A Hybrid High-Order Method for the Gross--Pitaevskii Eigenvalue Problem

TL;DR

This paper presents a hybrid high-order numerical method for accurately approximating the ground state of the nonlinear Gross--Pitaevskii eigenvalue problem, providing guaranteed lower energy bounds without post-processing.

Contribution

The paper introduces a novel hybrid high-order method that achieves optimal convergence and directly provides guaranteed lower bounds for the ground state energy.

Findings

Proves optimal convergence rates for ground state, eigenvalue, and energy approximations.

Provides guaranteed and asymptotically exact lower energy bounds.

Achieves more accurate bounds without post-processing.

Abstract

We introduce a hybrid high-order method for approximating the ground state of the nonlinear Gross--Pitaevskii eigenvalue problem. Optimal convergence rates are proved for the ground state approximation, as well as for the associated eigenvalue and energy approximations. Unlike classical conforming methods, which inherently provide upper bounds on the ground state energy, the proposed approach gives rise to guaranteed and asymptotically exact lower energy bounds. Importantly, and in contrast to previous works, they are obtained directly without the need of post-processing, leading to more accurate guaranteed lower energy bounds in practice.