This paper outlines an energy-minimization finite-element approach to the computational modeling of equilibrium configurations for nematic liquid crystals with applied electric fields. The method targets minimization of system free energy based on the electrically augmented Frank-Oseen free energy model. The Hessian, resulting from the linearization of the first-order optimality conditions, is shown to be invertible when discretized by a mixed finite-element method under certain assumptions. This implies that the intermediate discrete linearizations are well-posed. Two numerical experiments are performed. The first compares the algorithm's solution of a classical Freedericksz transition problem to the known analytical solution and demonstrates the convergence of the algorithm to the true solution. The second experiment targets a problem with more complicated boundary conditions, simulating a nano-patterned surface. The algorithm accurately handles heterogeneous constant coefficients and efficiently resolves configurations resulting from classical and complicated boundary conditions relevant in ongoing research.