A New Sobolev Gradient Method for Direct Minimization of the Gross–Pitaevskii Energy with Rotation

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 H¹ and derive the corresponding gradient. Second, 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 is 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 L² or H¹ gradient methods, especially when high rotation rates are considered.