Axially symmetric problems (e.g. Laplace's equation in cylindrical co-ordinates) containing discs possess boundary singularities arising from the mixed boundary conditions that occur across the disc edge. A modified finite-difference method is presented which effectively eliminates the inaccuracies that occur in the standard numerical solution near such singularities. Techniques for developing the analytical forms of such singularities are given and modified finite-difference approximations are obtained. The steady-state diffusion of oxygen around a circular electrode is taken as the model problem and a modified quadrature method is presented for the calculation of the oxygen flux through the electrode
