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We present an effective harmonic density interpolation method for thenumerical evaluation of singular and nearly singular Laplace boundary integraloperators and layer potentials in two and three spatial dimensions. The methodrelies on the use of Green's third identity and local Taylor-likeinterpolations of density functions in terms of harmonic polynomials. Theproposed technique effectively regularizes the singularities present inboundary integral operators and layer potentials, and recasts the latter interms of integrands that are bounded or even more regular, depending on theorder of the density interpolation. The resulting boundary integrals can thenbe easily, accurately, and inexpensively evaluated by means of standardquadrature rules. A variety of numerical examples demonstrate the effectivenessof the technique when used in conjunction with the classical trapezoidal rule(to integrate over smooth curves) in two-dimensions, and with a Chebyshev-typequadrature rule (to integrate over surfaces given as unions of non-overlappingquadrilateral patches) in three-dimensions.

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