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

TL;DR

The paper introduces a harmonic density interpolation technique that regularizes singular boundary integral operators for Laplace potentials, enabling accurate and efficient numerical evaluation in 2D and 3D.

Contribution

It develops a novel harmonic density interpolation method that improves the regularization and numerical evaluation of Laplace layer potentials in multiple dimensions.

Findings

Effective regularization of singularities in boundary integral operators.

Accurate evaluation of layer potentials using standard quadrature rules.

Demonstrated success in 2D and 3D numerical examples.

Abstract

We present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials 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 operators and layer potentials, and recasts the latter in terms of integrands that are bounded or even more regular, depending on the order of the density interpolation. The resulting boundary 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.