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On the extraction technique
in boundary integral equations

Authors: C. Schwab and W. L. Wendland
Journal: Math. Comp. 68 (1999), 91-122
MSC (1991): Primary 45F15, 15N38, 45K05; Secondary 47G30, 58G15, 35J25
MathSciNet review: 1620247
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Abstract: In this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the Cauchy data of regular three-dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations. The method rests on the derivatives of the generalized Green's representation formula, which are expressed in terms of singular boundary integrals as Hadamard's finite parts. Their regularization, together with asymptotic pseudohomogeneous kernel expansions, yields a constructive method for obtaining generalized jump relations. These expansions are obtained via composition of Taylor expansions of the local surface representation, the density functions, differential operators and the fundamental solution of the original problem, together with the use of local polar coordinates in the parameter domain. For boundary integral equations obtained by the direct method, this method allows the recursive numerical extraction of potentials and their derivatives near and up to the boundary surface.

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  • 1. I. Babu\v{s}ka and A.K. Aziz: Survey lectures on the mathematical foundations of the finite element method. In: The Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations (ed. A. K. Aziz), Academic Press, New York (1972) 3-359. MR 54:9111
  • 2. Yu.D. Burago and V.G. Maz'ja: Potential Theory and Function Theory for Irregular Regions. Seminars in Math., V.A. Steklov Math. Inst., Leningrad, Vol. 3 (1967), English Transl. Consultants Bureau, New York (1969). MR 37:3031; MR 39:1633
  • 3. M. Costabel: Starke Elliptizität von Randintegraloperatoren erster Art, Habilitationsschrift Technische Hochschule Darmstadt, Fachbereich Mathematik, Preprint No. 868 (1984).
  • 4. M. Costabel: Boundary integral operators on Lipschitz domains - elementary results, SIAM J. Math. Anal. 19 (1987) 613-626. MR 89h:35090
  • 5. M. Costabel and M.Dauge: On representation formulas and radiation conditions. Math. Methods Appl. Sci. 20 (1997), 133-150. MR 97m:35031
  • 6. M. Costabel and W. L. Wendland: Strong ellipticity of boundary integral operators, Journ. Reine Angew. Math. 372 (1986) 34-63. MR 88c:35048
  • 7. T.A. Cruse and J.D. Richardson: Continuity of the elastic BIE formulation, (Preprint Dept. Mech. Engrg. Vanderbilt Univ. 1995).
  • 8. T.A. Cruse and Q. Huang: On the nonsingular traction-BIE in elasticity. Intern. J. Numer. Methods Engrg. 37 (1994), 2041-2072. MR 95a:73074
  • 9. R. Dautray and J.-L. Lions: Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 4: Integral Equations and Numerical Methods. Springer-Verlag, Berlin 1990. MR 91h:00004b
  • 10. C. Fiedler and L.Gaul: Limiting procedures and calculation of boundary stresses in three-dimenional boundary element method. In: Advances in Comp. Mechanics (M. Papadrakakis and B.H.V. Topping, eds.), Civil-Comp. Ltd., Edinburgh (1994), 311-322.
  • 11. M. Guiggiani: ``Accurate evaluation of stresses on the boundary using hypersingular integral equations'' in: Proc. of the First European Conference on Numerical Methods in Engineering, Brussels, 7-11 Sept. 1992. Ch. Hirsch (Ed.), Elsevier Science Publishers.
  • 12. M. Guiggiani, G. Krishnasamy, T. J. Rudolphi and F. J. Rizzo: A general algorithm for the numerical solution of hypersingular boundary integral equations, Trans. ASME J. Appl. Mech. 59 (1992), 604-614. MR 93f:73109
  • 13. F. Hartmann: Elastic potentials on piecewise smooth surfaces, J. Elasticity 12 (1982) 31-50. MR 83f:73025b
  • 14. G. C. Hsiao and W. L. Wendland: A finite element method for some integral equations of the first kind, J. Math. Anal. Appl. 58 (1977) 449-481. MR 57:1945
  • 15. G. C. Hsiao and W. L. Wendland: The Aubin-Nitsche lemma for integral equations, J. Integral Equations 3 (1981) 299-315. MR 83j:45019
  • 16. G. C. Hsiao and W. L. Wendland: On a boundary integral method for some exterior problems in elasticity, Trudy Tbiliss. Univ. Mat. Mekh. Astr. 18 (1985), 31-60. MR 88f:73019
  • 17. G. C. Hsiao and W. L. Wendland: Variational Methods for Boundary Integral Equations. In preparation.
  • 18. O. Huber, A. Lang and G. Kuhn: Evaluation of the stress tensor in 3D elastostatics by direct solving of hypersingular integrals. Comp. Mechanics 12 (1993) 39-50. MR 94e:73065
  • 19. F. John: Plane Waves and Spherical Means Applied to Partial Differential Equations, Wiley Interscience, New York 1955. MR 17:746d
  • 20. J.H. Kane and C. Balakrishna: Symmetric Galerkin boundary formulations employing curved elements. Intern. J. Numer. Methods Engrg. 36 (1993) 2157-2187. MR 94c:73067
  • 21. R. Kieser: Hypersinguläre Operatoren und einseitige Sprungrelationen in der Methode der Randelemente, Doctoral Dissertation, Stuttgart University 1991.
  • 22. R. Kieser, C. Schwab and W. L. Wendland: Numerical evaluation of singular and finite part integrals on curved surfaces using symbolic manipulation, Computing 49 (1992), 279-301. MR 93i:76037
  • 23. J.C. Lachat and J.O. Watson: Effective numerical treatment of boundary integral equations: A formulation for three-dimensional elastostatics. Intern. J. Numer. Methods Engrg. 10 (1979) 991-1005.
  • 24. D. Laugwitz: Differentialgeometrie. B.G. Teubner, Stuttgart, 1960. MR 22:7061
  • 25. J. L. Lions and E. Magenes: Non-Homogeneous Boundary Value Problems and Applications I, Springer-Verlag, Berlin, 1972. MR 50:2670
  • 26. T. Matsumoto and M. Tanaka: Boundary stress calculation using regularized boundary integral equation for displacement gradients. Intern. J. Numer. Methods Engrg. 36 (1993) 783-797.
  • 27. S.G. Mikhlin: Multidimensional Singular Integrals and Integral Equations. Pergamon Press, Oxford 1965. MR 32:2866
  • 28. C. Miranda: Partial Differential Equations of Elliptic Type. Springer-Verlag, Berlin 1970. MR 44:1924
  • 29. C. B. Morrey, Jr.: Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966. MR 34:2380
  • 30. J. Ne\v{c}as: Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris 1967. MR 37:3168
  • 31. J.C. Nedelec and J. Planchard: Une méthode variationelle d'élements finis pour la résolution numerique d'un problème exterieur dans $\bbbr^3$, Revue Franc. Automatique Inf. Rech. Operationelle 3 (1973) 105-129. MR 54:11992
  • 32. H. Schulz, Ch. Schwab and W.L. Wendland: An extraction technique for BEM. In: Lecture Notes on Numerical Fluid Mechanics, Vol. 54 (W. Hackbusch &
    G. Wittum, eds.), Vieweg Verlag (1996), pp. 219-231.
  • 33. C. Schwab: Variable order composite quadrature of singular and nearly singular integrals. Computing 53 (1994) 173-194. MR 96a:65035
  • 34. C. Schwab and W. L. Wendland: Kernel properties and representations of boundary integral operators, Mathematische Nachrichten 156 (1992) 187-218. MR 94g:65135
  • 35. C. Schwab and W. L. Wendland: On numerical cubatures of singular surface integrals in boundary element methods, Numerische Mathematik 62 (1992) 343-369. MR 93h:65035
  • 36. L. Schwartz: Théorie des Distributions. Hermann, Paris 1966 (3rd. ed.). MR 35:730
  • 37. F. Treves: Pseudodifferential and Fourier Integral Operators. Vol. 1. Plenum Press, New York 1980. MR 82i:35173
  • 38. B.R. Vainberg: Asymptotic Methods in Equations of Mathematical Physics. Gordon & Breach, New York 1989. MR 91h:35081
  • 39. W.L. Wendland: Die Behandlung von Randwertaufgaben im $\bbbr^3$ mit Hilfe von Einfach- und Doppelschichtpotentialen. Numer. Math. 11 (1968) 380-404. MR 37:7103
  • 40. W.L. Wendland: Strongly elliptic boundary integral equations. In: The State of the Art in Numerical Analysis (eds. A. Iserles, M. Powell), Clarendon Press, Oxford (1987) 511-561. MR 88m:65209

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Additional Information

C. Schwab
Affiliation: Seminar für Angewandte Mathematik, ETH Zürich, CH–8092 Zürich, Switzerland

W. L. Wendland
Affiliation: Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, D–70569 Stuttgart, Germany

Keywords: Boundary integral equation methods, derivatives of the Cauchy data, regularization of hypersingular potentials
Received by editor(s): April 29, 1997
Dedicated: This work is dedicated to Professor Dr. G. C. Hsiao on the occasion of his 60$^{th}$ birthday
Article copyright: © Copyright 1999 American Mathematical Society

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