Computation of the logarithmic potential in a neighborhood of a boundary curve (Q1082772)

The development of direct economic methods for the solution of Poisson's equation in planar domains of a complex geometry leads to the necessity of computing values of the logarithmic potential in a neighborhood of a boundary curve. The specifics of the problem require that the computation of the logarithmic potential be more economic than the Marsh algorithm for Poisson's equation in a rectangle and have second order accuracy for densities of low smoothness. In this paper a method for solving this problem is presented based on a jump theorem for a normal derivative of the logarithmic potential. The method has second order accuracy for densities with bounded third derivative and is not worse than the Marsh algorithm with respect to the asymptotic number of arithmetic operations.