Numerical studies on desingularized Cauchyʹs formula with applications to interior potential problems

Based on the Cauchyʹs formula, a pair of fully desingularized real boundary integral equations is proposed for solving interior boundary value problems in the potential theory. With Gaussian points as the collocation points of the boundary integral equation, an arbitrary high-order Gaussian quadrature can be used globally to discretize the integral equations. The numerical scheme is simple, e cient and accurate. Moreover, using Holderʹs condition of the analytic function, the discontinuities of the tangential derivatives of the analytic function across the corner point is studied in detail. A numerical treatment for using corner point as a collocation point of the Gaussian quadrature is also developed. Two examples are included to demonstrate the superiority of usage of the desingularized Cauchyʹs formula and the developed numerical scheme.