A Stochastic Mixed Formulation for Darcy-Forchheimer Flow on Smooth Surfaces with Mean-Square Stable Surface Finite Element Approximation

A Stochastic Mixed Formulation for Darcy-Forchheimer Flow on Smooth Surfaces with Mean-Square Stable Surface Finite Element Approximation | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Stochastic Mixed Formulation for Darcy-Forchheimer Flow on Smooth Surfaces with Mean-Square Stable Surface Finite Element Approximation Marcial Nguemfouo This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9303569/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract We propose a stochastic extension of the surface Darcy–Forchheimer problem on a smooth closed surface Γ ⊂ R3. The unknowns are a tangential velocity field and a scalar pressure, as in the deterministic mixed model, but the momentum balance is perturbed by an additive tangential Wiener forcing. The divergence constraint is preserved pathwise, and the pressure remains a Lagrange multiplier enforcing mass conservation. After rewriting the problem on the divergence-free manifold by means of a right inverse of the surface divergence, we prove well-posedness of a variational solution by monotone-operator methods in expectation and almost surely. We then introduce a lowest-order Raviart–Thomas / piecewise constant discretization on a polyhedral approximation Γh of the surface and couple it with an implicit Euler–Maruyama time stepping. Under standard geometric approximation assumptions, Lipschitz covariance regularity of the noise, and suitable regularity of the exact solution, we derive a strong mean-square error estimate of order O(h + τ 1/2) for the velocity in the natural mixed norm and of order O(h + τ 1/2) for the pressure in L2. The analysis isolates the nonlinear monotonicity term, the stochastic consistency defect, and the geometric perturbation terms induced by the surface approximation. We conclude with modeling perspectives for uncertainty quantification in flows on biological membranes, fractured interfaces, and porous shells. stochastic PDEs on surfaces Darcy–Forchheimer flow mixed finite elements monotone operators Euler–Maruyama scheme geometric consistency Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 09 Apr, 2026 Editor assigned by journal 09 Apr, 2026 Submission checks completed at journal 08 Apr, 2026 First submitted to journal 02 Apr, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9303569","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":622086799,"identity":"bf836a39-d7b5-4087-82a3-3ad70fc27599","order_by":0,"name":"Marcial Nguemfouo","email":"data:image/png;base64,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","orcid":"","institution":"University of Yaoundé I","correspondingAuthor":true,"prefix":"","firstName":"Marcial","middleName":"","lastName":"Nguemfouo","suffix":""}],"badges":[],"createdAt":"2026-04-02 13:08:34","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9303569/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9303569/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107121893,"identity":"685fab11-f23b-4d3c-8fe9-08ed723814db","added_by":"auto","created_at":"2026-04-17 04:40:42","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1194992,"visible":true,"origin":"","legend":"","description":"","filename":"DarcyFohheimerStochasticok.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9303569/v1_covered_a79119a1-c4ab-4acb-a884-1f649b69d8f2.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Stochastic Mixed Formulation for Darcy-Forchheimer Flow on Smooth Surfaces with Mean-Square Stable Surface Finite Element Approximation","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nonlinear-differential-equations-and-applications-nodea","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"ndea","sideBox":"Learn more about [Nonlinear Differential Equations and Applications NoDEA](http://link.springer.com/journal/30)","snPcode":"30","submissionUrl":"https://submission.nature.com/new-submission/30/3","title":"Nonlinear Differential Equations and Applications NoDEA","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"stochastic PDEs on surfaces, Darcy–Forchheimer flow, mixed finite elements, monotone operators, Euler–Maruyama scheme, geometric consistency","lastPublishedDoi":"10.21203/rs.3.rs-9303569/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9303569/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe propose a stochastic extension of the surface Darcy–Forchheimer problem on a smooth closed surface Γ ⊂ R3. The unknowns are a tangential velocity field and a scalar pressure, as in the deterministic mixed model, but the momentum balance is perturbed by an additive tangential Wiener forcing. The divergence constraint is preserved pathwise, and the pressure remains a Lagrange multiplier enforcing mass conservation. After rewriting the problem on the divergence-free manifold by means of a right inverse of the surface divergence, we prove well-posedness of a variational solution by monotone-operator methods in expectation and almost surely. We then introduce a lowest-order Raviart–Thomas / piecewise constant discretization on a polyhedral approximation Γh of the surface and couple it with an implicit Euler–Maruyama time stepping. Under standard geometric approximation assumptions, Lipschitz covariance regularity of the noise, and suitable regularity of the exact solution, we derive a strong mean-square error estimate of order O(h + τ 1/2) for the velocity in the natural mixed norm and of order O(h + τ 1/2) for the pressure in L2. The analysis isolates the nonlinear monotonicity term, the stochastic consistency defect, and the geometric perturbation terms induced by the surface approximation. 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