The posteriori analysis of the spectral element discretization of the wave equation

The posteriori analysis of the spectral element discretization of the wave equation | 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 posteriori analysis of the spectral element discretization of the wave equation Nejmeddine chorfi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4176116/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract This work focuses on discretizing a second-order linear wave equation using the implicit Euler scheme for time discretization and the spectral elements method for spatial discretization. We prove that optimal adaptivity can be achieved by combining adaptive time steps with adaptive spectral mesh. We introduce two sets of error indicators for time and space, respectively, and derive optimal estimates. Second order wave equation Implicit Euler method Spectral element method Posteriori analysis Error indicators Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 21 Apr, 2024 Reviews received at journal 20 Apr, 2024 Reviews received at journal 13 Apr, 2024 Reviewers agreed at journal 05 Apr, 2024 Reviewers agreed at journal 05 Apr, 2024 Reviewers invited by journal 05 Apr, 2024 Editor assigned by journal 03 Apr, 2024 Submission checks completed at journal 03 Apr, 2024 First submitted to journal 27 Mar, 2024 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-4176116","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":288109333,"identity":"d1863fab-27a5-469b-9c4a-485a5e145a76","order_by":0,"name":"Nejmeddine chorfi","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABD0lEQVRIie2PsUrEQBBAJwQ2FnOm3RDRX1g5UCQH/srarI2CIIRUEhCSwrO/zzjwByIprlmtcwS8NHYWwjWCCE5CsHEvWgrug4XHMI9lACyWv4kEYC625tLzmZd28vskyLD4MYE2ga8k5XI4OfTS5/lrHO2QOOuL5IlvBy/7JBPw8xtpSo6mhVrOHk+RxA1n+pKz8GxMooDrh7kpEZVU9SgrkYS5o0xeUSJCmoDg5+Zk1aj6o01WTZdwFujx+2BSgaqd7hfoE44Hw79oqZZTukXok+sQNSWo4gi1wo23LLSq3uLoWCzK+zUmku/l5V2NyWTXz2+NCQDKXpy0ly3Rzc3rLV7xbdJs3rZYLJb/yCcUDFz6of//lQAAAABJRU5ErkJggg==","orcid":"","institution":"King Saud University","correspondingAuthor":true,"prefix":"","firstName":"Nejmeddine","middleName":"","lastName":"chorfi","suffix":""}],"badges":[],"createdAt":"2024-03-27 12:44:24","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4176116/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4176116/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":54313316,"identity":"d2122ed8-578f-4eb4-a58e-6bef2ed44914","added_by":"auto","created_at":"2024-04-08 17:27:48","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":200193,"visible":true,"origin":"","legend":"","description":"","filename":"Ondemodifie.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4176116/v1_covered_add89ae0-fbf0-4de2-8724-297b308fe46e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The posteriori analysis of the spectral element discretization of the wave equation","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":"boundary-value-problems","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"bovp","sideBox":"Learn more about [Boundary Value Problems](http://boundaryvalueproblems.springeropen.com)","snPcode":"13661","submissionUrl":"https://submission.nature.com/new-submission/13661/3","title":"Boundary Value Problems","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Second order wave equation, Implicit Euler method, Spectral element method, Posteriori analysis, Error indicators","lastPublishedDoi":"10.21203/rs.3.rs-4176116/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4176116/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This work focuses on discretizing a second-order linear wave equation using the implicit Euler scheme for time discretization and the spectral elements method for spatial discretization. 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