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Fully discretized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem

Ziang Chen, Jianfeng Lu, Yulong Lu, Xiangxiong Zhang

TL;DR

The paper develops a fully discretized Sobolev gradient flow in the $H^1$-norm to compute the ground state of the Gross–Pitaevskii eigenvalue problem. By combining finite element discretization with quadrature (including $P^1$ on simplicial meshes and high-order $Q^k$ spectral elements), it proves global convergence to a discrete critical point and local exponential convergence to the ground state under a positive discrete eigengap, with a mesh-independent lower bound established for $P^1$ on unstructured meshes. The proposed modified $H^1$ scheme leverages an efficient inverse of a constant-coefficient operator, enabling GPU-accelerated, large-scale 2D/3D computations that validate high-order accuracy and scalability. Together, these results provide a rigorous and practical framework for accurate GP ground-state computations on modern architectures, with implications for Bose–Einstein condensate simulations and nonlinear eigenvalue problems.

Abstract

This paper studies the numerical approximation of the ground state of the Gross-Pitaevskii (GP) eigenvalue problem with a fully discretized Sobolev gradient flow induced by the $H^1$ norm. For the spatial discretization, we consider the finite element method with quadrature using $P^k$ basis on a simplicial mesh and $Q^k$ basis on a rectangular mesh. We prove the global convergence to a critical point of the discrete GP energy, and establish a local exponential convergence to the ground state under the assumption that the linearized discrete Schrödinger operator has a positive spectral gap. We also show that for the $P^1$ finite element discretization with quadrature on an unstructured shape regular simplicial mesh, the eigengap satisfies a mesh-independent lower bound, which implies a mesh-independent local convergence rate for the proposed discrete gradient flow. Numerical experiments with discretization by high order $Q^k$ spectral element methods in two and three dimensions are provided to validate the efficiency of the proposed method.

Fully discretized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem

TL;DR

The paper develops a fully discretized Sobolev gradient flow in the -norm to compute the ground state of the Gross–Pitaevskii eigenvalue problem. By combining finite element discretization with quadrature (including on simplicial meshes and high-order spectral elements), it proves global convergence to a discrete critical point and local exponential convergence to the ground state under a positive discrete eigengap, with a mesh-independent lower bound established for on unstructured meshes. The proposed modified scheme leverages an efficient inverse of a constant-coefficient operator, enabling GPU-accelerated, large-scale 2D/3D computations that validate high-order accuracy and scalability. Together, these results provide a rigorous and practical framework for accurate GP ground-state computations on modern architectures, with implications for Bose–Einstein condensate simulations and nonlinear eigenvalue problems.

Abstract

This paper studies the numerical approximation of the ground state of the Gross-Pitaevskii (GP) eigenvalue problem with a fully discretized Sobolev gradient flow induced by the norm. For the spatial discretization, we consider the finite element method with quadrature using basis on a simplicial mesh and basis on a rectangular mesh. We prove the global convergence to a critical point of the discrete GP energy, and establish a local exponential convergence to the ground state under the assumption that the linearized discrete Schrödinger operator has a positive spectral gap. We also show that for the finite element discretization with quadrature on an unstructured shape regular simplicial mesh, the eigengap satisfies a mesh-independent lower bound, which implies a mesh-independent local convergence rate for the proposed discrete gradient flow. Numerical experiments with discretization by high order spectral element methods in two and three dimensions are provided to validate the efficiency of the proposed method.
Paper Structure (42 sections, 26 theorems, 132 equations, 6 figures, 3 tables)

This paper contains 42 sections, 26 theorems, 132 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Gross-Pitaevskii eigenvalue problem
  3. Related work
  4. Contribution of the present work
  5. Organization of the rest of the paper
  6. The classical finite element method with quadrature
  7. Finite element Galerkin method
  8. Finite element method with quadrature
  9. The matrix-vector form
  10. The discrete energy and discrete $L^2$ norm
  11. Properties of the discrete energy
  12. The monotonicity of the discrete Laplacian
  13. Ground state of the discrete energy
  14. fully discretized Sobolev gradient descent and modified $H^1$ scheme
  15. The Sobolev gradient descent
  16. ...and 27 more sections

Key Result

Theorem 3.1

For the $P^1$ finite element scheme with quadrature on a simplicial mesh satisfying simpicialmesh, which includes the classical second order finite difference scheme, $A_{\mathbf u}=-\Delta_h+\mathbb V+\beta \mathop{\mathrm{diag}}\nolimits(\mathbf u^2)$ is an M-matrix thus monotone. As a result, it

Figures (6)

  • Figure 1: A 2D example with $\beta=10$ of second order finite difference on a $800\times 800$ grid. The modified $H^1$ norm has parameter $\alpha=0.15$ in \ref{['modified-H1-scheme']}. The $H^1$ seminorm scheme is \ref{['modified-H1-scheme']} with $\alpha=0$. The CPU time for the $H^1$ scheme with $\alpha=0.15$ and the fixed step size $1$ to converge is 3 seconds, and the CPU time for the $H^1$ scheme with $\alpha=0.15$ or $\alpha=0$ with optimal step size to converge is more than 6 seconds. For $L^2$, $A_u$, and $A_0$ schemes (see henning2020sobolev for definition), preconditioned conjugate gradient (PCG) is used for inverting a matrix like $-\Delta_h+V(\boldsymbol{x})$ and $(-\Delta_h)^{-1}$ is used as a preconditioner. The PCG converges within 30 iterations for all linear systems in this test.
  • Figure 2: A 2D example with $\beta=5$ of second order discrete Laplacian on a $300\times 300$ grid. The modified $H^1$ norm has parameter $\alpha=0.15$ and step size $1$ in \ref{['modifiednorm']}. The Backward Forward Euler with stabilization \ref{['BFSP']} uses the optimal parameters $\alpha$ in bao2006efficient and the largest stable step size $0.1$, which is also the most efficient step size. The initial condition is the ground state for $\beta=0$.
  • Figure 3: The performance of the modified $H^1$ scheme \ref{['modified-H1-scheme']} solving 3D Gross-Pitaevskii nonlinear eigenvalue problem with potential \ref{['3d-potential-gpu']}. The left shows that the performance is independent of discretization and grid size. Thus parameters can be tuned on a coarse grid as shown in the right.
  • Figure 4: The potential function \ref{['3d-potential-gpu']} and its ground state.
  • Figure 5: A 3D example for a combined harmonic and optical lattice potential. Left is the isosurface of the ground state for isovalue $0.002$, and right is the slice view of the ground state. For $\beta=1600$, using $Q^{40}$ spectral element method on a $5^3$ mesh, \ref{['modified-H1-scheme']} with $\alpha=10$ and $\tau=0.1$ and $\mathbf u^0\equiv 1$ converges after 665 iterations. The online computation time is 6 seconds on Nvidia A100. $E(\mathbf u^*_h)=33.80227900547$ and $\lambda_h^*=80.89511440602$, consistent with the results in bao2006efficient.
  • ...and 1 more figures

Theorems & Definitions (53)

  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Theorem 5.2
  • Corollary 5.3
  • ...and 43 more