Pré-Publication, Document De Travail Année : 2009

A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation

Ionut Danaila
Parimah Kazemi
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Résumé

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.

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Est une version de hal-02011560 Article Ionut Danaila, Parimah Kazemi. A New Sobolev Gradient Method for Direct Minimization of the Gross–Pitaevskii Energy with Rotation. SIAM Journal on Scientific Computing, 2010, 32 (5), pp.2447-2467. ⟨hal-02011560⟩

publié dans SISC

Dates et versions

hal-00432061 , version 1 (13-11-2009)
hal-00432061 , version 2 (26-05-2010)

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Ionut Danaila, Parimah Kazemi. A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation. 2009. ⟨hal-00432061v2⟩
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