The accuracy of standard finite element and boundary element methods for elliptic boundary value problems on polygonal or polyhedral domains is significantly reduced due to the influence of corner singularities. Based on the validity of a singular decomposition of the solution, this pollution effect is described, with respect to different norms. Several procedures are proposed and analyzed to recover the full order of convergence for the Galerkin solution, as well as for the stress intensity factors.


Stress Intensity Factor Boundary Element Method Corner Point Representation Formula Singular Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kondrat’ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow. Math. Soc. 16 (1967), 227–313MATHGoogle Scholar
  2. 2.
    Maz’ja, V.G. and B.A. Plamenevskij: Coefficients in the asymp-totics of the solutions of elliptic boundary value problems, J. Sov. Math. 9 (1978), 750–764CrossRefGoogle Scholar
  3. 3.
    Fix, G.J., Gulati, S. and G.I. Wakoff: On the use of singular functions with finite element approximations, J. Comput. Physics 13 (1973). 209–228CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Blum, H. and M. Dobrowolski: On finite element methods for elliptic equations on domains with corners, Computing 28 (1982), 53–63CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Dobrowolski, M.: On the numerical treatment of elliptic equations with singularities, Report 8505, Universität der Bundeswehr München, FB Informatik, 1985Google Scholar
  6. 6.
    Costabel, M. and E. Stephan: Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, Mathematical models and methods in mechanics, Banach Center Publ. 15, Warsaw 1985Google Scholar
  7. 7.
    Stephan, E.P. and W.L. Wendland: An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems, Appl. Anal. 18 (1984), 183–220CrossRefMathSciNetGoogle Scholar
  8. 8.
    Tolksdorf, P.: On the Dirichletproblem for quasilinear equations in domains with conical boundary points, Comm. PDE 8 (1983), 773–810CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Dobrowolski, M.: On finite element methods for nonlinear elliptic problems on domains with corners, in: Singularities and Constructive Methods for Their Treatment, Oberwolfach 1983, Lecture Notes in Math. 1121, Springer 1985Google Scholar
  10. 10.
    Melzer, H. and R. Rannacher: Spannungskonzentrationen in Eckpunkten der Kirchhoffschen Platte, Bauingenieur 55 (1980), 181–184Google Scholar
  11. 11.
    Blum, H. and R. Rannacher: À note on Herrmann’s second mixed plate element, Preprint, Universität Saarbrücken, in preparationGoogle Scholar
  12. 12.
    Babuska, I., Kellogg, R.B. and J. Pitkäranta: Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math. 33 (1979), 447–471MATHGoogle Scholar
  13. 13.
    Schatz, A.H. and L.B. Wahlbin: Maximum norm estimates in the finite element method on plane polygonal domains I,II, I: Math. Comp. 32 (1978), 73–109, II: Math. Comp. 33 (1979), 465–492MATHMathSciNetGoogle Scholar
  14. 14.
    Babuska, I. and W.C. Rheinboldt: Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15(4) (1978), 736–754CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Bank, R.E. and A. Weiser: Some a posteriori error estimates for elliptic partial differential equations, Math. Comp. 44 (1985), 1–19CrossRefMathSciNetGoogle Scholar
  16. 16.
    Rank, E. : A posteriori Fehlerabschätzungen und adaptive Netzverfeinerung für Finite-Element und Randintegralmethoden, Mitteilungen aus dem Institut für Bauingenieurwesen I, Heft 16, TU München 1985Google Scholar
  17. 17.
    Dobrowolski, M. : Numerical approximation of elliptic interface and corner problems, Habilitationsschrift, Universität Bonn 1981Google Scholar
  18. 18.
    Blum, H. : Der Einfluß von Eckensingularitäten bei der numerischen Behandlung der biharmonischen Gleichung, Bonner Math. Schr. 140, 1982Google Scholar
  19. 19.
    Moussaoui, M.A. : Sur l’approximation des solution du problème de Dirichlet dans un ouvert avec coins, in: Singularities and Constructive Methods for Their Treatment, Oberwolfach 1983, Lecture Notes in Math. 1121, Springer 1985Google Scholar
  20. 20.
    Blum, H. , Lin, Q. and R. Rannacher: Asymptotic error expansion and Richardson extrapolation for linear finite elements, Numer. Math. 49 (1986), 11–37CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Blum, H. and R. Rannacher: Extrapolation techniques for reducing the pollution effect of reentrant corners in the finite element method, submitted to Numer. Math.Google Scholar
  22. 22.
    Blum, H.: On the approximation of linear elliptic systems on polygonal domains, in: Singularities and Constructive Methods for Their Treatment, Oberwolfach 1983, Lecture Notes in Math. 1121, Springer 1985Google Scholar
  23. 23.
    Grisvard, P.: Elliptic problems in nonsmooth domains, Pitman 1985MATHGoogle Scholar
  24. 24.
    Wendland, W.L. : Boundary element methods and their asymptotic convergence, in: Theoretical Acoustics and Numerical Techniques (Ed. P. Filippi). CISM Courses and Lectures, Springer 1983Google Scholar
  25. 25.
    Babuska, I.: Finite element method for domains with corners, Computing 6 (1970), 264–273CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Zenger, C. and H. Gietl: Improved difference schemes for the Dirichlet problem of Poisson’s equation in the neighbourhood of corners, Numer. Math. 30 (1978), 315–332MATHMathSciNetGoogle Scholar
  27. 27.
    Schatz, A.H. : Lectures on the treatment of corner singularities, Universität Bonn 1980 (unpublished)Google Scholar
  28. 28.
    Schatz, A.H. : A weak discrete maximum principle and stability in the finite element method in L on plane polygonal domains, Math. Comp. 34 (1980), 77–91MATHMathSciNetGoogle Scholar
  29. 29.
    Frehse, J. and R. Rannacher: Eine L1-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente, Bonner Math. Schr. 89 (1976), 92–114MathSciNetGoogle Scholar
  30. 30.
    Wendland, W.L., Stephan, E. and G.C. Hsiao: On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Meth. in the Appl. Sci. 1 (1979), 265–321CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Costabel, M. : Starke Elliptizität von Randintegraloperatoren erster Art, Habilitationsschrift, TH Darmstadt 1984Google Scholar
  32. 32.
    Costabel, M. : Principles of boundary element methods, Preprint 998, TH Darmstadt 1986Google Scholar
  33. 33.
    Kuhn, G. : Boundary element technique in elastostatics and linear fracture mechanics, this volumeGoogle Scholar
  34. 34.
    Blum, H. : A simple and accurate method for the determination of stress intensity factors and solutions for problems on domains with corners, in: The Mathematics of Finite Elements and Applications IV (Ed. J.R. Whiteman), Academic Press 1982Google Scholar
  35. 35.
    Dobrowolski, M. : On quasilinear elliptic equations in domains with conical boundary points, Report 8506, Universität der Bundeswehr München, FB Informatik, 1985Google Scholar
  36. 36.
    Nitsche, J.A. : Zur lokalen Konvergenz von Projektionen auf finite Elemente, Approximation Theory, Bonn 1976, Lecture Notes in Math. 556, Springer 1976Google Scholar

Copyright information

© Springer-Verlag Wien 1988

Authors and Affiliations

  • H. Blum
    • 1
  1. 1.Universität des SaarlandesSaarbrückenGermany

Personalised recommendations