  For a Bose-Einstein condensate placed in a rotating trap and confined in the
z axis, we set a framework of study for the Gross-Pitaevskii energy in the
Thomas Fermi regime. We investigate an asymptotic development of the energy,
the critical velocities of nucleation of vortices with respect to a small
parameter $\ep$ and the location of vortices. The limit $\ep$ going to zero
corresponds to the Thomas Fermi regime. The non-dimensionalized energy is
similar to the Ginzburg-Landau energy for superconductors in the high-kappa
high-field limit and our estimates rely on techniques developed for this latter
problem. We also take the advantage of this similarity to develop a numerical
algorithm for computing the Bose-Einstein vortices. Numerical results and
energy diagrams are presented.
