Mixed finite elements for the Gross-Pitaevskii eigenvalue problem: a priori error analysis and guaranteed lower energy bound

TL;DR

This work introduces a lowest-order Raviart-Thomas mixed finite element discretisation for the Gross-Pitaevskii eigenvalue problem and proves an a priori error analysis with first-order convergence for the primal and dual variables and second-order convergence for the energy and eigenvalue. A key result is a guaranteed, asymptotically exact lower bound on the ground-state energy, obtained via a post-processing of the discrete energy and supported by a commuting-property framework between $\pi_h$, $\Pi_h$, and the discrete gradient $G_h$. The authors establish convergence of the discrete ground state to the continuous one, quantify error decompositions, and demonstrate second-order energy and eigenvalue accuracy under additional regularity; they also provide extensive numerical experiments validating optimal rates and the lower-bound guarantee. Collectively, the results offer a reliable, mathematically rigorous alternative to conforming discretisations for simulating Bose-Einstein condensates and related nonlinear eigenvalue problems, with practical implications for accurate energy bounds in computational quantum systems.

Abstract

We establish an a priori error analysis for the lowest-order Raviart-Thomas finite element discretisation of the nonlinear Gross-Pitaevskii eigenvalue problem. Optimal convergence rates are obtained for the primal and dual variables as well as for the eigenvalue and energy approximations. In contrast to conformal approaches, which naturally imply upper energy bounds, the proposed mixed discretisation provides a guaranteed and asymptotically exact lower bound for the ground state energy. The theoretical results are illustrated by a series of numerical experiments.

Figures (6)

• Figure 5.1: Projection of the potential onto the space of piecewise constants (left) and ground states for the harmonic potential for the parameters $\kappa = 10,100,1000$ (second to last plot).
• Figure 5.2: Error plots of the primal and dual variables (left) and of the energy and eigenvalue (right). The expected orders of convergence are indicated by black dotted lines.
• Figure 5.3: Ground state energy approximations for the harmonic potential (left) and the difference between the reference energy and the energy approximations in a double-logarithmic plot (right). The blue and red curves correspond to the discrete energy and the post-processed discrete energy, respectively.
• Figure 5.4: Disorder potential with coarse mesh used to construct the hierarchy of meshes in the background (left). Approximation of the highly localized ground state (right).
• Figure 5.5: Ground state energy approximations for the disorder potential (left) and the difference between the reference energy and the energy approximations in a double-logarithmic plot (right). The blue and red curves correspond to the discrete energy and the post-processed discrete energy, respectively.

Theorems & Definitions (29)

• Lemma 2.1: Commuting property
• proof
• Theorem 3.1: Lower bound
• proof
• Lemma 4.1: Energy error characterization
• proof
• Remark 4.2: Tilde notation
• Theorem 4.3: Plain convergence of mixed method
• proof
• Corollary 4.4: Uniform boundedness