A novel proof of natural boundary conditions for the Poisson equation

The finite-element formulation of Poisson´s equation has implicit natural boundary conditions that are not immediately obvious through introductory-course teaching methods. To prove the existence of these conditions, it is necessary to resort to more elaborate Galerkin techniques. In practice, to avoid going into the complex proofs of the Galerkin methods, introductory courses in numerical methods for electrical engineers gloss over the boundary conditions. The author presents a self-contained proof for the natural boundary conditions of the variational expression for elliptical field problems without recourse to the theory of weighted residuals. This allows educators to treat natural boundary conditions with greater rigor and thereby impart a better understanding of them in early courses in finite elements