Gradient Flow Finite Element Discretisations with Energy-Based $hp$-Adaptivity for the Gross-Pitaevskii Equation with Angular Momentum
Abstract
This article deals with the stationary Gross-Pitaevskii non-linear eigenvalue problem in the presence of a rotating magnetic field that is used to model macroscopic quantum effects such as Bose-Einstein condensates (BECs). In this regime, the ground-state wave-function can exhibit an a priori unknown number of quantum vortices at unknown locations, which necessitates the exploitation of adaptive numerical strategies. To this end, we consider the conforming finite element method in combination with a discrete Sobolev gradient descent, which is guided by the energy-topology of the problem, to address the nonlinearity. In addition, a key novelty of this work is an -adaptive strategy that is solely based on energy decay rather than a posteriori error estimators for the refinement process. Numerical results demonstrate that the -adaptive strategy is highly efficient in terms of accuracy to compute the ground-state wave function and energy for several test problems where we observe exponential convergence.
Cite
@article{arxiv.2412.17680,
title = {Gradient Flow Finite Element Discretisations with Energy-Based $hp$-Adaptivity for the Gross-Pitaevskii Equation with Angular Momentum},
author = {Pascal Heid and Paul Houston and Benjamin Stamm and Thomas P. Wihler},
journal= {arXiv preprint arXiv:2412.17680},
year = {2025}
}