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Equadiff 6 pp 345-352 | Cite as

Singularities in two- and three-dimensional elliptic problems and finite element methods for their treatment

  • J. R. Whiteman
Lectures Presented In Sections Section C Numerical Methods
Part of the Lecture Notes in Mathematics book series (LNM, volume 1192)

Keywords

Polygonal Domain Adaptive Mesh Refinement Singular Element Poisson Problem Polyhedral Domain 
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References

  1. [1]
    Ciarlet P.G., The finite Element Method for Elliptic Problems. Nort-Holland, Amsterdam, 1979.Google Scholar
  2. [2]
    Schatz A., An introduction to the analysis of the error in the finite element method for second order elliptic boundary value problems. pp. 94–139 of P.R. Turner (ed.) Numerical Analysis Lancaster 1984. Lecture Notes in Mathematics 129, Springer-Verlag, Berlin, 1985.CrossRefGoogle Scholar
  3. [3]
    Nitsche J.A., L=convergence of finite element approximations, mathematical aspects of finite element methods. Lecture Notes in Mathematics 606, Springer-Verlag, Berlin, 1977.Google Scholar
  4. [4]
    Grisvard P., Behaviour of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain. pp. 207–274 of B. Hubbard (ed.), Numerical Solution of Partial Differential Equations III, SYNSPADE 1975. Academic Press, New York, 1976.CrossRefGoogle Scholar
  5. [5]
    Schaltz A. and Wahlbin L., Maximum norm estimates in the finite element method on plane polygonal domains. Parts I and II. Math. Comp. 32, 73–109, 1978, and Math. Comp. 33, 465–492, 1979.MathSciNetzbMATHGoogle Scholar
  6. [6]
    Oden J.T. and O'Leary J., Some remarks on finite element approximations of crack problems and an analysis of hybrid methods. J. Struct. Mech. 64, 415–436, 1978.CrossRefGoogle Scholar
  7. [7]
    Fix G., Higher order Rayleigh Ritz approximations. J. Math. Mech. 18, 645–657, 1969.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Barnhill R.E. and Whiteman J.R., Error analysis of Galerkin methods for Dirichlet problems containing boundary singularities. J. Inst. Math. Applics.15, 121–125, 1975.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Stephan E. and Whiteman J.R., Singularities of the Laplacian at corners and edges of three dimensional domains and their treatment with finite element methods. Technical Report BICOM 81/1, Institute of Computational Mathematics, Brunel Uninersity, 1981.Google Scholar
  10. [10]
    Akin J.E., Generation of elements with singularities. Int. J. Numer. Method. Eng. 10, 1249–1259, 1976.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Blackburn W.S., Calculation of stress intensity factors at crack tips using special finite elements. pp. 327–336 of J.R. Whiteman (ed.), The Mathematics of Finite Elements and Applications. Academic Press, London, 1973.CrossRefGoogle Scholar
  12. [12]
    Stern M. and Becker E., A conforming crack tip element with quadratic variation in the singular fields. Int. J. Numer. Meth. Eng. 12, 279–288, 1978.CrossRefzbMATHGoogle Scholar
  13. [13]
    O'Leary J.R., An error analysis for singular finite elements. TICOM Report 81–4, Texas Institute of Computational Mechanics, University of Texas at Austin, 1981.Google Scholar
  14. [14]
    Babuska I. and Osborn J., Finite element methods for the solution of problems with rough input data. pp. 1–18 of P. Grisvard, W. Wendland and J.R. Whiteman (eds.), Singularities and Constructive Methods for Their Treatment. Lecture Notes in Mathematics 1121, Springer Verlag, Berlin, 1985.CrossRefGoogle Scholar
  15. [15]
    Babuska I. and Rheinboldt W.C., Error estimates for adaptive finite element computations. SIAM J. Num. Anal. 15, 736–754, 1978.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Babuska I. and Rheinboldt W.C., Reliable error estimation and mesh adaptation for the finite element method. pp. 67–108 of J.T. Oden (ed.), Computational Methods in nonlinear Mechanics. North-Holland, Amsterdam, 1979.Google Scholar
  17. [17]
    Craig A.W., Zhu J.Z. and Zienkiewicz O.C., A-posterironi error estimation, adaptive mesh refinement and multigrid methods using hierarchical finite element bases. pp. 587–594 of J.R. Whiteman (ed.), The Mathematics of Finite Elements and Applications V., Mafelap 1984. Academic Press, London, 1985.Google Scholar
  18. [18]
    Bank R.E. and Sherman A.H., The use of adaptive grid refinement for badly behaved elliptic partical differential equations. pp. 18–24 of Computers in Simulation XXII. North Holland, Amsterdam, 1980.Google Scholar
  19. [19]
    Rivara M.C., Dynamic implementation of the h-version of the finite element method. pp. 595–602 of J.R. Whiteman (ed.), The Mathem. of Finite Elements and Applic. V., MAFELAP 1984, Academic Press, London, 1985.Google Scholar
  20. [20]
    Destuynder P, Djaoua M. and Lescure S., On numerical methods for fracture mechanics. pp. 69–84 of P. Grisvard, W.L. Wendland and J.R. Whiteman (eds.), Singularities and constructive methods for their treatment. Lecture Notes in Mathematics 1121, Springer Verlag, Berlin, 1985.CrossRefGoogle Scholar
  21. [21]
    Levine N., Superconvergence recovery of the gradient from piecewise linear finite element approximations. Technical Report 6/83, Dept. of Mathematics, University of Reading, 1983.Google Scholar
  22. [22]
    Stephan E., A modified Fix method for the mixed boundary value problem of the Laplacian in a polyhedral domain. Preprint Nr. 538. Fachbereich Mathematik, T.H. Darmstadt, 1980.Google Scholar
  23. [23]
    Walden H. and Kellogg R.B., Numerical determination of the fundamental eigenvalue for the Laplace operator on a sphercial domain. J. Engineering Mathematics 11, 299–318, 1977.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Beagles A.E. and Whiteman J.R., Treatment of a re-entrant vertex in a three dimensional Poisson problem. pp. 19–27 of P. Grisvard, W.H. Wendland and J.R. Whiteman (eds.), Singularities and Constructive Methods for Their Treatment. Lecture Notes in Mathematics 1121, Springer Verlag, Berlin, 1985.CrossRefGoogle Scholar
  25. [25]
    Beagles A.E. and Whiteman J.R., Finite element treatment of boundary singularities by argumentation with non-exact singular functions. Technical Report BICOM 85/1, Institute of Computational Mathematics, Brunel University, 1985.Google Scholar

Copyright information

© Equadiff 6 and Springer-Verlag 1986

Authors and Affiliations

  • J. R. Whiteman
    • 1
  1. 1.Brunel UniversityUxbridgeEngland

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