Adaptive integration over three dimensional smooth convex sets

This work involves developing a method to numerically integrate smooth functions over a smooth, bounded, convex set in three dimensional Euclidean space. The integrand is assumed to have continuous sixth partial derivatives, and the boundary of the set is assumed to be a surface with bounded, continuously differentiable curvatures.The scheme consists of using an orthogonal mesh of width h to decompose the region into some standard group of geometrical subregions for which numerical integration formulae are developed. Error formulae for these standard subregions are derived by approximating the integrand with its local Taylor polynomial approximation.The method is adaptive in that only those subregions with insufficiently small error are further subdivided. The global error is shown to be of the fourth order in h, ensuring that subsequent refinements will result in a satisfactory total error. The scheme is illustrated with examples run on a programme written in the language UBASIC