2024/03/09 by Ziang Chen, Jianfeng Lu, Chen, Ziang +5 · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2403.06028
openalex publication_date 2024/03/09 · openalex created_date 2024/03/13 · openalex updated_date 2026/07/28
This paper studies the numerical approximation of the ground state of the Gross-Pitaevskii (GP) eigenvalue problem with a fully discretized Sobolev gradient flow induced by the H1 norm. For the spatial discretization, we consider the finite element method with quadrature using Pk basis on a simplicial mesh and Qk basis on a rectangular mesh. We prove the global convergence to a critical point of the discrete GP energy, and establish a local exponential convergence to the ground state under the assumption that the linearized discrete Schrödinger operator has a positive spectral gap. We also show that for the P1 finite element discretization with quadrature on an unstructured shape regular simplicial mesh, the eigengap satisfies a mesh-independent lower bound, which implies a mesh-independent local convergence rate for the proposed discrete gradient flow. Numerical experiments with discretization by high order Qk spectral element methods in two and three dimensions are provided to validate the efficiency of the proposed method.