The penalty‐Galerkin finite‐element method is used to simulate the flow of a polystyrene melt over a rectangular slot placed perpendicular to the flow direction. The White‐Metzner constitutive equation is used with a Carreau model viscosity function and a shear rate‐dependent relaxation time defined so that the primary normal stress difference is exactly reproduced by the model in simple shear flow. Values of the stress field predicted by the simulation are compared with those obtained experimentally by means of flow birefringence. As observed by others, the limiting elasticity value as determined by the Weissenberg number (We) for convergence of the algorithm decreased with increased refinement of the mesh. However, good agreement is still found between predicted values of stress using the coarse mesh and those measured by means of flow birefringence. This work suggests that there may be an optimum mesh for a given flow and constitutive equation which will still give physically realistic results. The Weissenberg number for the melt used in the experimental study asymptotically approached a value of about 1.5 with increasing shear stress, suggesting that it may not be necessary to reach excessively high values of We for simulations involving some polymer melts.