Curved Finite Element Methods for the Solution of Integral Singular Equations on Surfaces in R3

  • J. C. Nedelec
Conference paper
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 134)

Abstract

In NEDELEC-PLANCHARD [N.P.], we have shown that the singular integral equation on a closed surface Γ of R3:
$$\frac{1}{{4\pi }}\int\limits_\Gamma {\frac{{q(x)}}{{|x - y|}}} dx = u(y);y \in \Gamma $$
(1)
admits an unique solution, and is variational and coercive in the Hilbert space H−1/2(Γ).

In this paper, with the help of curved finite elements, we introduce an approximate surface Γh and an approximate problem on Γh , which solution is qh . Then, we study the error of approximation in some Hilbert spaces, between q , solution of (0), and qh , and also, between the two corresponding potentials u and uh .

Keywords

Assure 

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Bibliography

  1. [A]
    J.H. ARGYRIS: “Energy theorems and structural analysis.” — Butterworths Scientific Publications, London, 1971.Google Scholar
  2. [B.B.]
    P.L. BUTZER, H. BERENS: “Semigroups of operators and approximations.” — Springer Verlag, Berlin, 1967.CrossRefGoogle Scholar
  3. [C.1]
    P.G. CIARLET: “La méthode des éléments finis appliquée aux coques.” — LABORIA, Rapport de Recherche N° 113, 1975.Google Scholar
  4. [C.2]
    P.G. CIARLET: Cours de l’Ecole d’Eté de P“ontréal, 1975.Google Scholar
  5. [C.R.1]
    P.G. CIARLET, P.A. RAVIART: “General Lagrange and Hermite interpolation in Rn with applications to finite element methods.” — Arch. Rat. Mech. Anal., 46, 1972, 177–199.MathSciNetMATHCrossRefGoogle Scholar
  6. [C.R.2]
    P.G. CIARLET, P.A. PAVIART: “Interpolation theory over curved elements with applications to finite element methods.” — Comp. Meth. in Appl. Mech. and Eng., 1, 1972, 217–249.MATHCrossRefGoogle Scholar
  7. [Cr]
    T.A. CRUSE: “Application of the boundary integral equation method to three-dimensional stress analysis.” — Computer Structury, 3, 1973, 509.CrossRefGoogle Scholar
  8. [D.L.]
    M. DUBOIS, J.C. LACHAT: “The integral formulation of boundary value problems. Variational methods in Engineering. ” — University of Southampton, 1972, 989.Google Scholar
  9. [H]
    B. HANOUZET: “Espace de Sobolev avec poids. Application au problème de Dirichlet dans un demi-espace.” Rend. del Semin. Path. della Univ. di Padova, XLVI, 1971, 247–272.Google Scholar
  10. [H.1]
    J.L. HESS: “Review of integral equation techniques for solving potential flow problems with emphasis on the surface source method.” — Comp. Meth. in Appl. Mech. and Eng., 5, 1975, 145–176.MATHCrossRefGoogle Scholar
  11. [H.2]
    J.L. HESS: “Improved solution for potential flow about arbitrary axisymtuetric bodies by the use of a higher-order surface source method.” — Comp. Meth. in Appl. Mech. and Eng., 5, 1975, 297–308.MATHCrossRefGoogle Scholar
  12. [H.S.]
    J.L. HESS, A.M.C. SMITH: “Calculation of nonlifting potential flow about arbitrary three-dimensional bodies.” — J. of Ship Research, 8, 1964, 22–44.Google Scholar
  13. [H.W.]
    G.C. HSIAO, W. WENDLAND: “A finite element method for some integral equations of the first kind.” — To appear.Google Scholar
  14. [Ho]
    HORMANDER: “Linear partial differential operators.” — Springer Verlag, Berlin, 1963.CrossRefGoogle Scholar
  15. [K]
    V.D. KUPRADZE: “Potential methods in the theory of elasticity.” — Darvey, 1965.Google Scholar
  16. [L]
    J.L. LACHAT: “Application de la méthode des éléments finis à l’élasticité plane et aux corps de révolution.” — Mémoires Techniques du C.E.T.I.M., N° 9, Sept. 1971.Google Scholar
  17. [L.W.]
    J.L. LACHAT, J.O. WATSON: “A second generation boundary integral equation program for three-dimensional elastic analysis.”— C.E.T.I.M., 1974.Google Scholar
  18. [LF]
    J. LELONG-FERRAND: “Géométrie différentielle.” Masson, Paris, 1963.Google Scholar
  19. [L.1]
    M.N. LEROUX: “Résolution numérique du problème du potentiel dans le plan par une méthode variationnelle d’éléments finis.” — Thèse de 3ème Cycle, Université de Rennes, 1974.Google Scholar
  20. [L.2]
    M.N. LEROUX: “Equations intégrales pour le problème du potentiel électrique dans le plan. ” C.R.A.S. Paris, Série Math. (A), T. 278, 18.02. 1974, 541–544.Google Scholar
  21. [L.M.]
    J.L. LIONS, E. MAGENES: “Problèmes aux limites non homogènes.” T. 1, Dunod, Paris, 1968.MATHGoogle Scholar
  22. [M]
    S.G. MIKHLIN: “Linear integral equations.” — T. 2, Gordon and Breach, Science Publ. Inc., New York, 1960.Google Scholar
  23. [N.P.]
    J.C. NEDELEC, J. PLANCHARD: “Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dans R3.” R.A.I.R.O., 7, 1973, R3, 105–129.MathSciNetGoogle Scholar
  24. [S]
    R. SEELEY: Cours du C.I.M.E., Stresa 1968 — Ed. Cremonese, Roma, 1969.Google Scholar
  25. [St]
    A.H. STROUD “Approximaté calculation of multiple integrals.” Prentice Hall Inc., 1972.Google Scholar
  26. [Z]
    O.C. ZIENKIEWICZ: “The finite element method in engineering science.” Mc Graw Hill, London, 1971.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  • J. C. Nedelec
    • 1
  1. 1.Centre de Mathématiques AppliquéesEcole PolytechniquePalaiseauFrance

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