Let \(\Omega \subset \mathbb{R}^ 2\) be a bounded and simply-connected polygonal domain. The author considers the numerical solution of the nonlinear boundary value problem \(\Delta u(P) = 0\) for \(P \in \Omega\) with \((\partial u/\partial n_ p)(P) = -g(P,u(P)) + f(P)\) for \(P \in \partial\Omega\) where \(n_ p\) denotes the exterior unit normal. Using Green's representation formula for harmonic functions the boundary value problem is transformed into a nonlinear integral equation for \(u(P)\) with \(P \in \partial\Omega\). The existence of a solution of the nonlinear boundary value problem is assumed. The collocation method is formulated. An existence and uniqueness theorem for the collocation solution is given and the convergence is proved. Superconvergence at the collocation points can be achieved. Numerical examples are given.