[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85242-en":3,"doc-seo-85242-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85242,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","A Four-Field Auxiliary Reformulation of Cahn-Hilliard Time Step Analysis and Conforming Finite Element Discretization","A four-field auxiliary reformulation is proposed for a time-discrete Cahn–Hilliard step. The construction leverages a two-dimensional Rafetseder–Zulehner scalar trace structure to express −Δc as 2p + div u via an auxiliary scalar p and auxiliary vector u. The resulting mixed problem forms a Stokes/elasticity-type auxiliary system, while the phase-field evolution remains the standard mass-conserving gradient flow. The approach yields a continuous four-field formulation equivalent to the classical convex-splitting mixed time step, including conforming finite element discretization and stability estimates, verified by numerical experiments on convergence, mass conservation, energy decay, and consistency of the auxiliary block.","arXiv :2607 . 10724v1 [math .NA] 12 Jul 2026  \nA FOUR-FIELD AUXILIARY REFORMULATION OF ACAHN–HILLIARD TIME STEP: ANALYSIS AND CONFORMING FINITE ELEMENT DISCRETIZATION  \nMARVIN FRITZ  \nFaculty of Mathematics, University of Vienna, Vienna, Austria  \nABSTRACT. We propose a four-field auxiliary reformulation of a time-discrete Cahn– Hilliard step. The construction is motivated by the scalar trace structure underlying two-dimensional Rafetseder–Zulehner decompositions and represents the scalar quantity −∆c in the form 2p+div u through an auxiliary scalar field p and an auxiliary vector field  \nu. The resulting auxiliary problem is a mixed second-order Stokes/elasticity-type system, while the evolution equation for the phase field retains its standard mass-conserving gradient-flow structure. We derive a continuous four-field formulation that is equivalent to the classical convex-splitting mixed time step. We also state a conforming finite element discretization and prove one-step spatial estimates for the phase-field variables together with a stable auxiliary block estimate containing an explicit weak-Laplacian recovery defect. The numerical experiments verify the expected phase-field convergence in a manufactured setting, mass conservation, energy decay, and projected consistency of the auxiliary block.  \n1 Introduction  \nThe Cahn–Hilliard equation is a classical diffuse-interface model for phase separation in binary mixtures [8] . In its standard form, it is written as a second-order system for the phase field c and the chemical potential µ .  \n∂tc = m∆µ, µ = −ε2 ∆c + f′(c) ,  \nwhere m > 0 denotes the mobility, ε > 0 is the interfacial parameter, and f is a bulk freeenergy density. Eliminating the chemical potential yields a fourth-order parabolic equation whose principal part is biharmonic. A direct weak treatment of that fourth-order equation naturally leads to H 2-regularity and therefore suggests C 1-conforming discretizations [6] .  \nFor biharmonic-type operators, several alternatives to conforming C 1 methods are available, including discontinuous Galerkin methods, C0 interior penalty methods, and mixed formulations with auxiliary variables; see, for instance, [2, 3, 11] . Among the mixed approaches, the Rafetseder–Zulehner framework provides a representation of the bending-moment tensor through one scalar and one vector field, thereby reducing the problem to a sequence of second-order subproblems; see [26–28] . This point of view is particularly attractive because it requires only H 1-type regularity for the auxiliary variablesand has proved effective even in settings where nontrivial boundary coupling must be treated explicitly [26, 28] . At the same time, it is well known that naive decouplings ofbiharmonic problems may fail on nonconvex domains, so the precise scalar trace relation is a structural issue rather than a cosmetic one [31] .  \nIn the numerical analysis of the Cahn–Hilliard equation, a large literature is available for standard finite element discretizations, mixed methods, and energy-stable time discretizations. Classical finite element studies go back at least to Elliott and French [15], with further analysis for logarithmic free energies in [12] . Second-order splitting strategies were studied in [16], and Eyre’s convex-splitting idea [17] has become one of the standard references for unconditionally gradient-stable time marching. Further important developments include finite element methods for degenerate mobilities [1], nonlocal diffusion [4], mixed finite element error analysis [18], and discontinuous Galerkin discretizations [25]; see also the broad numerical overviews for Allen–Cahn and Cahn–Hilliard equations [7, 30] .  \nThe present paper develops a four-field auxiliary reformulation of one time-discrete Cahn–Hilliard step that employs the two-dimensional scalar trace structure associated with Rafetseder–Zulehner-type decompositions. We retain the standard mass-conserving evolution equation and repres","cbCaigdHXvkiTkyN","https://ap.wps.com/l/cbCaigdHXvkiTkyN","pdf",3795153,1,28,"English","en",105,"# Introduction\n## Four-field auxiliary reformulation and motivation\n## Time-discrete Cahn–Hilliard and convex splitting\n## Mixed formulations and Rafetseder–Zulehner framework\n## Paper organization\n# Auxiliary block construction\n# Conforming finite element discretization\n# Equivalence and numerical results","[{\"question\":\"What is the main contribution of the proposed four-field auxiliary reformulation?\",\"answer\":\"It rewrites a time-discrete Cahn–Hilliard step by introducing auxiliary unknowns p and u so that the higher-order quantity −Δc is represented through the identity −Δc = 2p + div u, leading to a mixed Stokes/elasticity-type auxiliary system while keeping the phase evolution in its standard mass-conserving gradient-flow form.\"},{\"question\":\"How does the method relate to the classical convex-splitting time step?\",\"answer\":\"The paper derives a continuous four-field formulation and shows it is equivalent to the classical convex-splitting mixed time step, without altering the underlying phase-field update.\"},{\"question\":\"What does the finite element analysis and verification focus on?\",\"answer\":\"It provides conforming finite element discretization and proves one-step spatial estimates for phase-field variables, including a stable auxiliary block estimate with an explicit weak-Laplacian recovery defect; numerical experiments then confirm expected phase-field convergence, mass conservation, energy decay, and projected consistency of the auxiliary block.\"}]",1784202000,71,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"a-four-field-auxiliary-reformulation-of-cahn-hilliard-time-step-analysis-and-conforming-finite-element-discretization","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/a-four-field-auxiliary-reformulation-of-cahn-hilliard-time-step-analysis-and-conforming-finite-element-discretization/85242/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 is the main contribution of the proposed four-field auxiliary reformulation?","Question",{"text":75,"@type":76},"It rewrites a time-discrete Cahn–Hilliard step by introducing auxiliary unknowns p and u so that the higher-order quantity −Δc is represented through the identity −Δc = 2p + div u, leading to a mixed Stokes/elasticity-type auxiliary system while keeping the phase evolution in its standard mass-conserving gradient-flow form.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method relate to the classical convex-splitting time step?",{"text":80,"@type":76},"The paper derives a continuous four-field formulation and shows it is equivalent to the classical convex-splitting mixed time step, without altering the underlying phase-field update.",{"name":82,"@type":73,"acceptedAnswer":83},"What does the finite element analysis and verification focus on?",{"text":84,"@type":76},"It provides conforming finite element discretization and proves one-step spatial estimates for phase-field variables, including a stable auxiliary block estimate with an explicit weak-Laplacian recovery defect; numerical experiments then confirm expected phase-field convergence, mass conservation, energy decay, and projected consistency of the auxiliary block.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]