High-efficiency approximation of non-local dispersion model for light interaction with metallic nanostructures by modified two-grid method

High-efficiency approximation of non-local dispersion model for light interaction with metallic nanostructures by modified two-grid method | 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 High-efficiency approximation of non-local dispersion model for light interaction with metallic nanostructures by modified two-grid method Mei Wang, Changhui Yao This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3053056/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract In this paper, we design a modified two-grid method (MTGM) for the Maxwell’s system by adding one correction on the coarse mesh, called postprocessing technique, to the classical two-grid method (TGM), which makes MTGM run smoothing for the edge element like Lagrange elements. The considered Maxwell’s system describes the non-local dispersion model for light interaction with metallic nanostructures. The main contributions of this paper have three parts. Firstly, we give the integral expansion formulas in order to set up supercloseness in the first step of MTGM. Secondly, we take a group of superconvergent solutions on the coarse mesh into the second step as the correction values. Such an algorithm overcomes the difficulties that the edge element can not be applied to the numerical electromagnetic system by TGM directly. Thirdly, we employ the Crank-Nicolson fully discrete scheme to obtain a convergent rate O(τ 2 +h+H 2 ) by using the lowest mixed N´ed´elec − Raviart − Thomas finite element, where τ is the time mesh size, and H, h is the course mesh size and fine mesh size in space, respectively. In the end, we present two numerical examples to verify our algorithm, which demonstrates that the MTGM can save about 30% CPU time. Non-local dispersion model Maxwell’s system Modified two-grid method Post-processing Superconvergence Full Text Additional Declarations No competing interests reported. Supplementary Files Highlight.pdf Cite Share Download PDF Status: Posted Version 1 posted 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-3053056","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":209302451,"identity":"2be3de99-2e9a-4138-a164-8ec0b1013234","order_by":0,"name":"Mei Wang","email":"","orcid":"","institution":"Zhengzhou College of Finance and Economics","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Mei","middleName":"","lastName":"Wang","suffix":""},{"id":209302452,"identity":"5c26cef5-8496-4720-b436-7576ca04d598","order_by":1,"name":"Changhui Yao","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVRIiWNgGAWjYDACCRBRYUGyljMSpGphbCNFC//sHsMPH+dJ2POzH2D88IPBTp6B/ewB/JbcOWMsOXObROLMngRmyR6GZMMGnrwEvFoMJHIMpHm3SSQY3GBgY2ZgYE5gkOAxIKTF+DfvHAl7e4iWeqK0mEnzNkgwbpAAazlMWIvEjbQyyxnHJBJnnElsluwxOG7YxpODXwv/jOTNNz7U2Njztx8++OFHRbU8P/sZ/FoYGDhgChgbgO5kYGAjoB4I2B8QVjMKRsEoGAUjGwAAKcs2GcOFLEUAAAAASUVORK5CYII=","orcid":"","institution":"Zhengzhou University","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Changhui","middleName":"","lastName":"Yao","suffix":""}],"badges":[],"createdAt":"2023-06-12 11:44:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3053056/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3053056/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103034531,"identity":"30959ffa-3f18-45ed-bedd-edde9ab69cf3","added_by":"auto","created_at":"2026-02-20 00:38:57","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":303489,"visible":true,"origin":"","legend":"","description":"","filename":"20220624.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3053056/v1_covered_c3eaba58-d7c0-40ff-bb8b-76a33aa0bf0b.pdf"},{"id":38569658,"identity":"b0ca3401-95d4-4bc2-8a1c-0a20e1c71c57","added_by":"auto","created_at":"2023-06-15 02:10:18","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":56681,"visible":true,"origin":"","legend":"","description":"","filename":"Highlight.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3053056/v1/f78379ebf1783a08ed54d8c9.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"High-efficiency approximation of non-local dispersion model for light interaction with metallic nanostructures by modified two-grid method","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"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":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Non-local dispersion model, Maxwell’s system, Modified two-grid method, Post-processing, Superconvergence","lastPublishedDoi":"10.21203/rs.3.rs-3053056/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3053056/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this paper, we design a modified two-grid method (MTGM) for the Maxwell’s system by adding one correction on the coarse mesh, called postprocessing technique, to the classical two-grid method (TGM), which makes MTGM run smoothing for the edge element like Lagrange elements. The considered Maxwell’s system describes the non-local dispersion model for light interaction with metallic nanostructures. The main contributions of this paper have three parts. Firstly, we give the integral expansion formulas in order to set up supercloseness in the first step of MTGM. Secondly, we take a group of superconvergent solutions on the coarse mesh into the second step as the correction values. Such an algorithm overcomes the difficulties that the edge element can not be applied to the numerical electromagnetic system by TGM directly. Thirdly, we employ the Crank-Nicolson fully discrete scheme to obtain a convergent rate O(τ\u003csup\u003e2 \u003c/sup\u003e+h+H\u003csup\u003e2\u003c/sup\u003e) by using the lowest mixed N´ed´elec − Raviart − Thomas finite element, where τ is the time mesh size, and H, h is the course mesh size and fine mesh size in space, respectively. 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