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

TL;DR

The paper develops an hp-adaptive conforming finite element method to solve the stationary Gross–Pitaevskii equation in a rotating frame, where vortices appear at unknown locations. It combines a projected Sobolev gradient flow with an energy-driven hp refinement strategy, enabling simultaneous evolution and mesh adaptivity without residual-based estimators. The authors prove convergence of the discrete gradient-flow iteration and demonstrate exponential convergence of the ground-state energy with respect to degrees of freedom, including challenging cases with angular momentum that generate vortex lattices. The results validate highly accurate ground-states and energies for diverse potentials, highlighting the method’s efficiency and reliability for Bose–Einstein condensate simulations.

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.

Figures (8)

• Figure 1: Local element patches in a triangular mesh $\mathcal{T}_N$, $N\ge 0$, in two--dimensions. Left: Mesh patch $\mathcal{T}^\kappa_N$ about an element $\kappa$, which consists of $\kappa$ and its face-wise neighbours. Right: Mesh patch $\mathcal{T}^{\kappa,{\tt ref}}_{N}$ which is constructed based on isotropically refining $\kappa$ (red refinement) and on a green refinement of its neighbours.
• Figure 2: Non-linear GPE with harmonic confinement potential. (a) Computed ground-state $u_N$; (b) Comparison of the error with respect to the square root of the number of degrees of freedom; (c) Final $hp$-mesh.
• Figure 3: Non-linear GPE with optical lattice potential. (a) Computed ground-state $u_N$; (b) Comparison of the error with respect to the square root of the number of degrees of freedom; (c) Final $hp$-mesh.
• Figure 4: Non-linear GPE with a nonsymmetric potential $V$. (a) Computed ground-state $u_N$; (b) Comparison of the error with respect to the square root of the number of degrees of freedom; (c) Final $hp$-mesh.
• Figure 5: GPE with angular momentum: $\omega = 0.5$, $\beta = 10$. (a) Computed ground-state density $|u_N|^2$; (b) Comparison of the error with respect to the square root of the number of degrees of freedom; (c) Final $hp$-mesh.

Theorems & Definitions (12)

• Proposition 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof
• Corollary 2.4
• Remark 2.5
• Remark 2.6
• Theorem 2.7
• proof