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Transformation of hypersingular integrals and black-box cubature. (English) Zbl 0958.65123
The authors discuss the numerical computation of hypersingular integrals, as they appear in the Galerkin 3-D boundary element implementation. Emphasis is given to the adaptation of existing quadrature methods, for weakly singular integrals, to the problem.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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[1] Donald G. Anderson, Gaussian quadrature formulae for \int \(_{0}\)\textonesuperior -\?\?(\?)\?(\?)\?\?, Math. Comp. 19 (1965), 477 – 481. · Zbl 0132.36803
[2] K. Atkinson. Solving Integral Equations on Surfaces in Space. In G. Hämmerlin and K. Hoffmann, editors, Constructive Methods for the Practical Treatment of Integral Equations, pages 20-43. Birkhäuser: ISNM, 1985. CMP 19:10
[3] Martin Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613 – 626. · Zbl 0644.35037
[4] Stefan Erichsen and Stefan A. Sauter, Efficient automatic quadrature in 3-d Galerkin BEM, Comput. Methods Appl. Mech. Engrg. 157 (1998), no. 3-4, 215 – 224. Seventh Conference on Numerical Methods and Computational Mechanics in Science and Engineering (NMCM 96) (Miskolc). · Zbl 0943.65139
[5] I. Graham, W. Hackbusch, and S. Sauter. Discrete boundary element methods on general meshes in 3d. Technical Report 97/19, University of Bath, U.K., 1997. Mathematics Preprint, to appear in Numer. Math. · Zbl 0966.65090
[6] M. Guiggiani. Direct Evaluation of Hypersingular Integrals in 2D BEM. In W. Hackbusch, editor, Proc. Of 7th GAMM Seminar on Numerical Techniques for BEM, Kiel 1991, pages 23-34, Braunschweig, 1991. Vieweg. · Zbl 0743.65013
[7] M. Guiggiani and A. Gigante, A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method, Trans. ASME J. Appl. Mech. 57 (1990), no. 4, 906 – 915. · Zbl 0735.73084
[8] Wolfgang Hackbusch, Integral equations, International Series of Numerical Mathematics, vol. 120, Birkhäuser Verlag, Basel, 1995. Theory and numerical treatment; Translated and revised by the author from the 1989 German original. · Zbl 0681.65099
[9] Wolfgang Hackbush and Stefan A. Sauter, On the efficient use of the Galerkin method to solve Fredholm integral equations, Proceedings of ISNA ’92 — International Symposium on Numerical Analysis, Part I (Prague, 1992), 1993, pp. 301 – 322. · Zbl 0791.65101
[10] Hou De Han, A boundary element method for Signorini problems in three dimensions, Numer. Math. 60 (1991), no. 1, 63 – 75. · Zbl 0723.65092
[11] Hou De Han, The boundary integro-differential equations of three-dimensional Neumann problem in linear elasticity, Numer. Math. 68 (1994), no. 2, 269 – 281. · Zbl 0806.73008
[12] R. Kieser. Über einseitige Sprungrelationen und hypersinguläre Operatoren in der Methode der Randelemente. PhD thesis, Mathematisches Institut A, Universität Stuttgart, Germany, 1990.
[13] José M. Pérez-Jordá, Emilio San-Fabián, and Federico Moscardó, A simple, reliable and efficient scheme for automatic numerical integration, Comput. Phys. Comm. 70 (1992), no. 2, 271 – 284.
[14] C. Lage. Software Development for Boundary Element Mehtods: Analysis and Design of Efficient Techniques (in German). PhD thesis, Lehrstuhl Prakt. Math., Universität Kiel, 1995.
[15] J. N. Lyness, Applications of extrapolation techniques to multidimensional quadrature of some integrand functions with a singularity, J. Computational Phys. 20 (1976), no. 3, 346 – 364. · Zbl 0336.65015
[16] J. N. Lyness, An error functional expansion for \?-dimensional quadrature with an integrand function singular at a point, Math. Comp. 30 (1976), no. 133, 1 – 23. · Zbl 0343.65007
[17] J. N. Lyness and G. Monegato, Quadrature error functional expansions for the simplex when the integrand function has singularities at vertices, Math. Comp. 34 (1980), no. 149, 213 – 225. · Zbl 0442.41023
[18] Solomon G. Mikhlin and Siegfried Prössdorf, Singular integral operators, Springer-Verlag, Berlin, 1986. Translated from the German by Albrecht Böttcher and Reinhard Lehmann. · Zbl 0612.47024
[19] J.-C. Nédélec, Curved finite element methods for the solution of integral singular equations on surfaces in \?³, Computing methods in applied sciences and engineering (Second Internat. Sympos., Versailles, 1975) Springer, Berlin, 1976, pp. 374 – 390. Lecture Notes in Econom. and Math. Systems, Vol. 134.
[20] J.-C. Nédélec, Integral equations with nonintegrable kernels, Integral Equations Operator Theory 5 (1982), no. 4, 562 – 572. · Zbl 0479.65060
[21] O. A. Oleĭnik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, vol. 26, North-Holland Publishing Co., Amsterdam, 1992. · Zbl 0768.73003
[22] S. Sauter and C. Lage. Transformation of hypersingular integrals and black-box cubature (extended version). Technical Report 97-17, Universität Kiel, 1997. (Available via WWW-address: http://www.numerik.uni-kiel.de/reports/1997/). · Zbl 0958.65123
[23] S. A. Sauter. Über die effiziente Verwendung des Galerkinverfahrens zur Lösung Fredholmscher Integralgleichungen. PhD thesis, Inst. f. Prakt. Math., Universität Kiel, 1992. · Zbl 0850.65366
[24] S. A. Sauter and A. Krapp, On the effect of numerical integration in the Galerkin boundary element method, Numer. Math. 74 (1996), no. 3, 337 – 359. · Zbl 0878.65104
[25] C. Schwab and W. L. Wendland, Kernel properties and representations of boundary integral operators, Math. Nachr. 156 (1992), 187 – 218. · Zbl 0805.35168
[26] 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. · Zbl 0761.65012
[27] Tobias von Petersdorff and Christoph Schwab, Fully discrete multiscale Galerkin BEM, Multiscale wavelet methods for partial differential equations, Wavelet Anal. Appl., vol. 6, Academic Press, San Diego, CA, 1997, pp. 287 – 346.
[28] 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.
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