We investigate the differenced collocation method for solving the direct boundary integral method for the mixed boundary value problem for Laplace's equation on a smooth domain. The problem in proving results for the collocation method is that the prin­ cipal part of the system of equations contains a first kind integral equation with a logarithmic kernel. The collocation method for solving such equations has only been partially justified. The major difficulty is proving the stability of the method. To analyse the collocation method we use a modified form of the collocation method, which we call the differenced collocation method. This modified collocation method permits us to prove stability results under weak assumptions on the mesh, which allows for the use of graded or adaptive meshes to overcome the degradation in numerical schemes caused by the singularities. As a by-product of our analysis we shall also prove corresponding convergence results for the collocation method. Before we consider the mixed problem we first analyse the somewhat simpler Dirichlet problem for Laplace's equation on a smooth domain. This problem still contains a first kind integral equation with a logarithmic kernel, however the solution will be smooth provided the boundary data is smooth. After we have analysed this problem we then consider the mixed boundary value problem. An adaptive boundary element procedure is also proposed that is based on a non-residual a-posteriori error estimate. This a-posteriori error estimate is constructed only from information about the jumps in the approximation. It is therefore cheap and easy to compute, offering a considerable saving over the existing residual based methods. We present numerical examples for this procedure and compare our algorithm with residual based adaptive schemes as well as with graded meshes.