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

Moritz Hauck, Yizhou Liang+1 more|arXiv (Cornell University)|May 27, 2024

Nonlinear Photonic Systems4 citations

TL;DR

This paper proposes a positivity-preserving finite element method for the Gross-Pitaevskii ground state using mass lumping to preserve discrete uniqueness and positivity. It proves that all non-negative discrete excited states coincide with the ground state, ensuring global convergence of gradient flow solvers and establishing optimal convergence rates via rigorous a priori error analysis.

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.

Motivation & Objective

• To develop a finite element discretization that preserves the positivity and uniqueness up to sign of the continuous Gross-Pitaevskii ground state.
• To establish discrete uniqueness by proving that any non-negative discrete excited state must coincide with the discrete ground state.
• To ensure global convergence of gradient flow solvers by leveraging discrete positivity and uniqueness.
• To perform a rigorous a priori error analysis showing optimal convergence rates for all discrete unknowns.

Proposed method

• The method employs mass lumping in a finite element discretization to transfer continuous positivity and uniqueness properties to the discrete setting.
• It uses a variational formulation of the Gross-Pitaevskii eigenvalue problem as a constrained minimization of the Gross-Pitaevskii energy functional.
• The discrete ground state is computed via gradient flow iterations, with convergence guaranteed by the proven discrete uniqueness and positivity.
• A priori error estimates are derived using elliptic regularity, nodal interpolation, and norm equivalence between discrete and continuous norms.
• The analysis relies on Riesz representation and embedding theorems to bound $ L^∞ $-norms of discrete solutions uniformly in the mesh size.
• The method ensures that all non-negative discrete solutions are the ground state, enabling identification of the limit of iterative solvers.

Experimental results

Research questions

• RQ1 Can a finite element discretization preserve the positivity and uniqueness up to sign of the continuous Gross-Pitaevskii ground state?
• RQ2 Does every non-negative discrete excited state in the finite element setting coincide with the discrete ground state?
• RQ3 Can global convergence of gradient flow solvers be guaranteed under discrete positivity and uniqueness?
• RQ4 What is the optimal convergence rate of the proposed non-standard finite element method for all unknowns?
• RQ5 How can uniform $ L^∞ $-bounds for discrete solutions be established under energy boundedness?

Key findings

• The discrete ground state is uniquely positive up to sign, and all non-negative discrete excited states coincide with it, ensuring that gradient flow limits are the ground state.
• The method achieves optimal convergence rates in all norms: $ \|u - u_h\|_{H^1} \lesssim h $, $ \|u - u_h\|_{L^2} \lesssim h^2 $, and $ \|u - u_h\|_{L^\infty} \lesssim h^{2-d/2} $ for $ d \in \{1,2,3\} $.
• Uniform $ L^\infty $-bounds for discrete solutions are established under energy boundedness, with $ \|u_h\|_{L^\infty} \lesssim 1 $ and $ \|\hat{u}_h\|_{L^\infty} \lesssim 1 $.
• The proposed method ensures global convergence of gradient flow solvers to the discrete ground state due to discrete uniqueness and positivity.
• The error analysis confirms optimal convergence for the energy, gradient, and state approximation, with bounds depending on the regularity of the exact solution.
• The method is robust under mass lumping, preserving key theoretical properties of the continuous problem in the discrete setting.