{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,2]],"date-time":"2026-09-02T15:23:24Z","timestamp":1788362604050,"version":"build-2803163510"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>In this paper we improve traditional steepest descent methods for the direct minimization of the Gross\u2013Pitaevskii (GP) energy with rotation at two levels. We first define a new inner product to equip the Sobolev space $H^1$ and derive the corresponding gradient. Second, for the treatment of the mass conservation constraint, we use a projection method that avoids more complicated approaches based on modified energy functionals or traditional normalization methods. The descent method with these two new ingredients is studied theoretically in a Hilbert space setting, and we give a proof of the global existence and convergence in the asymptotic limit to a minimizer of the GP energy. The new method is implemented in both finite difference and finite element two-dimensional settings and is used to compute various complex configurations with vortices of rotating Bose\u2013Einstein condensates. The new Sobolev gradient method shows better numerical performances compared to classical $L^2$ or $H^1$ gradient methods, especially when high rotation rates are considered.<\/jats:p>","DOI":"10.1137\/100782115","type":"journal-article","created":{"date-parts":[[2010,8,12]],"date-time":"2010-08-12T18:05:04Z","timestamp":1281636304000},"page":"2447-2467","source":"Crossref","is-referenced-by-count":90,"title":["A New Sobolev Gradient Method for Direct Minimization of the Gross\u2013Pitaevskii Energy with Rotation"],"prefix":"10.1137","volume":"32","author":[{"given":"Ionut","family":"Danaila","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Parimah","family":"Kazemi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,8,12]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.1060182"},{"key":"R2","unstructured":"R. Adams,\n                      Sobolev Spaces\n                      , Academic Press, New York, 1975."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.68.023603"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.69.033608"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.64.063603"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994264249"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1002\/num.20347"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1126\/science.269.5221.198"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"W. Bao and Q. Du,\n                      Computing the ground state solution of Bose\u2013Einstein condensates by a normalized gradient flow\n                      , SIAM J. Sci. 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Ohtsuke, FreeFem++ (manual), http:\/\/www.freefem.org\/."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.66.053606"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.65.023603"},{"key":"R22","unstructured":"P. Kazemi and M. Eckart,\n                      Minimizing the Gross-Pitaevskii energy functional with the Sobolev gradient\u2014Analytical and numerical results\n                      , Int. J. Comput. Methods, to appear."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-006-1524-9"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.84.806"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.83.3358"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"J. W. Neuberger,\n                      Sobolev Gradients and Differential Equations\n                      , Lecture Notes in Math. 1670, Springer, Berlin, Heidelberg, 1997.","DOI":"10.1007\/BFb0092831"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2003040"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2008.12.016"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/100782115","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:40:57Z","timestamp":1787330457000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/100782115"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":28,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/100782115"],"URL":"https:\/\/doi.org\/10.1137\/100782115","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}