On uniform convergence of approximations of the double layer potential near the boundary of a two-dimensional domain

On the basis of piecewise quadratic interpolation, semi-analytical approximations of the double layer potential near and on the boundary of a two-dimensional domain are obtained. To calculate the integrals formed after the interpolation of the density function, exact integration with respect to the variable $\rho=\left(r^2-d^2\right)^{1/2}$ is used, where $d$ and $r$ are the distances from the observed point to the boundary of the domain and to the boundary point of integration, respectively. The study proves the stable convergence of such approximations with the cubic velocity uniformly near the boundary of the class $C^5$, and also on the boundary itself. It is also proved that the use of standard quadrature formulas for calculating the integrals does not violate the uniform cubic convergence of approximations of the direct value of the potential on the boundary of the class $C^6$. With some simplifications, it is proved that the use of standard quadrature formulas for calculating the integrals entails the absence of uniform convergence of potential approximations inside the domain near any boundary point. The theoretical conclusions are confirmed by the results of the numerical solution of the Dirichlet problem for the Laplace equation in a circular domain.