The behavior of an electrochemical thin film under input voltage (potentiostatic) conditions is numerically investigated. Thin films are used in micro-batteries and proton-exchange-membrane fuel cells: these devices are expected to play a significant role in the next generation energy systems for use in vehicles as a replacement to combustion engines. The electrochemical investigation of thin films is a relevant topic for a wide range of applications such as hydrogels, ionic polymer metal composites, biological membranes, and treatment of tumors. In this work, a continuum-based model is presented in order to describe the behavior of thin membranes. The electrochemical behavior of thin membranes is usually hard to investigate with experiments. Therefore, numerical simulations are carried out in order to enable a better understanding of the chemical reactions occurring within microscopic regions at the electrode/electrolyte interfaces. Diffusive-migrative ionic fluxes and electric field distribution are considered. A one-dimensional domain is employed. The fully-coupled electrochemical field is given by the Poisson-Nernst-Planck equations. The model involves initial and interface/boundary conditions appropriate for an electrolytic/galvanic cell. The latter are the Stern layer conditions for polarization (or diffuse charge) effects and the Frumkin-Butler-Volmer equations for electrochemical kinetics of chemical reactions. Time-dependent numerical simulations within a finite element framework are performed using the commercial tools MATLAB and COMSOL Multiphysics. The results are consistent with the physical behavior of electrolytic cells under potentiostatic conditions. The time evolution of the main electrochemical parameters is in accordance with the imposed boundary/interface conditions. Interestingly, the ion flux and the electric field show slight asymmetries at the boundaries. Moreover, the model well predicts the behavior of systems, such as redox flow cells or rechargeable batteries, that can either run under applied voltage or applied current conditions. In fact, the field equations and the boundary conditions, presented here for electrolytic cells under applied voltage, can be applied also for galvanic cells under applied current. Equations and boundary conditions for applied voltage and applied current working conditions are presented in a compact form in order to emphasize differences and similarities.