Application of class Sm variable transformations to numerical integration over surfaces of spheres

摘要

Class Sm variable transformations with integer m for finite-range integrals were introduced by the author (Numerical Integration IV, International series of Numerical Mathematics, Basel, 1993, pp. 359–373) about a decade ago. These transformations “periodize” the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy, especially with even m. In a recent work by the author (Math. Comp. (2005)), these transformations were extended to arbitrary m, and their role in improving the convergence of the trapezoidal rule for different classes of integrands was studied in detail. It was shown that, with m chosen appropriately, exceptionally high accuracy can be achieved by the trapezoidal rule. In the present work, we make use of these transformations in the computation of integrals on surfaces of spheres in conjunction with the product trapezoidal rule. We treat integrands that have point singularities of the single-layer and double-layer types. We propose different approaches and provide full analyses of the errors incurred in each. We show that surprisingly high accuracies can be achieved with suitable values of m. We also illustrate the theoretical results with numerical examples. Finally, we also recall analogous procedures developed in another work by the author (Appl. Math. Comput. (2005)) for regular integrands.