Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T18:13:06.250Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical integration on the sphere

Published online by Cambridge University Press:  17 February 2009

Kendall Atkinson
Kendall Atkinson
Affiliation:
Mathematics Department, University of Iowa, Iowa City, Iowa 52242, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This is a discussion of some numerical integration methods for surface integrals over the unit sphere in R3. Product Gaussian quadrature and finite-element type methods are considered. The paper concludes with a discussion of the evaluation of singular double layer integrals arising in potential theory.

Type
Research Article
Information
The ANZIAM Journal , Volume 23 , Issue 3 , January 1982 , pp. 332 - 347
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Atkinson, K., “The numerical solution of Laplace's equation in three dimensions”, SIAM J.Numer. Anal. (in press).Google Scholar
[2]Atkinson, K., “The numerical solution of Laplace's equation in three dimensions, part II”, in Albrecht, J. and Collatz, L. (eds.), Numerical treatment of integral equations (Birkhauser-Verlag, Basel, 1980), 1–23.Google Scholar
[3]Birtles, A., Mayo, B., and Bennett, A., “Computer technique for solving 3-dimensional electron-optics and capacitance problems”, Proc. IEEE 120 (1972), 213–220.Google Scholar
[4]Gunter, N., Potential theory (Ungar, New York, 1967).Google Scholar
[5]Jaswon, M. and Symin, O., Integral equation methods in potential theory and elastostatics (Academic Press, London, 1977).Google Scholar
[6]Keast, P. and diaz, J., “Quadrature rules for the surface of the s-dimensional sphere”, preprint, Univ. of Toronto, 1979.Google Scholar
[7]Lebedev, V., “Quadratures on a sphere”, USSR Comp. Math, and Phys. 16 (1976), 10–24.CrossRefGoogle Scholar
[8]Lyness, J., “An error functional expansion for N-dimensional quadrature with an integrand function singular at a point”, Math. Comp. 30 (1976), 1–23.Google Scholar
[9]MacRobert, T., Spherical harmonics (Pergamon Press, London, 1967).Google Scholar
[10]Mikhlin, S. G., Mathematical physics: an advanced course (North-Holland, Amsterdam, 1970).Google Scholar
[11]Ragozin, D., “Constructive polynomial approximation on spheres and projective spaces”, Trans. Amer. Math. Soc. 162 (1971), 157–170.Google Scholar
[12]Sobolev, S. L., “Cubature formulas on the sphere invariant under finite groups of rotations”, Soviet Math. 3 (1962), 1307–1310.Google Scholar
[13]Stroud, A., Approximate calculation of multiple integrals (Prentice-Hall, New Jersey, 1971).Google Scholar
[14]Wait, R., “Use of finite elements in multi-dimensional problems in practice’, in Walsh, J. and Delves, M. (eds.), Numerical solution of integral equations (Oxford University Press, 1974), 300–311.Google Scholar