We apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations. The derivative matrix for a general sequence of collocation points is explicitly constructed. We explore the effect of several factors on the performance of these methods. We show that global methods cannot be interpreted in terms of local methods. For example, the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary. This difference does not depend on the density of the collocation points near the boundaries. Hence, intuition based on finite difference methods can lead to false results.