A multigrid numerical solution algorithm has been developed for the laminar (Stokes) flow of a compressible medium in a thin film. The solver has been applied to two model problems each representative of lubrication problems in a specific way. For both problems the solutions of the Stokes equations are compared with the solutions of the Reynolds equation. The configurations of both model problems were chosen such that based on the ratio film thickness to contact length (H/L) the difference between the Reynolds and the Stokes solutions will be very small, so the geometry of the gap itself does not lead to a significant cross film dependence of the pressure. It is shown that in this situation the compressibility can still lead to a cross-film pressure dependence which is predicted by the Stokes solution and not by the Reynolds solution. The results demonstrate that limitations exist to the validity of the Reynolds equation related to the compressibility of the medium.