Analysis of Atkinson's variable transformation for numerical integration over smooth surfaces in ℝ3

Recently, a variable transformation for integrals over smooth surfaces in R3 was introduced in a paper by Atkinson. This interesting transformation, which includes a “grading” parameter that can be fixed by the user, makes it possible to compute these integrals numerically via the product trapezoidal rule in an efficient manner. Some analysis of the approximations thus produced was provided by Atkinson, who also stated some conjectures concerning the unusually fast convergence of his quadrature formulas observed for certain values of the grading parameter. In a recent report by Atkinson and Sommariva, this analysis is continued for the case in which the integral is over the surface of a sphere and the integrand is smooth over this surface, and optimal results are given for special values of the grading parameter. In the present work, we give a complete analysis of Atkinson’s method over arbitrary smooth surfaces that are homeomorphic to the surface of the unit sphere. We obtain optimal results that explain the actual rates of convergence, and we achieve this for all values of the grading parameter.