[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84099-en":3,"doc-seo-84099-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84099,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","Numerical Vortex Resolution for the Gross-Pitaevskii Equation in the Rapid Rotation Thomas-Fermi Scaling","The paper analyzes finite element approximations of Gross–Pitaevskii ground states in the rapid-rotation Thomas–Fermi regime, where healing length and vortex core size scale like ε≪1 and the confinement can degenerate near a critical angular velocity. The study identifies ε-dependence of ground states and shows that local flatness of the energy landscape governs numerical resolution. Mesh size conditions are derived to ensure discrete ground states that are quasi-best approximations, with H1 error asymptotically behaving like h/ε2, but requiring stronger resolution than h≲ε via the first spectral gap of a Riemannian Hessian, linking vortex core size, spectral stability, and discretization accuracy.","arXiv :2607 .06362v1 [math .NA] 7 Jul 2026  \nNumerical vortex resolution for the Gross–Pitaevskii equation in the rapid rotation Thomas–Fermi scaling ∗  \nPatrick Henning,1 Anna Persson2 and Christos Pilichos2  \nAbstract  \nIn this paper we analyze finite element approximations of ground states of the Gross– Pitaevskii equation in the rapid rotation Thomas–Fermi scaling. In this regime, the healing length and vortex core size are of order ε ≪ 1, while the effective confinement potential may degenerate as the angular velocity approaches a critical value. In this setting, we analyze the ε-dependence of the ground states and show that the local flatness of the energy landscape plays a decisive role for numerical resolution. More precisely, we establish mesh size conditions that guarantee the existence of discrete ground states in finite element spaces which are quasi-best approximations of an exact ground state. In particular, we prove that the absolute H 1-error behaves asymptotically like h/ε2 . However, to enter this asymptotic regime, the mesh size must satisfy a significantly stronger resolution condition than the natural requirement h ≲ ε . The additional restriction is governed by the first spectral gap of the Riemannian Hessian of the energy functional at the ground state, which measures the local flatness of the energy surface. With this, our results provide an explanation of the mesh resolution required to capture vortex structures in rapidly rotating Bose–Einstein condensates and highlight the interplay between vortex core size, spectral stability, and discretization accuracy.  \n1 Introduction  \nAt temperatures close to absolute zero, dilute bosonic gases may undergo a phase transition to a Bose–Einstein condensate (BEC), a state of matter in which a macroscopic fraction of the particles occupies the same quantum state; see, e.g., [13, 25, 42] . This collective behavior leads to striking quantum effects on a macroscopic scale, among which superfluidity (the ability of the fluid to flow without dissipation) is one of the most prominent features [41] . When a condensate is set into rotation, superfluidity manifests itself through the formation of quantized vortices. As the rotation frequency increases, these vortices arrange in regular lattice patterns and may eventually fill large portions of the condensate.  \nA widely used mean-field description of rotating Bose–Einstein condensates is provided by the Gross–Pitaevskii energy functional [30, 43, 48], whose ground states are obtained by minimizing the energy under a normalization constraint. These ground states describe stationary configurations of the condensate and capture vortex structures generated by rotation. In the rapid-rotation Thomas–Fermi regime, corresponding to strong interactions and angular velocities that compete with trapping effects at leading order [1, 21], the healing length and vortex core size become small and are characterized by a parameter ε ≪ 1. At the same time, the effective confinement may weaken as the angular velocity approaches a critical value, leading to locally flat energy landscapes near the ground state. Hence, significant challenges arise from the presence of multiple vortices, small length scales, and locally flat energy landscapes.  \n∗ P. Henning acknowledges the support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the project grant 551527112 . A. Persson acknowledges support by the Swedish research council through the project grant 2022-03543 .  \n1 Department of Mathematics, Ruhr-University Bochum, DE-44801 Bochum, Germany.  \nEmail: [patrick.henning@rub.de](patrick.henning@rub.de)  \n2 Department of Information Technology; Division of Scientific Computing, Uppsala University, SE-751 05 Uppsala, Sweden. Email: [apersson@it.uu.se](apersson@it.uu.se), [christos.pilichos@it.uu.se](christos.pilichos@it.uu.se)  \nThe numerical computation of such ground states must address these challenges and involves tw","cbCaili5Tzy2mJ5H","https://ap.wps.com/l/cbCaili5Tzy2mJ5H","pdf",1109951,4,1,44,"English","en",105,"# Abstract\n# Introduction\n## Rapid-rotation Bose–Einstein condensates and vortices\n## Gross–Pitaevskii ground states and the ε-regime\n## Numerical challenges and focus of the work\n## Related finite element error analyses (non-rotating case)","[{\"question\":\"What numerical challenge is emphasized for the Gross–Pitaevskii equation in the rapid-rotation regime?\",\"answer\":\"The paper emphasizes resolving small vortex cores and accurately capturing reduced stability caused by locally flat energy landscapes near the ground state.\"},{\"question\":\"How does the paper characterize the H1-error for discrete ground states?\",\"answer\":\"It establishes that the absolute H1-error behaves asymptotically like h/ε2 under the derived mesh conditions.\"},{\"question\":\"Why is the required mesh resolution stronger than the natural condition h≲ε?\",\"answer\":\"Because the extra restriction is determined by the first spectral gap of the Riemannian Hessian of the energy functional at the ground state, which measures how locally flat the energy surface is.\"}]",1784192786,111,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"numerical-vortex-resolution-for-the-gross-pitaevskii-equation-in-the-rapid-rotation-thomas-fermi-scaling","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/numerical-vortex-resolution-for-the-gross-pitaevskii-equation-in-the-rapid-rotation-thomas-fermi-scaling/84099/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-28","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What numerical challenge is emphasized for the Gross–Pitaevskii equation in the rapid-rotation regime?","Question",{"text":75,"@type":76},"The paper emphasizes resolving small vortex cores and accurately capturing reduced stability caused by locally flat energy landscapes near the ground state.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper characterize the H1-error for discrete ground states?",{"text":80,"@type":76},"It establishes that the absolute H1-error behaves asymptotically like h/ε2 under the derived mesh conditions.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is the required mesh resolution stronger than the natural condition h≲ε?",{"text":84,"@type":76},"Because the extra restriction is determined by the first spectral gap of the Riemannian Hessian of the energy functional at the ground state, which measures how locally flat the energy surface 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