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

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.