Mechanical quadrature methods and their splitting extrapolations for solving boundary integral equations of axisymmetric Laplace mixed boundary value problems

The mechanical quadrature methods (MQM) and splitting extrapolation methods (SEM) are applied to the boundary integral equations (BIE) of axisymmetric mixed boundary value problems governed by Laplace's equation. By ring potential theory, the double integral equations of axisymmetric Laplace problems can be converted into the single integral equations. For solving the BIE, the MQM based on the quadrature rules for computing the singular periodic functions are presented, which possesses a high order accuracy O(h03) and low computing complexities. Moreover, using the SEM based on the multi-parameter asymptotic error expansion, we cannot only improve the accuracy order of approximation, but also give a posteriori error estimate. Several numerical examples show that the accuracy order of approximation is very high, and the SEM and a posteriori error estimate are also very effective.