A high-order quadrature rule 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 {\it on each patch}, which allows for a natural harmonic polynomial extension in a {\it volumetric neighborhood} of the patch. 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 often 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 on 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, leading to improved efficiency and robustness. The algorithmic steps are discussed in detail. The accuracy and efficiency of the recursive reduction quadrature are illustrated via several numerical examples.

翻译：本文构建了一种高阶积分法则，用于计算三维光滑曲面上的拉普拉斯单层与双层势及其法向导数。该构造方法始于对每个曲面片上的密度函数进行调和近似，从而可在曲面片的体邻域内自然建立调和多项式延拓。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，而非通常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采用奇异性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每个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与近奇异面积分方案所需的自适应积分过程，从而提升了计算效率与鲁棒性。文中详细讨论了算法步骤，并通过若干数值算例验证了递归降维积分法的精度与效率。