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Mathematics of Computation

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On the extraction technique in boundary integral equations
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by C. Schwab and W. L. Wendland PDF
Math. Comp. 68 (1999), 91-122 Request permission

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.
References
  • Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
  • Ju. S. Burago and V. G. Maz′ja, Certain questions of potential theory and function theory for regions with irregular boundaries, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 3 (1967), 152 (Russian). MR 0227447
  • M. Costabel: Starke Elliptizität von Randintegraloperatoren erster Art, Habilitationsschrift Technische Hochschule Darmstadt, Fachbereich Mathematik, Preprint No. 868 (1984).
  • Martin Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613–626. MR 937473, DOI 10.1137/0519043
  • Martin Costabel and Monique Dauge, On representation formulas and radiation conditions, Math. Methods Appl. Sci. 20 (1997), no. 2, 133–150. MR 1430037, DOI 10.1002/(SICI)1099-1476(19970125)20:2<133::AID-MMA841>3.0.CO;2-Y
  • M. Costabel and W. L. Wendland, Strong ellipticity of boundary integral operators, J. Reine Angew. Math. 372 (1986), 34–63. MR 863517, DOI 10.1515/crll.1986.372.34
  • T.A. Cruse and J.D. Richardson: Continuity of the elastic BIE formulation, (Preprint Dept. Mech. Engrg. Vanderbilt Univ. 1995).
  • Q. Huang and T. A. Cruse, On the non-singular traction-BIE in elasticity, Internat. J. Numer. Methods Engrg. 37 (1994), no. 12, 2041–2072. MR 1279502, DOI 10.1002/nme.1620371204
  • Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 3, Springer-Verlag, Berlin, 1990. Spectral theory and applications; With the collaboration of Michel Artola and Michel Cessenat; Translated from the French by John C. Amson. MR 1064315
  • 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.
  • 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.
  • 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), no. 3, 604–614. MR 1183216, DOI 10.1115/1.2893766
  • F. Hartmann, The physical nature of elastic layers, J. Elasticity 12 (1982), no. 1, 19–29 (English, with German summary). MR 651117, DOI 10.1007/BF00043703
  • George C. Hsiao and Wolfgang L. Wendland, A finite element method for some integral equations of the first kind, J. Math. Anal. Appl. 58 (1977), no. 3, 449–481. MR 461963, DOI 10.1016/0022-247X(77)90186-X
  • G. C. Hsiao and W. L. Wendland, The Aubin-Nitsche lemma for integral equations, J. Integral Equations 3 (1981), no. 4, 299–315. MR 634453
  • G. C. Hsiao and W. L. Wendland, On a boundary integral method for some exterior problems in elasticity, Trudy Tbiliss. Univ. Mat. Mekh. Astronom. 18 (1985), 31–60 (English, with Russian and Georgian summaries). MR 874318
  • G. C. Hsiao and W. L. Wendland: Variational Methods for Boundary Integral Equations. In preparation.
  • O. Huber, A. Lang, and G. Kuhn, Evaluation of the stress tensor in $3$D elastostatics by direct solving of hypersingular integrals, Comput. Mech. 12 (1993), no. 1-2, 39–50. MR 1229600, DOI 10.1007/BF00370484
  • Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
  • J. H. Kane and C. Balakrishna, Symmetric Galerkin boundary formulations employing curved elements, Internat. J. Numer. Methods Engrg. 36 (1993), no. 13, 2157–2187. MR 1229221, DOI 10.1002/nme.1620361303
  • R. Kieser: Hypersinguläre Operatoren und einseitige Sprungrelationen in der Methode der Randelemente, Doctoral Dissertation, Stuttgart University 1991.
  • U. Kosel and K.-J. Röhlig, Bifurkationsprobleme bei Kanalströmungen zäher Flüssigkeiten, Z. Angew. Math. Mech. 72 (1992), no. 6, T428–T436 (German). Bericht über die Wissenschaftliche Jahrestagung der GAMM (Kraków, 1991). MR 1178317
  • 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.
  • Detlef Laugwitz, Differentialgeometrie, Mathematische Leitfäden, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1960 (German). MR 0116266
  • J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
  • 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.
  • S. G. Mikhlin, Multidimensional singular integrals and integral equations, Pergamon Press, Oxford-New York-Paris, 1965. Translated from the Russian by W. J. A. Whyte; Translation edited by I. N. Sneddon. MR 0185399
  • Carlo Miranda, Partial differential equations of elliptic type, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York-Berlin, 1970. Second revised edition. Translated from the Italian by Zane C. Motteler. MR 0284700
  • Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511, DOI 10.1007/978-3-540-69952-1
  • Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
  • J.-C. Nédélec and J. Planchard, Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dans $R^{3}$, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 105–129 (French, with English summary). MR 424022
  • 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.
  • C. Schwab, Variable order composite quadrature of singular and nearly singular integrals, Computing 53 (1994), no. 2, 173–194 (English, with English and German summaries). MR 1300776, DOI 10.1007/BF02252988
  • C. Schwab and W. L. Wendland, Kernel properties and representations of boundary integral operators, Math. Nachr. 156 (1992), 187–218. MR 1233945, DOI 10.1002/mana.19921560113
  • C. Schwab and W. L. Wendland, On numerical cubatures of singular surface integrals in boundary element methods, Numer. Math. 62 (1992), no. 3, 343–369. MR 1169009, DOI 10.1007/BF01396234
  • Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, IX-X, Hermann, Paris, 1966 (French). Nouvelle édition, entiérement corrigée, refondue et augmentée. MR 0209834
  • François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 1, University Series in Mathematics, Plenum Press, New York-London, 1980. Pseudodifferential operators. MR 597144, DOI 10.1007/978-1-4684-8780-0
  • B. R. Vaĭnberg, Asymptotic methods in equations of mathematical physics, Gordon & Breach Science Publishers, New York, 1989. Translated from the Russian by E. Primrose. MR 1054376
  • Wolfgang Wendland, Die Behandlung von Randwertaufgaben im $R_{3}$ mit Hilfe von Einfach- und Doppelschichtpotentialen, Numer. Math. 11 (1968), 380–404 (German). MR 231550, DOI 10.1007/BF02161886
  • W. L. Wendland, Strongly elliptic boundary integral equations, The state of the art in numerical analysis (Birmingham, 1986) Inst. Math. Appl. Conf. Ser. New Ser., vol. 9, Oxford Univ. Press, New York, 1987, pp. 511–562. MR 921677
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Additional Information
  • C. Schwab
  • Affiliation: Seminar für Angewandte Mathematik, ETH Zürich, CH–8092 Zürich, Switzerland
  • Email: schwab@sam.math.ethz.ch
  • W. L. Wendland
  • Affiliation: Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, D–70569 Stuttgart, Germany
  • Email: wendland@mathematik.uni-stuttgart.de
  • 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
  • © Copyright 1999 American Mathematical Society
  • Journal: Math. Comp. 68 (1999), 91-122
  • MSC (1991): Primary 45F15, 15N38, 45K05; Secondary 47G30, 58G15, 35J25
  • DOI: https://doi.org/10.1090/S0025-5718-99-01044-3
  • MathSciNet review: 1620247