The 3D Nonconforming Virtual Element Method with enhanced spaces: numerical investigation and implementation guide

The 3D Nonconforming Virtual Element Method with enhanced spaces: numerical investigation and implementation guide | 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 The 3D Nonconforming Virtual Element Method with enhanced spaces: numerical investigation and implementation guide Manuel Luigi Trezzi, Franco Dassi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7122590/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 05 Dec, 2025 Read the published version in Numerical Algorithms → Version 1 posted 9 You are reading this latest preprint version Abstract In this paper, we describe the 3D Nonconforming Virtual Element Method, where the local spaces are enhanced in order to build an L 2 projection of virtual functions onto polynomials of degree k. This higher order projection provides a more accurate representation of the load term and, more importantly, it enables the construction of the reaction term. We conduct a convergence analysis of the L 2 norm and H 1 seminorm errors with respect to mesh size and approximation degree. We also compare our results with those obtained using the conforming Virtual Element Method. Nonconforming Virtual Element Method Diffusion reaction problems Polyhedral meshes Implementation details Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 05 Dec, 2025 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 27 Sep, 2025 Reviews received at journal 26 Sep, 2025 Reviews received at journal 17 Sep, 2025 Reviewers agreed at journal 06 Aug, 2025 Reviewers agreed at journal 05 Aug, 2025 Reviewers invited by journal 15 Jul, 2025 Editor assigned by journal 15 Jul, 2025 Submission checks completed at journal 14 Jul, 2025 First submitted to journal 14 Jul, 2025 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. 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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-7122590","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":485698116,"identity":"3c6c87b2-97c8-43a9-b940-3f9f4480b5ee","order_by":0,"name":"Manuel Luigi Trezzi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVRIiWNgGAWjYNCCAiidUMHAwMbA2MbAYEBIiwGDBETLGbgWQnpgWkCKGUC68Fmj2372mMQHA4Y68/b2xx8ezjucxyd2uO0BQ8EfnFrMzuSlSc4A2iJz5kCCQeK2w8Vs0ontBvgcZnYgx0yaB6hFQiLhQAJQS2KbdGKbBF4t59+YSf8BaZF/2HAgcQ4xWm4AbWEA28LM2JDYQJSWN8aWPQYSkjN40pgZEo6lg7S0GyQYGONxWI7hjR8VNvwS7Mcff/xRY504f3b6swcf/sjh1AIFEmj8BEIaRsEoGAWjYBTgBQC75UxGVwMtyAAAAABJRU5ErkJggg==","orcid":"","institution":"University of Milano-Bicocca","correspondingAuthor":true,"prefix":"","firstName":"Manuel","middleName":"Luigi","lastName":"Trezzi","suffix":""},{"id":485698117,"identity":"005cb6bc-8c15-49df-9156-fcb43637c281","order_by":1,"name":"Franco Dassi","email":"","orcid":"","institution":"University of Milano-Bicocca","correspondingAuthor":false,"prefix":"","firstName":"Franco","middleName":"","lastName":"Dassi","suffix":""}],"badges":[],"createdAt":"2025-07-14 15:08:32","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7122590/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7122590/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-025-02280-w","type":"published","date":"2025-12-05T15:57:53+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":97724071,"identity":"b4090ac1-1f78-4fd6-ad3c-b6fa4c64feaa","added_by":"auto","created_at":"2025-12-08 16:11:35","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1367184,"visible":true,"origin":"","legend":"","description":"","filename":"NonConformi3dforSub.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7122590/v1_covered_431b95fa-4e85-486a-96fa-ca9ffe684de9.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The 3D Nonconforming Virtual Element Method with enhanced spaces: numerical investigation and implementation guide","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Nonconforming Virtual Element Method, Diffusion reaction problems, Polyhedral meshes, Implementation details","lastPublishedDoi":"10.21203/rs.3.rs-7122590/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7122590/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this paper, we describe the 3D Nonconforming Virtual Element Method, where the local spaces are enhanced in order to build an L\u003csup\u003e2\u003c/sup\u003e projection of virtual functions onto polynomials of degree k. 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