Some error estimates for the numerical approximation of surface integrals

Authors:
Kurt Georg and Johannes Tausch

Journal:
Math. Comp. **62** (1994), 755-763

MSC:
Primary 65D30

MathSciNet review:
1219704

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Recently, the first author introduced a new approach to the numerical quadrature of surface integrals in the context of boundary element methods. It is assumed that a global parametrization *m* of the surface is only indirectly given (e.g., via an iterative method) and that *m* is not accessible analytically. Of particular interest are parametrizations which are based on automatic triangulations of surfaces. In order to avoid an explicit reference to the partial derivatives of *m*, modified trapezoidal and midpoint rules were introduced. The present paper discusses some error estimates for these methods. The estimates are surprisingly difficult since -terms have to be shown to cancel; this does not occur in the expansion of the standard rules.

**[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. MR**1067149****[2]**-,*Two-grid iteration method for linear integral equations of the second kind on piecewise smooth surfaces in*, 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. MR**1087769**, 10.1137/0912024**[4]**K. Georg and R. Widmann,*Adaptive quadratures over surfaces*, Colorado State University, 1993, preprint.**[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. MR**1010893****[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. MR**507552**, 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. MR**557543**

Retrieve articles in *Mathematics of Computation*
with MSC:
65D30

Retrieve articles in all journals with MSC: 65D30

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1994-1219704-1

Keywords:
Numerical surface integration,
error estimates of quadrature rules,
boundary element method,
piecewise linear surface approximation,
extrapolation method,
quadrature formula,
trapezoidal rule,
midpoint rule

Article copyright:
© Copyright 1994
American Mathematical Society