We present a velocity-oriented discrete analog of the partial differential equations governing porous media flow: the edge-based face element method. Conventional finite element techniques calculate pressures in the nodes of the grid. However, such methods do not satisfy the requirement of flux continuity at the faces. In contrast, the edge-based method calculates vector potentials along the edges, leading to continuity of fluxes. The method is algebraically equivalent with the popular block-centered finite difference method and with the mixed-hybrid finite element method, but is algorithmically different and has the same robustness as the more conventional node-based velocity-oriented method. The numerical examples are computed analytically and may, therefore, be considered as an 'heuristic proof' of the theory and its practical applicability for reservoir engineering and geohydrology.