High-Order Meshfree Surface Integration, Including Singular Integrands

We develop and test high-order methods for integration on surface point clouds. The task of integrating a function on a surface arises in a range of applications in engineering and the sciences, particularly those involving various integral methods for partial differential equations. Mesh-based methods require a curved mesh for high-order convergence, which can be difficult to reliably obtain on many surfaces, and most meshfree methods require the ability to integrate a set of functions (such as radial basis functions) exactly on the domain of interest; these integrals are generally not known in closed form on most surfaces. We describe two methods for integrating on arbitrary, piecewise-smooth surfaces with or without boundary. Our approaches do not require a particular arrangement of points or an initial triangulation of the surface, making them completely meshfree. We also show how the methods can be extended to handle singular integrals while maintaining high accuracy without changing the point density near singularities.