High-Order Numerical Integration over Discrete Surfaces

We consider the problem of numerical integration of a function over a discrete surface to high-order accuracy. Surface integration is a common operation in numerical computations for scientific and engineering problems. Integration over discrete surfaces (such as a surface triangulation) is typically limited to only first or second-order accuracy due to the piecewise linear approximations of the surface and the function. We present a novel method that can achieve third and higher order accuracy for integration over discrete surfaces. Our method combines a stabilized least squares approximation, a blending procedure based on linear shape functions, and high-degree quadrature rules. We present theoretical analysis of the accuracy of our method as well as experimental results of up to sixth order accuracy with our method. 1. Introduction. Surface integration is a fundamental operation for many scientific and engineering problems. It is a core procedure for a variety of numerical methods such as the boundary integral method, finite element method, surface finite elements, integral transforms, finite volume method etc. For example, in the boundary integral method the solution is obtained by solving an integral equation, which in turn is solved by forming a linear system using collocation methods. The entries in the linear system are surface integrals. In geometric processing, computing the surface area and solid volume are fundamental primitives, both of which require surface integration. In computational fluid dynamics, the computation of the flux across a curved interface between different materials requires surface integrals. Surface finite element method, which is a special case of the finite element method for solving partial differential equations on surfaces, also requires surface integrals. The application of surface integration is also found in fluid-structure interactions, where the integral of the pressure over the structure is the force applied by the fluid to the structure. Surface integrals also appear in surface, interface and colloidal sciences as well as in the semiconductor industry and for pharmaceutical manufacturing. They are helpful in understanding various processes such as adhesion and fracture processes as well as in the manipulation of nanoscale objects. For these applications, the standard method for surface integration is to integrate over the individual triangles, but its accuracy is limited by the piecewise linear approximations to the geometry and the function. The relative error is reduced by either global or adaptive mesh refinement until a set tolerance is reached. A natural but yet fundamental question is whether we can …