Numerical integration over smooth surfaces in via class variable transformations. Part I: Smooth integrands

Abstract

Class Sm 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. For example, if the integrand function is smooth on the interval of integration including the endpoints, and vanishes at the endpoints, then excellent results are obtained by taking 2m to be an odd integer. In the present work, we consider 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, and we treat the cases in which the integrands are smooth. We propose two approaches, one in which the product trapezoidal rule is applied with the integrand as is, and another, in which the integrand is preprocessed before the rule is applied. We give thorough analyses of the errors incurred in both approaches, which show that surprisingly high accuracies can be achieved with suitable values of m. We also illustrate the theoretical results with numerical examples.