Geometry of the 3D Schroedinger problem and comparison with Finite Elements discretization

The numerical modeling of nanoscale electron devices needs the development of accurate and efficient numerical methods, in particular, for the numerical solution of the Schr\uf6dinger problem. If FEMs allow an accurate geometric representation of the device, they lead to a discrete counterpart of Schr\uf6dinger problem in terms of a computationally heavy generalized eigenvalue problem. Exploiting the geometric structure behind the Schr\uf6dinger problem, we will construct a numerically efficient discrete counterpart of it, yielding to a standard eigenvalue problem. We will also show how the two approaches are only partially akin to each other even when lumping is applied