- A finite element analysis based on the penalty function method is presented for the solution of the Navier-Stokes equations and the energy equation governing steady laminar motion of a viscous incompressible fluid. Numerical results are presented for the two-dimensional window cavity (R//A equals 10**3, 10**4, 10**5, 10**6) and three-dimensional window cavity (R//A equals 10**3 and 10**4) problems and a concentric cylindrical annulus problem. From the three-dimensional simulation of the flows, it is concluded that the fixed wall in the three-dimensional cavity has the (kinematic as well as thermal) effect of reducing the strength of the flow field. Due to the lack of quantitative results on the three-dimensional window cavity problem in the open literature, comparison of the results to assess the quality is not possible. However, it is noted that the global behaviour of solutions predicted by the penalty finite element model is very accurate despite the use of coarse meshes. In order to analyse higher Rayleigh number flows, mesh refinements must be made. Toward this end it is more practical to use out-of-core solution methods to meet the storage demands.