[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82629-en":3,"doc-seo-82629-105":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},82629,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Stream Function – Pressure Virtual Element Methods for the Stokes–Darcy Interface Problem","This paper introduces a novel Virtual Element Method (VEM) for the coupled Stokes–Darcy system in primal–primal form. In the free-flow Stokes domain, a stream-function formulation is employed that automatically satisfies incompressibility and lowers computational cost. At the interface, the method enforces mass conservation, normal stress balance, and the Beavers–Joseph–Saffman slip condition, coupling a biharmonic stream-function equation with Darcy’s pressure equation. The approach targets general polygonal meshes, handling irregular interface geometry without remeshing or adaptive refinement. Numerical simulations validate accuracy, including dead-end filtration and network flow in bioartificial organs.","arXiv :2607 .01622v2 [math .NA] 3 Jul 2026  \nNoname manuscript No.  \n(will be inserted by the editor)  \nStream function – pressure virtual element methods for the Stokes–Darcy interface problem  \nFranco Dassi · Rekha Khot · David Mora · Andr´es  \nE. Rubiano · Ricardo Ruiz-Baier  \nReceived: date / Accepted: date  \nAbstract This paper introduces a novel Virtual Element Method (VEM) for the coupled Stokes– Darcy system in primal-primal form. In the free-flow Stokes domain, we implement a stream function formulation that inherently satisfies the incompressibility constraint and reduces computational cost. Across the interface, mass conservation, normal stress balance, and the Beavers–Joseph–Saffman slip condition are enforced to couple the biharmonic stream function equation with the Darcy’s pressure equation. Leveraging VEM’s ability to handle general polygonal meshes, the proposed method naturally accommodates irregular interface geometries without requiring remeshing or adaptive refinement. The accuracy of the method is validated through several numerical simulations that include applications to dead-end filtration, and network flow in bioartificial organs.  \nKeywords Stream function formulation · Interface problem · Coupled Stokes–Darcy · Virtual Element Methods · A priori error analysis  \nMathematics Subject Classification (2020) 65N30 · 65N99 · 65Z05  \n1 Introduction  \nThe coupling of free-flow with porous media flow is a fundamental challenge in fluid mechanics, with several applications ranging from groundwater hydrology and reservoir engineering to industrial filtration and biomedical modelling (see e.g. [15,25,28,35,49]) . The interfacial coupling is achieved through continuity of normal velocities (mass conservation), balance of normal stresses, and the Beavers–Joseph– Saffman tangential slip condition [7,50] . While more general frameworks account for porous media with cracks, and the incorporation of other linear and nonlinear equations in the coupled problem, such as Navier–Stokes, Brinkman and Forchheimer (see, e.g., [12,24,30,33,56]), the classical Stokes–Darcy system maintains the essential features of the interface mechanisms and remains the most important benchmark for modelling fluid-porous interfaces.  \nA wide range of primal and mixed Finite Elements (FEs), discontinuous Galerkin (dG), and Hybrid High–Order (HHO) methods have been developed for the Stokes–Darcy problem; a non-exhaustive list of representative references includes [2,5,6,13,16,17,19,29,32,37,52,55] . Most of these approaches rely on velocity–pressure, velocity–pressure–pseudostress, or vorticity–pressure formulations in the Stokes  \nF. Dassi  \nDipartimento di Matematica e Applicazioni, Universit`a degli studi di Milano Bicocca, Via Roberto Cozzi 55, 20125, Milano, [Italy. E-mail: Franco.Dassi@unimib.it](Italy. E-mail: Franco.Dassi@unimib.it)  \nR. Khot  \nDepartment of Mathematics, Indian Institute of Technology Palakkad, Kanjikode 678623, Kerala, India. E-mail: [RekhaKhot@iitpkd.ac.in](RekhaKhot@iitpkd.ac.in)  \nD. Mora  \nGIMNAP, Departamento de Matem´atica, Universidad del B´ıo-B´ıo, 4051381 Concepci´on, Chile; and CI2 MA, Universidad de Concepci´on, Concepci´on, [Chile. E-mail: DMora@ubiobio.cl](Chile. E-mail: DMora@ubiobio.cl)  \nA. E. Rubiano · R. Ruiz-Baier  \nSchool of Mathematics, Monash University, 9 Rainforest Walk, Melbourne, VIC 3800, Australia. E-mail: An[dres.RubianoMartinez@monash.edu](dres.RubianoMartinez@monash.edu), [Ricardo.RuizBaier@monash.edu](Ricardo.RuizBaier@monash.edu)  \nregion and either primal or mixed formulations in the Darcy region. These formulations require the design of compatible discrete spaces that satisfy appropriate inf–sup stability conditions and accurately enforce the interface coupling conditions. While conforming finite element methods often employ matching discretisations across the interface, dG and HHO formulations can naturally accommodate non-matching meshes through the weak enforcement of the coupling","cbCaiiLIbzY3LhXN","https://ap.wps.com/l/cbCaiiLIbzY3LhXN","pdf",2595551,1,23,"English","en",105,"# Introduction\n## Main contributions\n# Virtual Element Method background\n## Stream function reformulation for incompressibility\n# Coupling conditions at the interface\n## Mass conservation and normal stress balance\n## Beavers–Joseph–Saffman slip condition\n# Numerical validation\n## Dead-end filtration applications\n## Bioartificial organ network flow","[{\"question\":\"What is the main goal of the proposed method?\",\"answer\":\"The paper develops a VEM for the coupled Stokes–Darcy system using a stream-function–pressure (primal–primal) formulation and focuses on enforcing the Stokes–Darcy interface mechanisms efficiently.\"},{\"question\":\"How does the method handle incompressibility in the Stokes region?\",\"answer\":\"A stream-function formulation is used in the free-flow Stokes domain, which inherently satisfies the incompressibility constraint and reduces computational cost.\"},{\"question\":\"Which physical conditions are enforced across the Stokes–Darcy interface?\",\"answer\":\"Across the interface, the method enforces mass conservation (continuity of mass), normal stress balance, and the Beavers–Joseph–Saffman tangential slip condition.\"}]",1784181915,58,{"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},"stream-function-pressure-virtual-element-methods-for-the-stokesdarcy-interface-problem","",{"@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/stream-function-pressure-virtual-element-methods-for-the-stokesdarcy-interface-problem/82629/",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 goal of the proposed method?","Question",{"text":75,"@type":76},"The paper develops a VEM for the coupled Stokes–Darcy system using a stream-function–pressure (primal–primal) formulation and focuses on enforcing the Stokes–Darcy interface mechanisms efficiently.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method handle incompressibility in the Stokes region?",{"text":80,"@type":76},"A stream-function formulation is used in the free-flow Stokes domain, which inherently satisfies the incompressibility constraint and reduces computational cost.",{"name":82,"@type":73,"acceptedAnswer":83},"Which physical conditions are enforced across the Stokes–Darcy interface?",{"text":84,"@type":76},"Across the interface, the method enforces mass conservation (continuity of mass), normal stress balance, and the Beavers–Joseph–Saffman tangential slip 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