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

摘要

We present a simple high-order singularity subtraction technique for thenumerical evaluation of Laplace boundary integral operators and layerpotentials in two-spatial dimensions. The proposed technique allows integraloperators and layer potentials to be expressed in terms of integrands ofprescribed smoothness. The resulting boundary integrals can then be easily,accurately, and inexpensively evaluated by means of standard quadrature rules.The method relies on the use of Green's third identity and a point-wiseTaylor-like expansion of the density function in terms of harmonic polynomials.Numerical examples demonstrate the effectiveness of the technique when used inconjunction with the classical trapezoidal rule. Accurate numerical simulationsof the electrical response of closely packed biological cells furtherillustrate the applicability of the proposed methodology.