Evaluation of Green's boundary formula and its first- and second-order derivatives in Legendre series

This work presents a numerical method for evaluating Green's boundary formula and its first- and second-order derivatives in two dimensions. Analysis includes both the normal and tangential derivatives, which contain logarithmic and strong singularities, as well as hypersingularities of order up to r(-3). The singular integrands are expanded by the Laurent series. Regularized equations are obtained by further expressing the unknown functions as a truncated Fourier-Legendre series, and applying the subtraction-addition technique. The resulting singular integrals are then evaluated analytically. The proposed regularization is conducted before any discretization. The regularized equations can be solved by directly applying standard quadrature rules over the entire integration domain. Test examples consist of the Neumann and Dirichlet problems related to an inverse elliptic cylinder, and a mixed boundary value problem related to a square cylinder. Comparing numerical results with analytical solutions demonstrates a very good accuracy, confirming the effectiveness of the proposed method. Copyright (C) 2009 John Wiley & Sons, Ltd.