In this work, a finite element simulation of a commercial thermoelectric cell, working as a cooling heat pump, is presented. The finite element is three-dimensional, non-linear in its formulation (using quadratic temperature-dependence on material properties) and fully coupled, including the Seebeck, Peltier, Thomson and Joule effects. Another special interface finite element is developed to prescribe the electric intensity, taking advantage of repetitions and symmetries. A thorough study of the distributions of voltage, temperature and the corresponding fluxes is presented, and the performance of the cell is compared with those of the manufacturer and simplified analytical formulations, showing a good agreement with the former. Combining the finite element model with the Monte Carlo technique, a Sensitivity Analysis is presented to take into account the performance variables dependence on the material properties, geometrical parameters and prescribed values. This analysis, which can be considered a first step to optimize these devices, concludes that the temperature-dependence of the material properties such as electric conductivity and Seebeck coefficient is very relevant on the cell performance.