On the Convergence of Sobolev Gradient Flow for the Gross–Pitaevskii Eigenvalue Problem

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 667-691, April 2024.
Abstract. We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross–Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross–Pitaevskii energy functional with respect to the [math]-metric and two other equivalent metrics on [math], including the iterate-independent [math]-metric and the iterate-dependent [math]-metric. We first prove the energy dissipation property and the global convergence to a critical point of the Gross–Pitaevskii energy for the discrete-time [math] and [math]-gradient flow. We also prove local exponential convergence of all three schemes to the ground state.