Gradient Flow Finite Element Discretisations with Energy-Based $hp$-Adaptivity for the Gross-Pitaevskii Equation with Angular Momentum

TL;DR

This paper introduces an energy-based $hp$-adaptive finite element method for the Gross-Pitaevskii equation with angular momentum, efficiently capturing quantum vortices and achieving exponential convergence.

Contribution

It presents a novel $hp$-adaptive strategy based on energy decay, improving the accuracy and efficiency of numerical solutions for the Gross-Pitaevskii equation with vortices.

Findings

The $hp$-adaptive method achieves exponential convergence.

Numerical results accurately compute ground states with vortices.

Energy decay guides effective adaptive refinement.

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 $hp$-adaptive strategy that is solely based on energy decay rather than a posteriori error estimators for the refinement process. Numerical results demonstrate that the $hp$-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.