The two-dimensional, confined, laminar flow over a downstream facing step was studied using a finite element, Navier-Stokes equation solver. The weak form of the stationary, incompressible Navier-Stokes equations in primitive variable form was obtained using the conventional Galerkin technique for mixed problems. Biquadratic Lagrange interpolating polynomials were used to construct the basis functions that generated the finite-dimensional subspace containing the approximate solutions to the velocity field, while the pressure field was represented by a discontinuous, piecewise-linear approximation. This particular combination of solution subspaces was previously shown in a mathematically rigorous fashion to yield stable, consistent solutions to the Navier-Stokes equations. The results of the computations were benchmarked against the experimental data of Denham and Patrick, and also compared to earlier calculations by Ecer and Thomas, both of whom utilized alternative, unconventional formulations. These comparisons indicate that with the proper choice of basis functions, a conventional Galerkin scheme can yield results that are in as good and in many cases better agreement with the available experimental data than those of unconventional schemes that rely upon an infusion of artificial dissipation to enhance their numerical stability. The computational algorithm was also used to ascertain the cause of the noticeable lack of development and skewness that characterized the experimental data of Denham and Patrick both at and upstream of the step. The results of this study indicated that as suspected by Denham and Patrick, the skewness as well as the lack of development of the velocity profiles near the step were caused by the geometry of the test apparatus upstream of the step rather than by the presence of the step itself. The numerical experiments conducted here have been carefully documented so as to facilitate future comparisons intended to assess the relative efficiency of the present method of computation. / Doctor of Philosophy