A posteriori estimates for FE-solutions of variational inequalities

  • H. Blum
  • A. Schröder
  • F. T. Suttmeier
Conference paper


The concept of weighted residual based estimators for a posteriori error control is extended in two respects: first, variational inequalities arising from problems in structural mechanics are treated by a suitable adaptation of Natterer’s duality argument; second, the arguments are carried over to p- and hp-finite element spaces, for which we propose an adaptive hp-algorithm. Several numerical tests confirm the robustness and efficiency of our results.


Variational Inequality Posteriori Error Obstacle Problem Posteriori Error Estimate Finite Element Space 
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  1. [1]
    Babuška, I., Gui, W. (1986): The h,p and hp versions of the finite element method in 1 dimension. III. The adaptive h-p version. Numer. Math. 49, 659–683MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Babuška, I., Guo, B. (1986): The hp version of the finite element method. I. The basic approximation results. Comput Mech. 1, 21–41MATHCrossRefGoogle Scholar
  3. [3]
    Backes, E. (1997): Gewichtete a posteriori Fehleranalyse bei der adaptiven Finite-Elemente-Methode: Ein Vergleich zwischen Residuen-und Bank-Weiser-Schätzer. Diplomarbeit. Institut für Angewandte Mathematik, Universität Heidelberg, HeidelbergGoogle Scholar
  4. [4]
    Becker, R. (1995): An adaptive finite element method for the incompressible Navier-Stokes equations on time-dependent domains. Dissertation. Institut für Angewandte Mathematik, Universität Heidelberg, HeidelbergGoogle Scholar
  5. [5]
    Becker, R., Rannacher, R. (1996): A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math. 4, 237–264MathSciNetMATHGoogle Scholar
  6. [6]
    Blum, H., Schröder, A., Suttmeier, F.T. (2002): A posteriori error estimates for FE-schemes of higher order for variational equations and inequalities, in preparationGoogle Scholar
  7. [7]
    Blum, H., Suttmeier, F.T. (2000): An adaptive finite element discretisation for a simplified Signorini problem. Calcolo 37, 65–77MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Blum, H., Suttmeier, F.T. (2000): Weighted error estimates for finite element solutions of variational inequalities. Computing 65, 119–134MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Brézis, H., Stampacchia, G. (1968): Sur la régularité de la solution d’inéquations elliptiques. Bull. Soc. Math. France 96, 153–180MathSciNetMATHGoogle Scholar
  10. [10]
    Carstensen, C., Scherf, O., Wriggers, P. (1999): Adaptive finite elements for elastic bodies in contact. SIAM J. Sci. Comput. 20, 1605–1626MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Duvaut, G., Lions, J.-L. (1976): Inequalities in mechanics and physics. Springer, BerlinMATHCrossRefGoogle Scholar
  12. [12]
    Eriksson, K., Estep, D., Hansbo, P., Johnson, C. (1995): Introduction to adaptive methods for differential equations. Acta Numerica. 1995. Cambridge University Press, Cambridge, pp.105–158Google Scholar
  13. [13]
    Hansbo, P., Johnson, C. (1999): Adaptive finite element methods for elastostatic contact problems. In: Bern, M. et al. (eds.): Grid generation and adaptive algorithms. Springer, New York, pp. 135–149CrossRefGoogle Scholar
  14. [14]
    Johnson, C., Hansbo, P. (1992): Adaptive finite element methods in computational mechanics. Comput Methods Appl. Mech. Engrg. 101, 143–181MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Kanschat, G. (1996): Parallel and adaptive Galerkin methods for radiative transfer problems. Dissertation. Institut für Angewandte Mathematik, University of Heidelberg, HeidelbergGoogle Scholar
  16. [16]
    Lions, J.-L., Stampacchia, G. (1967): Variational inequalities. Comm. Pure Appl. Math. 20, 493–519MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Natterer, F.(1976): Optimale L 2-Konvergenz finiter Elemente bei Variationsungleichungen. In: Frehse, J. et al. (eds.): Finite Elemente. (Bonner Mathematische Schriften 89) Institut für Angewandte Mathematik, Universität Bonn, Bonn, pp. 1–12Google Scholar
  18. [18]
    Rannacher, R. (1999): Error control in finite element computations. An introduction to error estimation and mesh-size adaptation. In: Bulgak, H., Zenger, C. (eds.): Error control and adaptivity in scientific computing. Kluwer, Dordrecht, pp. 247–278CrossRefGoogle Scholar
  19. [19]
    Rannacher, R., Suttmeier, F.T. (1999): A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity. Comput. Methods Appl. Mech. Engrg. 176, 333–361MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Schwab, C. (1998): p-and hp-finite element methods. Clarendon Press, OxfordGoogle Scholar
  21. [21]
    Suttmeier, F.T. (1996): Adaptive finite element approximation of problems in elastoplasticity theory. Dissertation. Institut für Angewandte Mathematik, University of Heidelberg, HeidelbergGoogle Scholar
  22. [22]
    Tricomi, F. (1955): Vorlesungen über Orthogonalreihen. Springer, BerlinMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2003

Authors and Affiliations

  • H. Blum
    • 1
  • A. Schröder
    • 1
  • F. T. Suttmeier
    • 1
  1. 1.Fachbereich MathematikUniversität DortmundDortmundGermany

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