Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensions

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.