A Harmonic Density Interpolation (HDI) method for high-order evaluation of Laplace layer potentials in 3D

This project is a Matlab implementation of a HDI method for high-order evaluation of Laplace layer potential and integral operators in three-spatial dimensions. The HDI method is a simple but effective numerical procedure for the numerical evaluation of singular and nearly singular boundary integrals. The method relies on the use of Green?s third identity and local Taylor interpolations of density functions in terms of harmonic polynomials. This procedure effectively regularizes the singularities present in boundary integral operators and layer potentials and recasts the latter in terms of integrands that are continuous or even more regular, depending on the order of singularity subtraction. The resulting boundary integrals can then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules. In this particular implementation we utilize Fejér first quadrature rule to integrate over surfaces given as unions of non-overlapping quadrilateral patches in three-dimensions.