On a Semi-analytical Method for Solution of the 2d Laplace Equation in Arbitrary Domains

On a Semi-analytical Method for Solution of the 2d Laplace Equation in Arbitrary Domains | 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 On a Semi-analytical Method for Solution of the 2d Laplace Equation in Arbitrary Domains David Matthew Kelly, Keith Roberts, Onur Kurum This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3950747/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 This study presents a semi-analytical approach to solve the Laplace equation in arbitrarily shaped two-dimensional domains. The method is meshless and addresses boundary value problems that include both pure Dirichlet and mixed Dirichlet-Neumann boundary conditions. The solution is obtained via a weighted superposition of harmonic polynomials which are obtained via an orthonormalization approach. We show that the approach is efficient in terms of number of operations. The numerical solution is convergent and exact (within machine precision) given a sufficient number of terms in the series. Moreover, the method offers several advantages over traditional approaches. Advantages include providing analytical expressions for the stream function and velocity components when solving potential flow problems. There are also important implications for the storage of model results; the method offers extremely low cost data storage. In the paper, several example applications involving arbitrary domains are presented. The results obtained are compared with known analytical solutions. Laplace Equation Meshless Method Series Solution Analytical Solution Harmonic Polynomials Potential Flow Full Text Additional Declarations No competing interests reported. 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-3950747","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":272726475,"identity":"0a19c7d5-2e26-4819-bbeb-af1f3be095ba","order_by":0,"name":"David Matthew Kelly","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6UlEQVRIiWNgGAWjYDCCMwwMHyAs5sYHQJKHjwgtjDNANA8DY7MBiGYjRUubBIhBUAvfmcMPG35U2Nnbsx9sq/yaYyfDxsD88NENPFokz7YZNvacSU7s4Ulsuy27LRnoMDZj4xw8WgzOM5g/ZmxjTuBhAGqR3MYM1MLDJo1fC/vHZsa2ense/odtxZLb6onQcrbHEKjlMGOPRGIb48dthwlrkTxzphDol+OJPTceNkszbjvOw8ZMwC98Z9I3AkOs2p69P/ngx5/bqu352ZsfPsanBQUw84BJYpWDAOMPUlSPglEwCkbBiAEAeFVJpYwWVUIAAAAASUVORK5CYII=","orcid":"","institution":"W.F. 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