTH AachenAachenGermany

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