Positivity preserving finite element method for the Gross-Pitaevskii ground state: discrete uniqueness and global convergence

TL;DR

This paper develops a mass-lumped linear finite element discretization for the Gross-Pitaevskii ground state that preserves key continuous properties, including positivity and uniqueness up to sign. It proves that the discrete ground state is the smallest eigenfunction of the nonlinear discrete problem via a convex reformulation and a discrete Picone inequality, enabling global convergence results for fully discrete Sobolev gradient flows. An a priori error analysis yields optimal convergence rates: $\|u-u_h\|_{H^1}\lesssim h$, $\|u-u_h\|_{L^2}\lesssim h^2$, and $|E-E_h|,|\lambda-\lambda_h|\lesssim h^2$, with uniform $L^{\infty}$ bounds. Numerical experiments on harmonic and disorder potentials confirm the theoretical predictions and illustrate practical benefits, such as diagonal complexity in nonlinear iterations and robust positivity.

Abstract

We propose a positivity preserving finite element discretization for the nonlinear Gross-Pitaevskii eigenvalue problem. The method employs mass lumping techniques, which allow to transfer the uniqueness up to sign and positivity properties of the continuous ground state to the discrete setting. We further prove that every non-negative discrete excited state up to sign coincides with the discrete ground state. This allows one to identify the limit of fully discretized gradient flows, which are typically used to compute the discrete ground state, and thereby establish their global convergence. Furthermore, we perform a rigorous a priori error analysis of the proposed non-standard finite element discretization, showing optimal orders of convergence for all unknowns. Numerical experiments illustrate the theoretical results of this paper.

Figures (5)

• Figure 6.1: Illustration of the harmonic potential on the left and and a discrete ground state approximation on the right.
• Figure 6.2: Error plots for the proposed method and the standard linear FEM for the harmonic potential. The relative $L^2$-approximation errors of the ground state and its gradient are shown on the left. On the right, the relative energy and eigenvalue approximation errors are shown.
• Figure 6.3: Energy and eigenvalue approximations shown on the left and right, respectively, computed using the proposed method and the standard linear FEM.
• Figure 6.4: Illustration of the disorder potential on the left and and a discrete ground state approximation on the right.
• Figure 7.1: Error plots of the proposed method and the standard linear FEM for the disorder potential. The relative $L^2$-approximation errors of the ground state and its gradient are shown on the left. On the right, the relative energy and eigenvalue approximation errors are shown.

Theorems & Definitions (15)

• Theorem 3.1: Uniqueness and positivity of discrete ground state
• proof
• Theorem 3.2: Discrete ground state eigenvalue
• proof
• Theorem 4.1: Non-negative discrete eigenstates
• proof
• Corollary 4.2: Global convergence to discrete ground state
• proof
• Remark 5.1: Tilde notation
• Theorem 5.2: A priori error analysis