Volume 20, pp. 104-118, 2005.

Quadrature over the sphere

Kendall Atkinson and Alvise Sommariva

Abstract

Consider integration over the unit sphere in ${\mathbf{R}}^3$, especially when the integrand has singular behaviour in a polar region. In an earlier paper [4], a numerical integration method was proposed that uses a transformation that leads to an integration problem over the unit sphere with an integrand that is much smoother in the polar regions of the sphere. The transformation uses a grading parameter $q$. The trapezoidal rule is applied to the spherical coordinates representation of the transformed problem. The method is simple to apply, and it was shown in [4] to have convergence $O\left(h^{2q}\right)$ or better for integer values of $2q$. In this paper, we extend those results to non-integral values of $2q$. We also examine superconvergence that was observed when $2q$ is an odd integer. The overall results agree with those of [11], although the latter is for a different, but related, class of transformations.

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Key words

spherical integration, trapezoidal rule, Euler-MacLaurin expansion

AMS subject classifications

65D32

Links to the cited ETNA articles

[4]	Vol. 17 (2004), pp. 133-150 Kendall Atkinson: Quadrature of singular integrands over surfaces