The simulation of a stationary fluid flow past an obstacle by means of a lattice Boltzmann method is discussed. The problem of finding appropriate boundary conditions on the boundaries of the truncated numerical domain is addressed by a method recently discussed in the literature, based on a truncated expansion of the solution. The iterative process at the heart of this method is coupled with the iteration steps of a progressive grid refinement technique that allows a rapid convergence towards a well resolved stationary state. It is shown that this combination results in a highly efficient numerical tool which can speed up the resolution process in a substantial manner.