On the uniform convergence of approximations to the tangential and normal derivatives of the single-layer potential near the boundary of a two-dimensional domain

Semi-analytical approximations to the tangential derivative (TD) and normal derivative (ND) of the single-layer potential (SLP) near the boundary of a two-dimensional domain, within the framework of the collocation boundary element method and not requiring approximation of the coordinate functions of the boundary, are proposed. To obtain approximations, analytical integration over the smooth component of the distance function and a special additive-multiplicative method for separation of singularities are used. It is proved that such approximations have a more uniform convergence near the domain boundary compared to similar approximations of the TD and ND of SLP based on a simple multiplicative method of separation of singularities. One of the reasons for the highly nonuniform convergence of traditional approximations to TD and ND of SLP based on the Gaussian quadrature formulas is established.