[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86379-en":3,"doc-seo-86379-105":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},86379,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","On h-adaptive Structure-Preserving for the Cahn–Hilliard–Navier–Stokes Equations with Degenerate Mobility","We develop structure-preserving discontinuous Galerkin discretizations for the Cahn–Hilliard–Navier–Stokes system with degenerate mobility, targeting robust boundedness and stability under challenging interface dynamics. The proposed SWIPD-L and SIPGD-L schemes incorporate parametrized mobility fluxes and edge-wise mobility treatments, yielding coercivity for the generalized trilinear form. The methods preserve mass conservation, enforce energy dissipation, and satisfy a discrete maximum principle. Comparisons with existing SIPG-L and SWIP-L confirm similar stability, while h-adaptive tests show significant computational savings without accuracy loss.","arXiv :2602 .22861v3 [math .NA] 13 Jul 2026  \nOn 􀁨 􀁰-adaptive Structure-Preservion for the Cahn–Hilliard–Navier–Stokes Equations with Degenerate Mobility  \nJimmy Kornelije Gunnarsson [0009−0000−7610−6360] and Robert Klfkorn [0000−0001−9664−0333]  \nAbstract We develop structure-preserving discontinuous Galerkin methods for the Cahn-Hilliard-Navier-Stokes equations with degenerate mobility. The proposed SWIPDL and SIPGD-L methods incorporate parametrized mobility fluxes with edge-wise mobility treatments for enhanced coercivity-stability control. We prove coercivity for the generalized trilinear form and demonstrate optimal convergence rates while preserving mass conservation, energy dissipation, and the discrete maximum principle. Comparisons with existing SIPG-L and SWIP-L methods confirm similar stability. Validation on ℎ􀀿-adaptive meshes for both standalone Cahn-Hilliard and coupled systems shows significant computational savings without accuracy loss.  \n1 Introduction  \nConsider the spatial domain Ω ⊂ R􀀳 , 􀀳 = 2, 3, with Lipschitz continuous boundary 􀁭Ω and the time interval (0, 􀀩 ] for 􀀩 ∈ R+ . The Cahn-Hilliard (CH) equations model phase separation in binary mixtures for a phase-field 􀁫 : Ω × (0, 􀀩 ] → [−1, 1] following:  \n􀁭 􀁃􀁫 + ∇ · (􀁫u) − ∇ · 􀀥􀀴−1 (􀀢(􀁫)∇􀁨) = 0, (1)  \n􀁨 + 􀀘􀀽2Δ􀁫 − 􀀬′(􀁫) = 0, (2)  \nwhere 􀀥􀀴 > 0 is the Peclet number, 􀁵 is an advection field, 􀁨 is the chemical potential,􀀬 (􀁫) := 14 (1 − 􀁫2 )2 is the double-well potential, and 􀀢(􀁫) := max {1 − 􀁫2 , 0} is the  \nJimmy Kornelije Gunnarsson  \nCentre for Mathematical Sciences, Lund University, Box 117, 22100 Lund, Sweden e-mail: jimmy_[kornelije.gunnarsson@math.lu.se](kornelije.gunnarsson@math.lu.se)  \nRobert Klfkorn  \nCentre for Mathematical Sciences, Lund University, Box 117, 22100 Lund, Sweden e-mail: robertk@ [math.lu.se](math.lu.se)  \n2 Jimmy Kornelije Gunnarsson and Robert Klfkorn  \nmobility function. We equip Eq. (1) with homogeneous Neumann boundary conditions 􀁮 · ∇􀁫 = 0 and 􀁮 · ∇􀁨 = 0 on 􀁭Ω .  \nWe present below a special case of a fundamental theorem for boundedness.  \nTheorem 1 (Boundedness [5]) Suppose that Ω is convex and that the initial phasefield satisfies 􀁫0 ∈ 􀀝 1 (Ω) and ||􀁫0 || 􀀡 ∞ (Ω) ≤ 1. Then, if the mobility function 􀀢 (􀁫) is defined as above, and ∫Ω 􀀢 (􀁫0 ) + 􀀬(􀁫0 ) \u003C 􀀘 for 􀀘 > 0, then the weak solution 􀁫 ∈ 􀀝1 (Ω) of the CH equation satisfies a weak maximum principle, i.e., ||􀁫|| 􀀡 ∞ (Ω) ≤ 1 for 􀁃 ∈ (0, 􀀩] .  \nWhile Theorem 1, proven in [5, Thm. 1], guarantees a weak maximum principle atthe continuum level for a degenerate mobility, achieving discrete boundedness remains a significant challenge. The standard Finite Element Method (FEM), symmetric weighted interior penalty (SWIP), and symmetric interior penalty Galerkin (SIPG) discretizations fail to preserve boundedness without additional stabilization [9], even when using the degenerate mobility 􀀢 . Recent work, however, has demonstrated that boundedness can be numerically achieved through carefully designed limited Galerkin schemes, including FEM-L for FEM, and for Discontinuous Galerkin (DG) schemes with SIPGL, and SWIP-L [9] with the degenerate CH equations.  \nIn this work, we extend these structure-preserving methods by introducing new mobility fluxes using the DG formulations with intersection-wise mobility treatments. These extend the SWIP-L and SIPG-L methods introduced in [9], by allowing for better control of the coercivity-stability balance.  \nBoundedness of the CH equations is in particular important when coupled to the incompressible Navier-Stokes (NS) equations:  \n􀁤 (􀁫) 􀀀􀁭􀁃􀁵 + 􀁵 · ∇􀁵 􀀁 + 􀁐 · ∇􀁵 + ∇ · 􀀀􀀥I − 2􀁠 (􀁫)􀀙 (􀁵)􀀁 = − 􀀬􀀴1􀀘􀀽 􀁫∇􀁨 (3)  \n∇ · u = 0 (4)  \nwhere 􀁤 (􀁫) := 1+2􀁫 􀁤 1 + 1−2􀁫 􀁤 2 is the density, 􀁐 := 􀁤~~ ~~2−2􀁤~~ ~~1 􀀥􀀴 −1􀀢 (􀁫)∇􀁨 is the mass flux,􀀥 is the pressure, 􀁠 (􀁫) := 1+2􀁫 􀁠 1 + 1−2􀁫 􀁠 2 is the viscosity, and 􀀙 (􀁵) = 12 (∇􀁵 + ∇􀁵􀀩 ) is the rate of deformation tensor, where I denotes the identity tensor. Moreover, 􀁤 􀀹 and 􀁠 􀀹 are the characteristic densities and vi","cbCaighyda9iZIJ3","https://ap.wps.com/l/cbCaighyda9iZIJ3","pdf",742921,2,1,12,"English","en",105,"# Introduction\n# Discretization\n## h-adaptive Structure-Preserving Methods for the CHNS Equations","[{\"question\":\"What problem do the SWIPD-L and SIPGD-L methods address?\",\"answer\":\"They address discrete boundedness and stability for the Cahn–Hilliard(-Navier–Stokes) equations when the mobility is degenerate, where standard schemes may fail without additional stabilization.\"},{\"question\":\"Which structural properties does the proposed discretization preserve?\",\"answer\":\"The methods preserve mass conservation, enforce energy dissipation, and satisfy a discrete maximum principle, while also providing coercivity for the generalized trilinear form.\"},{\"question\":\"How is the mobility handled to improve stability?\",\"answer\":\"The schemes introduce parametrized mobility fluxes and edge-wise (intersection-wise) mobility treatments, allowing better control of the coercivity–stability balance compared with earlier SIPG-L and SWIP-L approaches.\"}]",1784211330,30,{"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},"on-h-adaptive-structure-preserving-for-the-cahnhilliardnavierstokes-equations-with-degenerate-mobility","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/on-h-adaptive-structure-preserving-for-the-cahnhilliardnavierstokes-equations-with-degenerate-mobility/86379/",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","2026-07-22","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 problem do the SWIPD-L and SIPGD-L methods address?","Question",{"text":75,"@type":76},"They address discrete boundedness and stability for the Cahn–Hilliard(-Navier–Stokes) equations when the mobility is degenerate, where standard schemes may fail without additional stabilization.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which structural properties does the proposed discretization preserve?",{"text":80,"@type":76},"The methods preserve mass conservation, enforce energy dissipation, and satisfy a discrete maximum principle, while also providing coercivity for the generalized trilinear form.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the mobility handled to improve stability?",{"text":84,"@type":76},"The schemes introduce parametrized mobility fluxes and edge-wise (intersection-wise) mobility treatments, allowing better control of the coercivity–stability balance compared with earlier SIPG-L and SWIP-L 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