We consider the generalized Ericksen model of liquid crystals, which is an energy with 8 independent ``elastic'' constants that depends on two order parameters $\mathbf{n}$ (director) and $s$ (variable degree of orientation). We will also discuss the modeling of weak anchoring conditions (both homeotropic and planar), and fully coupled electro-statics with flexo-electric and order-electric effects. In addition, we present a new finite element discretization for the energy, that can handle the degenerate elliptic part without regularization, with the following properties: it is stable and it $\Gamma$-converges to the continuous energy. Moreover, it does not require the mesh to be weakly acute (a critical assumption in previous work). A minimization scheme for computing discrete minimizers will be discussed. We will conclude with several simulations (in 2-D and 3-D), in non-trivial domains, that illustrate the effects of the different elastic constants and the electric field.