Numerical integration over smooth surfaces in [formula omitted] via class [formula omitted] variable transformations. Part II: Singular integrands

Abstract

Class S m variable transformations with integer m for finite-range integrals were introduced by the author 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, these transformations were extended to arbitrary real 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. The present work is Part II of a series of two papers dealing with the use of these transformations in the computation of integrals on surfaces of simply connected bounded domains in R 3 , in conjunction with the product trapezoidal rule. We assume these surfaces are smooth and homeomorphic to the surface of the unit sphere. In Part I, we treat the cases in which the integrands are smooth. In the present work, we treat integrands that have point singularities of the single-layer and double-layer types on these surfaces. We propose two methods, one in which the product trapezoidal rule is applied with a standard variable transformation from S m , and another in which the trapezoidal rule is applied with a rather unconventional transformation derived from S m and achieves higher accuracy than the former. We give thorough analyses of the errors incurred by both methods, which show that surprisingly high accuracies can be achieved with suitable values of m. We also illustrate the theoretical results with numerical examples.