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

TL;DR

This paper presents an a priori error analysis for a mixed finite element method applied to the Gross-Pitaevskii eigenvalue problem, providing guaranteed lower bounds for the ground state energy and demonstrating optimal convergence.

Contribution

It introduces a mixed Raviart-Thomas finite element discretization that yields guaranteed lower energy bounds and optimal convergence rates, contrasting with conformal methods.

Findings

Optimal convergence rates for eigenvalue and energy approximations

Guaranteed lower bounds for the ground state energy

Numerical experiments confirm theoretical results

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.