A high-order quadrature scheme is constructed for the evaluation of Laplace single and double layer potentials and their normal derivatives on smooth surfaces in three dimensions. The construction begins with a harmonic approximation of the density on each patch, which allows for a natural harmonic polynomial extension in a volumetric neighborhood of the patch in the ambient space. Then by the general Stokes theorem, singular and nearly singular surface integrals are reduced to line integrals preserving the singularity of the kernel, instead of the standard origin-centered 1-forms that require expensive adaptive integration. These singularity-preserving line integrals can be semi-analytically evaluated using singularity-swap quadrature. In other words, the evaluation of singular and nearly singular surface integrals is reduced to function evaluations at the vertices on the boundary of each patch. The recursive reduction quadrature largely removes adaptive integration that is needed in most existing high-order quadratures for singular and nearly singular surface integrals, resulting in exceptional performance. The scheme achieves twelve-digit accuracy uniformly for close evaluations and offers a speedup of five times or more in constructing the sparse quadrature-correction matrix compared to previous state-of-the-art quadrature schemes.

翻译：本文构建了一种高阶积分方案，用于评估三维光滑曲面上的拉普拉斯单层与双层势函数及其法向导数。该方案首先在每个曲面片上对密度函数进行调和逼近，从而在环境空间中曲面片的体邻域内自然构造调和多项式延拓。随后通过广义斯托克斯定理，将奇异及近奇异面积分转化为保持核函数奇异性的线积分，而非传统上需要昂贵自适应积分的以原点为中心的一次微分形式。这些保持奇异性的线积分可通过奇异点交换积分法进行半解析计算。换言之，奇异及近奇异面积分的计算被简化为对各曲面片边界顶点处的函数求值。递归降维积分法基本消除了现有大多数高阶奇异/近奇异面积分方案所需的自适应积分过程，从而获得卓越的计算性能。该方案在近距离评估中能保持十二位数字的均匀精度，且在构建稀疏积分校正矩阵时，相比现有最先进的积分方案可获得五倍以上的加速效果。