High-accuracy quadrature methods for solving nonlinear boundary integral equations of axisymmetric Laplace’s equation

Abstract

In this article, we study the numerical solutions of axisymmetric Laplace’s equation with nonlinear boundary conditions. The problem could be converted into nonlinear boundary integral equations by the ring potential theory. The mechanical quadrature method is presented for solving the equations, which possesses high-accuracy order of O(h3) and low computing complexities. Using the asymptotical compact theory and Stepleman theorem, an asymptotic expansion of the errors with odd powers is obtained. Moreover, the Richardson’s extrapolation algorithm is used to improve the accuracy order of O(h5). The efficiency of the algorithm is illustrated by numerical examples.