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Structure-preserving finite element methods for computing dynamics of rotating Bose-Einstein condensate

Meng Li, Junjun Wang, Zhen Guan, Zhijie Du

TL;DR

The authors address the challenge of simulating rotating Bose–Einstein condensates via the Gross–Pitaevskii equation with angular momentum rotation, focusing on structure-preserving finite element discretizations that conserve mass and energy even for nonconforming spaces. They develop Crank–Nicolson time discretizations and conforming/nonconforming fully discrete schemes, augmented by a stabilization technique for nonconforming elements, and provide an unconditional convergence analysis yielding optimal $L^2$ and $H^1$ error estimates along with high-order $H^1$ convergence. Theoretical results are complemented by extensive numerical tests that verify accuracy, mass/energy conservation, and vortex-lattice dynamics under varying trap frequencies, demonstrating the robustness and applicability of the methods. The work advances conservative numerical analysis for rotating quantum systems, enabling accurate simulations in rough potentials and complex geometries with adaptive mesh capabilities. Overall, the proposed structure-preserving schemes offer reliable, high-fidelity simulations of rotating BEC dynamics with rigorous proofs of stability and convergence.

Abstract

This work is concerned with the construction and analysis of structure-preserving Galerkin methods for computing the dynamics of rotating Bose-Einstein condensate (BEC) based on the Gross-Pitaevskii equation with angular momentum rotation. Due to the presence of the rotation term, constructing finite element methods (FEMs) that preserve both mass and energy remains an unresolved issue, particularly in the context of nonconforming FEMs. Furthermore, in comparison to existing works, we provide a comprehensive convergence analysis, offering a thorough demonstration of the methods' optimal and high-order convergence properties. Finally, extensive numerical results are presented to check the theoretical analysis of the structure-preserving numerical method for rotating BEC, and the quantized vortex lattice's behavior is scrutinized through a series of numerical tests.

Structure-preserving finite element methods for computing dynamics of rotating Bose-Einstein condensate

TL;DR

The authors address the challenge of simulating rotating Bose–Einstein condensates via the Gross–Pitaevskii equation with angular momentum rotation, focusing on structure-preserving finite element discretizations that conserve mass and energy even for nonconforming spaces. They develop Crank–Nicolson time discretizations and conforming/nonconforming fully discrete schemes, augmented by a stabilization technique for nonconforming elements, and provide an unconditional convergence analysis yielding optimal and error estimates along with high-order convergence. Theoretical results are complemented by extensive numerical tests that verify accuracy, mass/energy conservation, and vortex-lattice dynamics under varying trap frequencies, demonstrating the robustness and applicability of the methods. The work advances conservative numerical analysis for rotating quantum systems, enabling accurate simulations in rough potentials and complex geometries with adaptive mesh capabilities. Overall, the proposed structure-preserving schemes offer reliable, high-fidelity simulations of rotating BEC dynamics with rigorous proofs of stability and convergence.

Abstract

This work is concerned with the construction and analysis of structure-preserving Galerkin methods for computing the dynamics of rotating Bose-Einstein condensate (BEC) based on the Gross-Pitaevskii equation with angular momentum rotation. Due to the presence of the rotation term, constructing finite element methods (FEMs) that preserve both mass and energy remains an unresolved issue, particularly in the context of nonconforming FEMs. Furthermore, in comparison to existing works, we provide a comprehensive convergence analysis, offering a thorough demonstration of the methods' optimal and high-order convergence properties. Finally, extensive numerical results are presented to check the theoretical analysis of the structure-preserving numerical method for rotating BEC, and the quantized vortex lattice's behavior is scrutinized through a series of numerical tests.
Paper Structure (20 sections, 12 theorems, 153 equations, 6 figures)

This paper contains 20 sections, 12 theorems, 153 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Discretized schemes
  3. Time-discrete method
  4. Fully discrete method
  5. Main results
  6. Error analysis for the FEM
  7. Error analysis for the time-discrete method
  8. A truncated auxiliary problem
  9. Existence of truncated time-discrete method
  10. Uniform $L^\infty$-boundedness of the truncated approximation
  11. Convergence and boundedness of the time-discrete method
  12. Spatial error analysis
  13. Fully discrete method with truncation
  14. Proof of $L^2$-norm convergence
  15. Proof of $H^1$-norm high-order convergence
  16. ...and 5 more sections

Key Result

Theorem 2.1

The time-discrete system eqn:timediscrete is conservative in the senses of the total mass and energy:

Figures (6)

  • Figure 1: Figures of the functions $\hat{\mu}_A(r)$, $d\hat{\mu}_A(r)/dr$ and $d^2\hat{\mu}_A(r)/dr^2$
  • Figure 2: The error estimates and convergence rates for the conforming (left) and nonconforming (right) FEMs.
  • Figure 3: The discrete mass and its relative error for the conforming FEM.
  • Figure 4: The discrete mass and its relative error for the nonconforming FEM.
  • Figure 5: The discrete energy and its relative error for the conforming FEM.
  • ...and 1 more figures

Theorems & Definitions (33)

  • Definition 2.1
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 23 more