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

TL;DR

This paper introduces a novel Sobolev gradient method for efficiently minimizing the Gross-Pitaevskii energy with rotation, improving convergence and performance in simulating rotating Bose-Einstein condensates.

Contribution

The paper develops a new Sobolev gradient approach with a tailored inner product and a projection method for mass conservation, ensuring global convergence and enhanced numerical efficiency.

Findings

Better numerical performance at high rotation rates

Successful implementation in finite difference and finite element methods

Effective computation of vortex configurations in Bose-Einstein condensates

Abstract

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.