Abstract
A practical finite-temperature theory is developed for the superfluid regime of a weakly interacting Bose gas in an optical lattice with additional harmonic confinement. We derive an extended Bose-Hubbard model that is valid for shallow lattices and when excited bands are occupied. Using the Hartree-Fock-Bogoliubov-Popov mean-field approach, and applying local-density and coarse-grained envelope approximations, we arrive at a theory that can be numerically implemented accurately and efficiently. We present results for a three-imensional system, characterizing the importance of the features of the extended Bose-Hubbard model and compare against other theoretical results and show an improved agreement with experimental data.