A numerical model that employs the finite-element method and a fully-coupled implicit solution scheme via Newton's technique is presented for simulating the performance of polymer-electrolyte-membrane (PEM) fuel cells. With our model, solved are the multi-dimensional momentum, mass & species, and charge conservation equations that govern, respectively, pressure-gradient driven flows along the gas flow channels (GFCs) and within the gas diffusion layers (GDLs), species transport along GFCs and within GDLs, and proton and water transport within the membrane as well as the Butler-Volmer constitutive equations describing the hydrogen oxidation reaction (HOR) and oxygen reduction reaction (ORR). For simplicity, the present version of our model considers PEM fuel cell operation as isothermal and water present as vapor, and treats the anode and cathode catalyst layers as respective interfaces at which HOR and ORR take place. With our numerical approach, all governing equations are solved simultaneously and quadratic convergence is ensured due to the use of Newton's method with an analytical Jacobian. To demonstrate the utility of our computational approach, computed predictions of velocity field, contours of hydrodynamic pressure and molar concentrations of hydrogen, oxygen and water species, and current distribution and polarization (or I-V) curves from a two-dimensional case study of a simplified PEM fuel cell are presented. To help assess the validity of our PEM fuel cell model, measurements of current distribution and polarization curves were performed using a segmented PEM fuel cell, and the resultant experimental data as well as that from the literature are compared with computed predictions.