Semianalytic approximation of normal derivative of double layer potential near and at boundary of two–dimensional domain

Abstract

The normal derivatives (ND) of the double layer potential (DLP) are defined on a boundary of a domain by hyper--singular integrals. This is why, it is impossible to calculate ND DLP with a satisfactory accuracy either on the boundary or in its vicinity using traditional quadrature formulas, which allow one to calculate ND DLP with a good accuracy at a sufficient distance from the boundary. In the present paper, we obtain semi–analytical approximations of ND DLP for the two–dimensional Laplace equation, which uniformly converge with an almost cubic velocity in a closed near--boundary domain that includes the boundary. For this purpose, we use exact integration over the smooth component of the distance function near the observation point, an additive–multiplicative method for extracting a singularity, and a piecewise quadratic interpolation of slowly varying functions. We provide the results of calculating ND DLP in a closed near–boundary domain of a unit circle, which confirm the uniform almost cubic convergence of the proposed approximations.