Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D


We present a simple but effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer poten- tials in two and three spatial dimensions. The method relies on the use of Green’s third identity and local Taylor-like interpolations of density functions in terms of harmonic polynomials. The proposed technique effectively regularizes the singularities present in boundary integral opera- tors 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 bound- ary integrals can then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules. A variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule (to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type quadrature rule (to integrate over surfaces given as unions of non-overlapping quadrilateral patches) in three-dimensions.

C. Perez-Arancibia, L. M. Faria, C. Turc, Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D, submitted (arXiv preprint).