Numerical quadrature over smooth surfaces with boundaries

This paper describes a high order accurate method to calculate integrals over curved surfaces with boundaries. Given data locations that are arbitrarily distributed over the surface, together with some functional description of the surface and its boundary, the algorithm produces matching quadrature weights. This extends on the authors' earlier methods for integrating over the surface of a sphere and over arbitrarily shaped smooth closed surfaces by also considering domain boundaries. The core approach consists again of combining RBF-FD (radial basis function-generated finite difference) approximations for curved surface triangles, which together make up the full surface. The provided examples include both curved and flat domains. In the highly special case of equi-spaced nodes over a regular interval in 1-D, the method provides a new opportunity for improving on the classical Gregory enhancements of the trapezoidal rule.