[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83025-en":3,"doc-seo-83025-105":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},83025,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","A unified energy-stable finite element approximation for evolving fluidic biomembranes","A unified finite element method addresses the dynamics of fluidic biomembranes by coupling incompressible Navier–Stokes flow in the bulk with surface Navier–Stokes dynamics on an evolving membrane. Bending effects enter through the Willmore energy, producing nonlinear geometric forces. The formulation allows the bulk mesh velocity to differ from the fluid velocity and introduces a free tangential surface velocity, yielding a unified weak form. An evolution equation for curvature enables a surface ALE weak discretization, delivering unconditionally energy-stable linear fully discrete schemes using fitted or unfitted finite elements.","arXiv :2607 .05998v1 [math .NA] 7 Jul 2026  \nA unified energy-stable finite element approximation for evolving fluidic biomembranes  \nHarald Garcke∗ Robert N¨urnberg† Quan Zhao‡  \nAbstract  \nWe present a unified finite element method for the dynamics of fluidic biomembranes. The model is governed by the Navier–Stokes equations in the bulk coupled to the surface Navier–Stokes equations on the evolving biomembrane surface, with bending forces arising from the Willmore energy. By allowing the bulk mesh velocity to be independent of the fluid velocity and permitting a free tangential surface velocity, we are able to derive a unified weak formulation of the coupled bulk-surface Navier–Stokes system. To address the bending force, we consider an evolution equation for the curvature and propose a surface arbitrary Lagrangian–Eulerian (ALE) weak formulation. Discretization with either fitted or unfitted finite elements leads to well-posed fully discrete linear schemes that are unconditionally energy stable.  \nWe present a variety of numerical examples to demonstrate the favourable properties of the proposed methods.  \nKey words. Fluidic biomembrane, two-phase flow, bulk-surface Navier–Stokes, energy stability, surface incompressibility, volume preservation.  \n1 Introduction  \nFluidic biomembranes play an essential role in many biological processes, including vesicle transport, cell motility, and membrane-mediated interactions between proteins. These membranes are typically composed of lipid bilayers that exhibit fluid-like behaviour along the membrane surface while interacting dynamically with the surrounding viscous fluid. At the continuum level, the membrane can therefore be modelled as an incompressible  \n∗ Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, Germany (har[ald.garcke@ur.de](ald.garcke@ur.de))  \n†Dipartimento di Matematica, Universit`a di Trento, 38123 Trento, Italy (robert.nurnberg@unitn.it)  \n‡School of Mathematical Sciences, University of Science and Technology of China, 230026 Hefei, Anhui, China ([quanzhao@ustc.edu.cn](quanzhao@ustc.edu.cn))  \nviscous surface endowed with bending elastic energies [39] . The bending elasticity in its simplest version is commonly described by the Willmore energy [41],  \nEb = 12 ZΓ κ2 dHd−1 , (1.1)  \nwhere Γ is a hypersurface in Rd , d ∈ {2 , 3} , κ denotes its mean curvature, and dHd−1 represents integration with respect to the (d − 1)-dimensional Hausdorff measure in Rd. This gives rise to nonlinear geometric forces acting on the membrane (see (2.12), below) . As a consequence, the resulting model leads to a coupled bulk-surface system consisting of the incompressible Navier–Stokes equations in the bulk and the surface incompressible Navier–Stokes equations on the evolving membrane, together with highly nonlinear fourth-order geometric forces [2 , 7] . The presence of these high-order geometric forces, together with the strong coupling between membrane motion and bulk fluid dynamics, poses significant challenges for the numerical approximation of evolving fluidic biomembranes.  \nSince the geometric forces are derived from the Willmore energy, their discretizationis closely related to the numerical approximation of the Willmore flow and its volumeand area-preserving variant, the Helfrich flow. These include parametric finite element methods (FEM) [4 , 6 , 11 , 17 , 23 , 24 , 27 , 29 , 30 , 36 , 37], phase-field approximations [15 , 20], the thresholding method [18], and the level set approach [14] . For the numerical approximation of the dynamics of lipid membranes and vesicles in fluidic environments, there exists a large body of work (see [1 , 2 , 7 , 8 , 12 , 19 , 25 , 28 , 31–33, 35 , 38 , 40] and the references therein) . Despite these contributions, to the best of our knowledge, numerical analysis for fluidic biomembrane models remains very limited, particularly with respect to stability estimates, due to the complexity of the model. We also refer the reader to [7 , 8] for an unfitte","cbCainKkKZcU1CW7","https://ap.wps.com/l/cbCainKkKZcU1CW7","pdf",1793321,3,1,39,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What physical and mathematical model governs the fluidic biomembrane dynamics?\",\"answer\":\"The bulk is described by the Navier–Stokes equations, coupled to surface Navier–Stokes equations on the evolving biomembrane. Bending forces arise from the Willmore energy via mean-curvature-related geometric terms.\"},{\"question\":\"How does the method achieve a unified weak formulation for the coupled system?\",\"answer\":\"It decouples the bulk mesh velocity from the fluid velocity and allows a free tangential surface velocity, which makes it possible to derive a unified weak formulation for the coupled bulk–surface Navier–Stokes system.\"},{\"question\":\"How are bending forces treated to obtain stable discretizations?\",\"answer\":\"The approach introduces an evolution equation for curvature and develops a surface ALE (arbitrary Lagrangian–Eulerian) weak formulation, enabling linear fully discrete schemes that are unconditionally energy stable.\"}]",1784184728,98,{"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},"a-unified-energy-stable-finite-element-approximation-for-evolving-fluidic-biomembranes","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-unified-energy-stable-finite-element-approximation-for-evolving-fluidic-biomembranes/83025/",4,{"url":51,"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","2026-07-24","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 physical and mathematical model governs the fluidic biomembrane dynamics?","Question",{"text":75,"@type":76},"The bulk is described by the Navier–Stokes equations, coupled to surface Navier–Stokes equations on the evolving biomembrane. Bending forces arise from the Willmore energy via mean-curvature-related geometric terms.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method achieve a unified weak formulation for the coupled system?",{"text":80,"@type":76},"It decouples the bulk mesh velocity from the fluid velocity and allows a free tangential surface velocity, which makes it possible to derive a unified weak formulation for the coupled bulk–surface Navier–Stokes system.",{"name":82,"@type":73,"acceptedAnswer":83},"How are bending forces treated to obtain stable discretizations?",{"text":84,"@type":76},"The approach introduces an evolution equation for curvature and develops a surface ALE (arbitrary Lagrangian–Eulerian) weak formulation, enabling linear fully discrete schemes that are unconditionally energy stable.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]