Finite-difference methods are relatively inefficient in the neighbourhood of boundary singularities in elliptic problems. A combination of special treatment near the singularity, based on local satisfaction of the differential equation and boundary conditions, is here matched with finite-difference formulae in the rest of the field. The method is applied to a general self-adjoint equation with either Dirichlet, Neumann or mixed conditions on parts of the boundary consisting of two straight lines meeting at the singular point. A practical problem, formerly solved by more extensive labour, illustrates the power of the method.