{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T08:56:00Z","timestamp":1765961760307},"reference-count":42,"publisher":"American Mathematical Society (AMS)","issue":"351","license":[{"start":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T00:00:00Z","timestamp":1742860800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["HE 2464\/7-1"],"award-info":[{"award-number":["HE 2464\/7-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The ground states of Bose\u2013Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross\u2013Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular, we prove a priori error estimates of optimal order in the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n  <mml:semantics>\n    <mml:msup>\n      <mml:mi>L<\/mml:mi>\n      <mml:mn>2<\/mml:mn>\n    <\/mml:msup>\n    <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>- and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1\">\n  <mml:semantics>\n    <mml:msup>\n      <mml:mi>H<\/mml:mi>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:msup>\n    <mml:annotation encoding=\"application\/x-tex\">H^1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-norm, as well as for the ground state energy and the corresponding chemical potential. A central issue in the analysis of the problem is the missing uniqueness of ground states, which is mainly caused by the invariance of the energy functional under complex phase shifts. Our error analysis is therefore based on an Euler\u2013Lagrange functional that we restrict to certain tangent spaces in which we have local uniqueness of ground states. This gives rise to an error decomposition that is ultimately used to derive the desired a priori error estimates. We also present numerical experiments to illustrate various aspects of the problem structure.<\/p>","DOI":"10.1090\/mcom\/3962","type":"journal-article","created":{"date-parts":[[2024,3,9]],"date-time":"2024-03-09T00:18:24Z","timestamp":1709943504000},"page":"1-32","source":"Crossref","is-referenced-by-count":6,"title":["On discrete ground states of rotating Bose\u2013Einstein condensates"],"prefix":"10.1090","volume":"94","author":[{"given":"Patrick","family":"Henning","sequence":"first","affiliation":[]},{"given":"Mahima","family":"Yadav","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,3,25]]},"reference":[{"issue":"3","key":"1","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. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"5","key":"2","doi-asserted-by":"publisher","first-page":"1629","DOI":"10.1051\/m2an\/2022036","article-title":"Energy-adaptive Riemannian optimization on the Stiefel manifold","volume":"56","author":"Altmann, Robert","year":"2022","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1016\/j.jcp.2017.04.040","article-title":"Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by preconditioned nonlinear conjugate gradient methods","volume":"343","author":"Antoine, Xavier","year":"2017","journal-title":"J. Comput. Phys.","ISSN":"http:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"4","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1016\/j.jcp.2013.10.045","article-title":"Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose-Einstein condensates","volume":"258","author":"Antoine, Xavier","year":"2014","journal-title":"J. Comput. Phys.","ISSN":"http:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"5","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1007\/BF02165003","article-title":"Error-bounds for finite element method","volume":"16","author":"Babu\u0161ka, Ivo","year":"1970","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"5","key":"6","doi-asserted-by":"publisher","first-page":"1674","DOI":"10.1137\/S1064827503422956","article-title":"Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow","volume":"25","author":"Bao, Weizhu","year":"2004","journal-title":"SIAM J. Sci. Comput.","ISSN":"http:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"23","key":"7","doi-asserted-by":"publisher","first-page":"9778","DOI":"10.1016\/j.jcp.2008.07.017","article-title":"A generalized-Laguerre-Hermite pseudospectral method for computing symmetric and central vortex states in Bose-Einstein condensates","volume":"227","author":"Bao, Weizhu","year":"2008","journal-title":"J. Comput. Phys.","ISSN":"http:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"crossref","first-page":"57","DOI":"10.4310\/CMS.2005.v3.n1.a5","article-title":"Ground, symmetric and central vortex states in rotating Bose-Einstein condensates","volume":"3","author":"Bao, Weizhu","year":"2005","journal-title":"Commun. Math. Sci.","ISSN":"http:\/\/id.crossref.org\/issn\/1539-6746","issn-type":"print"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3934\/krm.2013.6.1","article-title":"Mathematical theory and numerical methods for Bose-Einstein condensation","volume":"6","author":"Bao, Weizhu","year":"2013","journal-title":"Kinet. Relat. Models","ISSN":"http:\/\/id.crossref.org\/issn\/1937-5093","issn-type":"print"},{"issue":"20","key":"10","doi-asserted-by":"publisher","first-page":"7874","DOI":"10.1016\/j.jcp.2010.07.001","article-title":"Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates","volume":"229","author":"Bao, Weizhu","year":"2010","journal-title":"J. Comput. Phys.","ISSN":"http:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"11","doi-asserted-by":"crossref","unstructured":"S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift f\u00fcr Physik 26 (1924), no. 1, 178\u2013181.","DOI":"10.1007\/BF01327326"},{"key":"12","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and hybrid finite element methods","volume":"15","author":"Brezzi, Franco","year":"1991","ISBN":"http:\/\/id.crossref.org\/isbn\/0387975829"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1016\/j.jcp.2008.09.018","article-title":"A minimisation approach for computing the ground state of Gross-Pitaevskii systems","volume":"228","author":"Caliari, Marco","year":"2009","journal-title":"J. Comput. Phys.","ISSN":"http:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"2","key":"14","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1093\/imanum\/drw053","article-title":"Two-grid methods for a class of nonlinear elliptic eigenvalue problems","volume":"38","author":"Canc\u00e8s, Eric","year":"2018","journal-title":"IMA J. Numer. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"1-3","key":"15","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1007\/s10915-010-9358-1","article-title":"Numerical analysis of nonlinear eigenvalue problems","volume":"45","author":"Canc\u00e8s, Eric","year":"2010","journal-title":"J. Sci. Comput.","ISSN":"http:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"key":"16","doi-asserted-by":"crossref","unstructured":"E. Canc\u00e8s and C. Le Bris, Can we outperform the DIIS approach for electronic structure calculations?, Int. J. Quantum Chem. 79 (2000), no. 2, 82\u201390.","DOI":"10.1002\/1097-461X(2000)79:2<82::AID-QUA3>3.0.CO;2-I"},{"issue":"4","key":"17","doi-asserted-by":"publisher","first-page":"1828","DOI":"10.1137\/130916096","article-title":"Adaptive finite element approximations for Kohn-Sham models","volume":"12","author":"Chen, Huajie","year":"2014","journal-title":"Multiscale Model. Simul.","ISSN":"http:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/s10444-011-9235-y","article-title":"Numerical analysis of finite dimensional approximations of Kohn-Sham models","volume":"38","author":"Chen, Huajie","year":"2013","journal-title":"Adv. Comput. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"4","key":"19","doi-asserted-by":"publisher","first-page":"493","DOI":"10.4208\/aamm.10-m1057","article-title":"Adaptive finite element approximations for a class of nonlinear eigenvalue problems in quantum physics","volume":"3","author":"Chen, Huajie","year":"2011","journal-title":"Adv. Appl. Math. Mech.","ISSN":"http:\/\/id.crossref.org\/issn\/2070-0733","issn-type":"print"},{"issue":"14","key":"20","doi-asserted-by":"publisher","first-page":"1723","DOI":"10.1002\/mma.1292","article-title":"Numerical approximations of a nonlinear eigenvalue problem and applications to a density functional model","volume":"33","author":"Chen, Huajie","year":"2010","journal-title":"Math. Methods Appl. Sci.","ISSN":"http:\/\/id.crossref.org\/issn\/0170-4214","issn-type":"print"},{"issue":"21-22","key":"21","doi-asserted-by":"publisher","first-page":"1846","DOI":"10.1016\/j.cma.2011.02.008","article-title":"Finite element approximations of nonlinear eigenvalue problems in quantum physics","volume":"200","author":"Chen, Huajie","year":"2011","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"http:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"22","series-title":"Classics in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719208","volume-title":"The finite element method for elliptic problems","volume":"40","author":"Ciarlet, Philippe G.","year":"2002","ISBN":"http:\/\/id.crossref.org\/isbn\/0898715148"},{"issue":"5","key":"23","doi-asserted-by":"publisher","first-page":"2447","DOI":"10.1137\/100782115","article-title":"A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation","volume":"32","author":"Danaila, Ionut","year":"2010","journal-title":"SIAM J. Sci. Comput.","ISSN":"http:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"6","key":"24","doi-asserted-by":"publisher","first-page":"B1102--B1129","DOI":"10.1137\/17M1121974","article-title":"Computation of ground states of the Gross-Pitaevskii functional via Riemannian optimization","volume":"39","author":"Danaila, Ionut","year":"2017","journal-title":"SIAM J. Sci. Comput.","ISSN":"http:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"19","key":"25","doi-asserted-by":"publisher","first-page":"6946","DOI":"10.1016\/j.jcp.2010.05.032","article-title":"A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates","volume":"229","author":"Danaila, Ionut","year":"2010","journal-title":"J. Comput. Phys.","ISSN":"http:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"10","key":"26","doi-asserted-by":"publisher","first-page":"787","DOI":"10.1016\/j.cpc.2007.04.007","article-title":"Ground state of the time-independent Gross-Pitaevskii equation","volume":"177","author":"Dion, Claude M.","year":"2007","journal-title":"Comput. Phys. Comm.","ISSN":"http:\/\/id.crossref.org\/issn\/0010-4655","issn-type":"print"},{"key":"27","doi-asserted-by":"crossref","unstructured":"C. D\u00f6ding and P. Henning, Uniform \ud835\udc3f^{\u221e}-bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation, ArXiv e-print  arXiv:2210.01553, to appear in IMA J. Numer. Anal., 2023.","DOI":"10.1093\/imanum\/drad081"},{"key":"28","unstructured":"A. Einstein, Quantentheorie des Einatomigen Idealen Gases, Sitzber. Kgl. Preuss. Akad. Wiss., 1924, pp. 261\u2013267."},{"key":"29","doi-asserted-by":"publisher","first-page":"Paper No. 110165, 15","DOI":"10.1016\/j.jcp.2021.110165","article-title":"Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation","volume":"436","author":"Heid, Pascal","year":"2021","journal-title":"J. Comput. Phys.","ISSN":"http:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"4","key":"30","doi-asserted-by":"publisher","first-page":"1525","DOI":"10.1137\/130921520","article-title":"Two-level discretization techniques for ground state computations of Bose-Einstein condensates","volume":"52","author":"Henning, Patrick","year":"2014","journal-title":"SIAM J. Numer. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"31","doi-asserted-by":"publisher","first-page":"1744","DOI":"10.1137\/18M1230463","article-title":"Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency","volume":"58","author":"Henning, Patrick","year":"2020","journal-title":"SIAM J. Numer. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"32","doi-asserted-by":"publisher","first-page":"923","DOI":"10.1137\/15M1009172","article-title":"The finite element method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation","volume":"55","author":"Henning, Patrick","year":"2017","journal-title":"SIAM J. Numer. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"33","doi-asserted-by":"publisher","first-page":"993","DOI":"10.1137\/22M1516300","article-title":"On optimal convergence rates for discrete minimizers of the Gross-Pitaevskii energy in localized orthogonal decomposition spaces","volume":"21","author":"Henning, Patrick","year":"2023","journal-title":"Multiscale Model. Simul.","ISSN":"http:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"issue":"4","key":"34","doi-asserted-by":"publisher","first-page":"A1978--A2001","DOI":"10.1137\/130910014","article-title":"An inverse iteration method for eigenvalue problems with eigenvector nonlinearities","volume":"36","author":"Jarlebring, Elias","year":"2014","journal-title":"SIAM J. Sci. Comput.","ISSN":"http:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"35","series-title":"Math\\'{e}matiques \\& Applications (Berlin) [Mathematics \\& Applications]","isbn-type":"print","volume-title":"Introduction \\`a la th\\'{e}orie des points critiques et applications aux probl\\`emes elliptiques","volume":"13","author":"Kavian, Otared","year":"1993","ISBN":"http:\/\/id.crossref.org\/isbn\/2287004106"},{"issue":"4","key":"36","doi-asserted-by":"publisher","first-page":"739","DOI":"10.1007\/s002110100358","article-title":"Error bars and quadratically convergent methods for the numerical simulation of the Hartree-Fock equations","volume":"94","author":"Maday, Yvon","year":"2003","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"37","doi-asserted-by":"crossref","unstructured":"M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Vortices in a Bose-Einstein condensate, Phys. Rev. Lett. 83 (1999), no. 13, 2498\u20132501.","DOI":"10.1103\/PhysRevLett.83.2498"},{"volume-title":"On generalized finite-element methods","year":"1995","author":"Melenk, Jens Markus","key":"38"},{"key":"39","series-title":"International Series of Monographs on Physics","isbn-type":"print","volume-title":"Bose-Einstein condensation","volume":"116","author":"Pitaevskii, Lev","year":"2003","ISBN":"http:\/\/id.crossref.org\/isbn\/0198507194"},{"issue":"2","key":"40","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1007\/s002110050088","article-title":"Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems","volume":"69","author":"Pousin, J.","year":"1994","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"2","key":"41","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1088\/0951-7715\/17\/2\/010","article-title":"An analysis of finite-dimensional approximations for the ground state solution of Bose-Einstein condensates","volume":"17","author":"Zhou, Aihui","year":"2004","journal-title":"Nonlinearity","ISSN":"http:\/\/id.crossref.org\/issn\/0951-7715","issn-type":"print"},{"issue":"4","key":"42","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1002\/mma.793","article-title":"Finite dimensional approximations for the electronic ground state solution of a molecular system","volume":"30","author":"Zhou, Aihui","year":"2007","journal-title":"Math. Methods Appl. Sci.","ISSN":"http:\/\/id.crossref.org\/issn\/0170-4214","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-351\/S0025-5718-2024-03962-5\/mcom3962_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-351\/S0025-5718-2024-03962-5\/S0025-5718-2024-03962-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T17:49:20Z","timestamp":1728582560000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-351\/S0025-5718-2024-03962-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,25]]},"references-count":42,"journal-issue":{"issue":"351","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["S0025-5718-2024-03962-5"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3962","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"type":"print","value":"0025-5718"},{"type":"electronic","value":"1088-6842"}],"subject":[],"published":{"date-parts":[[2024,3,25]]}}}