On the convergence of Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem

2023/01/24 by Chen, Ziang, Lu, Jianfeng, Lu, Yulong +1 · 7 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2301.09818

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 H¹₀-metric and two other equivalent metrics on H₀¹, including the iterate-independent a₀-metric and the iterate-dependent a_u-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 H¹ and a₀-gradient flow. We also prove local exponential convergence of all three schemes to the ground state.

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