On approximation of the normal derivative of the single layer heat potential near the boundary of a two-dimensional domain

On the basis of piecewise-quadratic interpolation, semi-analytical approximations of the normal derivative of the thermal potential of a simple layer are obtained, which converge with a cubic velocity uniformly near the boundary of a two-dimensional spatial region. With some simplifications, it is proved that the use of a number of standard quadrature formulas leads to a violation of the uniform convergence of approximations of the normal derivative near the boundary of the domain. The theoretical conclusions are confirmed by the results of calculating the normal derivative of the solution of the second boundary value problem of heat conduction in a circular region.