In this paper we propose a highly efficient and stable lattice Boltzmann method for solving low Reynolds number exterior flows using a preconditioning technique. The present method is based on replacing the constant preconditioning parameter (gamma) within uniform grids [Guo, Phys. Rev. E 70, 066706 (2004)] by a space- and time-dependent one in a nested mesh-refined domain. To do this, for the transition from a fine to the neighboring coarser grid, gamma has been divided by a factor K , which is large initially and anneals stepwise to a small value after some iterations. With this technique, more than one order of magnitude larger convergence rate can be achieved, and several orders of magnitude larger system size can be treated.