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 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.