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\n 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\n \n \n \n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n H^1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n norm. For the spatial discretization, we consider the finite element method with quadrature using\n \n \n \n \n P<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msup>\n P^k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n basis on a simplicial mesh and\n \n \n \n \n Q<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msup>\n Q^k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n 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 Schr\u00f6dinger operator has a positive spectral gap. We also show that for the\n \n \n \n \n P<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n P^1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n 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\n \n \n \n \n Q<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msup>\n Q^k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n spectral element methods in two and three dimensions are provided to validate the efficiency of the proposed method.\n <\/p>","DOI":"10.1090\/mcom\/4032","type":"journal-article","created":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T08:47:53Z","timestamp":1732697273000},"page":"2723-2760","source":"Crossref","is-referenced-by-count":9,"title":["Fully discretized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem"],"prefix":"10.1090","volume":"94","author":[{"given":"Ziang","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianfeng","family":"Lu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yulong","family":"Lu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiangxiong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2024,11,27]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1515\/9781400830244","volume-title":"Optimization algorithms on matrix manifolds","author":"Absil, P.-A.","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9780691132983"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1007\/s00211-021-01216-5","article-title":"The \ud835\udc3d-method for the Gross-Pitaevskii eigenvalue problem","volume":"148","author":"Altmann, Robert","year":"2021","journal-title":"Numer. 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