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Some error estimates for the numerical approximation of surface integrals. (English) Zbl 0803.65022

K. Georg [SIAM J. Sci. Stat. Comput. 12, No. 2, 443-453 (1991; Zbl 0722.65005)] introduced new methods for the approach to the numerical quadrature of surface integrals in the context of boundary element methods. The main purpose of the new methods is to avoid the handling of the partial derivatives of \({\mathfrak m}\) via finite differences or interpolation, where a parametrization \({\mathfrak m}\) of the surface is given indirectly only.
This paper presents modified trapezoidal and midpoint rules and discusses some error estimates for these methods.
Reviewer: D.Acu (Sibiu)

MSC:

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
41A63 Multidimensional problems
65N38 Boundary element methods for boundary value problems involving PDEs

Citations:

Zbl 0722.65005
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References:

[1] K. E. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace’s equation in three dimensions, Numerical solution of integral equations, Math. Concepts Methods Sci. Engrg., vol. 42, Plenum, New York, 1990, pp. 1 – 34. · Zbl 0737.65085
[2] -, Two-grid iteration method for linear integral equations of the second kind on piecewise smooth surfaces in \( {{\mathbf{R}}^3}\), Report 14, Univ. of Iowa, submitted to SIAM J. Numer. Anal., 1991.
[3] Kurt Georg, Approximation of integrals for boundary element methods, SIAM J. Sci. Statist. Comput. 12 (1991), no. 2, 443 – 453. · Zbl 0722.65005 · doi:10.1137/0912024
[4] K. Georg and R. Widmann, Adaptive quadratures over surfaces, Colorado State University, 1993, preprint. · Zbl 0749.65017
[5] Wolfgang Hackbusch, Integralgleichungen, Teubner Studienbücher Mathematik. [Teubner Mathematical Textbooks], B. G. Teubner, Stuttgart, 1989 (German). Theorie und Numerik. [Theory and numerics]; Leitfäden der Angewandten Mathematik und Mechanik [Guides to Applied Mathematics and Mechanics], 68. · Zbl 0681.65099
[6] J. N. Lyness, Quadrature over a simplex. II. A representation for the error functional, SIAM J. Numer. Anal. 15 (1978), no. 5, 870 – 887. · Zbl 0407.41013 · doi:10.1137/0715057
[7] J. Stoer and R. Bulirsch, Introduction to numerical analysis, Springer-Verlag, New York-Heidelberg, 1980. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. · Zbl 0423.65002
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