The basic mathematical results on the elliptic boundary value problem which corresponds to the equations involved in the numerical simulation of semiconductor devices are reviewed. Particularly, smoothness and structure of the solutions of the fundamental semiconductor equations are discussed. The singular perturbation approach to the numerical solution of the semiconductor equations is presented. The implications of the results obtained with the singular perturbation approach on the application of Finite Difference methods and Finite Element methods are discussed. Criteria for an optimal mesh generation strategy are given. An example shows the power of these concepts combined with modern numerical methods in comparison to classical approaches.