Skip to main content
SpringerLink
Log in

A residual–based error estimator for BEM–discretizations of contact problems

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary.

We develop an a posteriori error estimate for boundary element solutions of static contact problems without friction. The presented result is based on an error estimate for linear pseudodifferential equations and on a certain commutator property for pseudodifferential operators. A heuristic extension of the obtained error estimate to frictional contact problems is presented, too. Numerical examples indicate a good performance of the error estimator for both the frictionless and the frictional problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Ainsworth, M., Oden, J.T.: Local a posteriori error estimators for variational inequalities. Num. Meth. Part. Diff. Eq. 9, 23–33 (1993)

    MathSciNet  MATH  Google Scholar 

  2. Carstensen, C., Scherf, O., Wriggers, P.: Adaptive finite elements for elastic bodies in contact. SIAM J. Sci. Comput. 20(5), 1605–1626 (1999)

    MATH  Google Scholar 

  3. Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)

    MathSciNet  MATH  Google Scholar 

  4. Costabel, M., Dauge, M.: Invertibility of the biharmonic single layer potential operator. Integral Equations Operator Theory 24, 46–67 (1996)

    MathSciNet  MATH  Google Scholar 

  5. Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology. Vol 4: Integral Equations and Numerical Methods. Berlin: Springer–Verlag 1990

  6. Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer–Verlag, New York: Berlin-Heidelberg 1976

  7. Eck, C., Jarušek, J.: Existence results for the semicoercive static contact problem with Coulomb friction. Nonlinear Analysis TM&A 42, 961–976 (2000)

    Google Scholar 

  8. Eck, C., Jarušek, J.: Existence results for the static contact problem with Coulomb friction. Math. Mod. Meth. Appl. Sci. 8(3), 445–468 (1998)

    MATH  Google Scholar 

  9. Eck, C., Steinbach, O., Wendland, W.L.: A symmetric boundary element method for contact problems with friction. Math. Comp. Simulation 50(1–4), 43–61 (1999)

    Google Scholar 

  10. Eskin, G.I.: Boundary Value Problems for Elliptic Pseudodifferential Equations. AMS Trans. 52, Providence, Rhode Island, 1981

  11. Gwinner, J., Stephan, E.P.: A boundary element procedure for contact problems in plane linear elastostatics. RAIRO, Modelisation Math. Anal. Numer. 27(4), 457–480 (1993)

    Google Scholar 

  12. Han, H., Hsiao, G.C.: The boundary element method for a contact problem. In Theory and Applications of Boundary Element Methods, Proc. 2nd China–Jap. Symp., Beijing 1988, 33–38 (1990)

  13. Han, H.: A direct boundary element method for Signorini problems. Mathematics of Computation 55(191), 115–128 (1990)

    MATH  Google Scholar 

  14. Hsiao, G.C., Wendland, W.L.: A finite element method for some integral equation of the first kind. J. Math. Anal. Appl. 58, 449–481 (1977)

    MATH  Google Scholar 

  15. Hsiao, G.C., Wendland, W.L.: The Aubin–Nitsche lemma for integral equations. J. Integral~Equations 3, 299–315 (1981)

    MathSciNet  MATH  Google Scholar 

  16. Hsiao, G.C., Wendland, W.L.: On a boundary integral method for some exterior problems in elasticity. Proc. Tbilisi Univ. UDK 539.3 Mat. Mech. Astron. 257(18), 31–60 (1985)

    Google Scholar 

  17. Jarušek, J.: Contact problems with bounded friction, coercive case. Czechosl. Math. J. 33(108), 237–261 (1983)

    Google Scholar 

  18. Jarušek, J.: Contact problems with bounded friction, semicoercive case. Czechosl. Math. J. 34(109), 619–629 (1984)

    Google Scholar 

  19. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia, 1988

  20. Kinderlehrer, D.: Remarks about Signorini's problem in linear elasticity. Ann. Scuola Norm. Sup. Pisa 4(8), 605–645 (1981)

    Google Scholar 

  21. Kornhuber, R.: A posteriori error estimates for elliptic variational inequalities. Comp. Math. Appl. 31(8), 49–60 (1996)

    MATH  Google Scholar 

  22. Lee, C.Y., Oden, J.T.: A posteriori error estimation of h-p finite element approximations of frictional contact problems. Comput. Methods Appl. Mech. Eng. 113(1–2), 11–45 (1994)

    Google Scholar 

  23. Nečas, J., Jarušek, J., Haslinger, J.: On the solution of the variational inequality to the Signorini problem with small friction. Bollettino U.M.I., 17(5), 796–811 (1980)

    Google Scholar 

  24. Saranen, J., Wendland, W.L.: Local residual-type error estimates for adaptive boundary element methods on closed curves. Applicable Analysis 48, 37–50 (1993)

    MathSciNet  MATH  Google Scholar 

  25. Schulz, H.: Über lokale und globale Fehlerabschätzungen für adaptive Randele- mentmethoden. Doctoral Thesis, University of Stuttgart, 1997

  26. Spann, W.: Error estimates for the boundary element approximation of a contact problem in elasticity. Math. Methods Appl. Sci. 20(3), 205–217 (1997)

    Article  MATH  Google Scholar 

  27. Steinbach, O.: A note on initial higher order convergence results for boundary element methods with approximated boundary conditions. Num. Methods Partial Diff. Eq. 16, 581–588 (2000)

    Article  Google Scholar 

  28. Steinbach, O., Wendland, W.L.: The construction of efficient preconditioners in the boundary element method. Adv. Comp. Math. 9, 191–216 (1998)

    Google Scholar 

  29. Steinbach, O., Wendland, W.L.: On C. Neumann's method for second order elliptic systems in domains with non–smooth boundaries. J. Math. Anal. Appl. 262, 733–748, (2001)

    Article  MATH  Google Scholar 

  30. Taylor, M.E.: Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey, 1981

  31. Wendland, W.L., Yu, D.: Adaptive boundary element methods for strongly elliptic integral equations. Numer. Math. 53, 539–558 (1988)

    Google Scholar 

  32. Wendland, W.L., Yu, D.: Local error estimates of boundary element methods for pseudo–differential–operators of non-negative order on closed curves. J. Comp. Math. 10, 273–289 (1990)

    Google Scholar 

  33. Wriggers, P., Scherf, O., Carstensen, C.: An adaptive finite element model for friction- less contact problems. ZAMM 75, S37–S40 (1995)

    Google Scholar 

  34. Wriggers, P., Scherf, O.: An adaptive finite element algorithm for contact problems in plasticity. Comp. Mech. 17, 88–97 (1995)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Erlangen, Germany

    C. Eck

  2. Institut für Angewandte Analysis und Numerische Simulation, Angewandte Mathematik, Universität Stuttgart, Lehrstuhl 6, Germany

    W.L. Wendland

Authors
  1. C. Eck
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W.L. Wendland
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W.L. Wendland.

Additional information

Mathematics Subject Classification (1991): 35J85, 65N38, 73T05

Dedicated to Hans Grabmüller on the occasion of his 60th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eck, C., Wendland, W. A residual–based error estimator for BEM–discretizations of contact problems. Numer. Math. 95, 253–282 (2003). https://doi.org/10.1007/s00211-002-0425-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-002-0425-x

Keywords

Search

Navigation