  In this paper we improve traditional steepest descent methods for the direct
minimization of the Gross-Pitaevskii (GP) energy with rotation at two levels.
We first define a new inner product to equip the Sobolev space $H^1$ and derive
the corresponding gradient. Secondly, for the treatment of the mass
conservation constraint, we use a projection method that avoids more
complicated approaches based on modified energy functionals or traditional
normalization methods. The descent method with these two new ingredients is
studied theoretically in a Hilbert space setting and we give a proof of the
global existence and convergence in the asymptotic limit to a minimizer of the
GP energy. The new method is implemented in both finite difference and finite
element two-dimensional settings and used to compute various complex
configurations with vortices of rotating Bose-Einstein condensates. The new
Sobolev gradient method shows better numerical performances compared to
classical $L^2$ or $H^1$ gradient methods, especially when high rotation rates
are considered.

Quantum Gases;Numerical Analysis