Hessian discretisation method for fourth-order semi-linear elliptic equations: applications to the von Kármán and Navier–Stokes models
This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element method...
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| description | This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth-order semi-linear elliptic equations with trilinear non-linearity. Four properties, namely, the coercivity, consistency, limit-conformity and compactness, enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications, namely, the Navier–Stokes equations in stream function vorticity formulation and the von Kármán equations of plate bending, are discussed. Results of numerical experiments are presented for the Morley and Adini ncFEMs, and the GR method. |
| doi_str_mv | 10.1007/s10444-020-09837-4 |
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The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth-order semi-linear elliptic equations with trilinear non-linearity. Four properties, namely, the coercivity, consistency, limit-conformity and compactness, enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications, namely, the Navier–Stokes equations in stream function vorticity formulation and the von Kármán equations of plate bending, are discussed. Results of numerical experiments are presented for the Morley and Adini ncFEMs, and the GR method.</description><identifier>ISSN: 1019-7168</identifier><identifier>EISSN: 1572-9044</identifier><identifier>DOI: 10.1007/s10444-020-09837-4</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Coercivity ; Computational fluid dynamics ; Computational mathematics ; Computational Mathematics and Numerical Analysis ; Computational Science and Engineering ; Convergence ; Discretization ; Elliptic functions ; Exact solutions ; Finite element method ; Fluid flow ; Linear functions ; Linearity ; Mathematical analysis ; Mathematical and Computational Biology ; Mathematical Modeling and Industrial Mathematics ; Mathematics ; Mathematics and Statistics ; Mathematics, Applied ; Navier-Stokes equations ; Nonlinearity ; Numerical methods ; Operators (mathematics) ; Physical Sciences ; Science & Technology ; Visualization ; Vorticity</subject><ispartof>Advances in computational mathematics, 2021-04, Vol.47 (2), Article 20</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature 2021</rights><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>true</woscitedreferencessubscribed><woscitedreferencescount>2</woscitedreferencescount><woscitedreferencesoriginalsourcerecordid>wos000620563500003</woscitedreferencesoriginalsourcerecordid><citedby>FETCH-LOGICAL-c319t-72fc0bcfd741527a811b03401450192fcec95c4ad089420eac3935033f63b4093</citedby><cites>FETCH-LOGICAL-c319t-72fc0bcfd741527a811b03401450192fcec95c4ad089420eac3935033f63b4093</cites><orcidid>0000-0002-3339-3053</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10444-020-09837-4$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10444-020-09837-4$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>315,782,786,27931,27932,39265,41495,42564,51326</link.rule.ids></links><search><creatorcontrib>Droniou, Jérome</creatorcontrib><creatorcontrib>Nataraj, Neela</creatorcontrib><creatorcontrib>Shylaja, Devika</creatorcontrib><title>Hessian discretisation method for fourth-order semi-linear elliptic equations: applications to the von Kármán and Navier–Stokes models</title><title>Advances in computational mathematics</title><addtitle>Adv Comput Math</addtitle><addtitle>ADV COMPUT MATH</addtitle><description>This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth-order semi-linear elliptic equations with trilinear non-linearity. Four properties, namely, the coercivity, consistency, limit-conformity and compactness, enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications, namely, the Navier–Stokes equations in stream function vorticity formulation and the von Kármán equations of plate bending, are discussed. Results of numerical experiments are presented for the Morley and Adini ncFEMs, and the GR method.</description><subject>Coercivity</subject><subject>Computational fluid dynamics</subject><subject>Computational mathematics</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Computational Science and Engineering</subject><subject>Convergence</subject><subject>Discretization</subject><subject>Elliptic functions</subject><subject>Exact solutions</subject><subject>Finite element method</subject><subject>Fluid flow</subject><subject>Linear functions</subject><subject>Linearity</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Biology</subject><subject>Mathematical Modeling and Industrial Mathematics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Mathematics, Applied</subject><subject>Navier-Stokes equations</subject><subject>Nonlinearity</subject><subject>Numerical methods</subject><subject>Operators (mathematics)</subject><subject>Physical Sciences</subject><subject>Science & Technology</subject><subject>Visualization</subject><subject>Vorticity</subject><issn>1019-7168</issn><issn>1572-9044</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><sourceid>HGBXW</sourceid><recordid>eNqNkEtuFDEQhlsIJELgAqwssUQO5Vd3mx0aAUFEsADWlsddzTh02x3bE8Qua7acIGfJTTgJzjSCHcKS5bLq_-vxNc1jBicMoHuWGUgpKXCgoHvRUXmnOWKq41TXxN0aA9O0Y21_v3mQ8zkA6LZTR833U8zZ20AGn13C4rMtPgYyY9nFgYwx1btPZUdjGjCRjLOnkw9oE8Fp8kvxjuDF_uDKz4ldlsm79UdKJGWH5LLWe3tzneab60BsGMg7e-kx_bz68aHEL5jJHAec8sPm3minjI9-v8fNp1cvP25O6dn71282L86oE0wX2vHRwdaNQyeZ4p3tGduCkMCkqjvWJDqtnLQD9FpyQOuEFgqEGFuxlaDFcfNkrbukeLHHXMx53TDUloZLDZwLpURV8VXlUsw54WiW5GebvhkG5pa5WZmbytwcmBtZTU9X01fcxjE7j8HhH2OF3nJQbZ2mntsW_f-rN74cqG7iPpRqFas1V3n4jOnvDv8Y7xcujqk9</recordid><startdate>20210401</startdate><enddate>20210401</enddate><creator>Droniou, Jérome</creator><creator>Nataraj, Neela</creator><creator>Shylaja, Devika</creator><general>Springer US</general><general>Springer Nature</general><general>Springer Nature B.V</general><scope>BLEPL</scope><scope>DTL</scope><scope>HGBXW</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-3339-3053</orcidid></search><sort><creationdate>20210401</creationdate><title>Hessian discretisation method for fourth-order semi-linear elliptic equations: applications to the von Kármán and Navier–Stokes models</title><author>Droniou, Jérome ; Nataraj, Neela ; Shylaja, Devika</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-72fc0bcfd741527a811b03401450192fcec95c4ad089420eac3935033f63b4093</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Coercivity</topic><topic>Computational fluid dynamics</topic><topic>Computational mathematics</topic><topic>Computational Mathematics and Numerical Analysis</topic><topic>Computational Science and Engineering</topic><topic>Convergence</topic><topic>Discretization</topic><topic>Elliptic functions</topic><topic>Exact solutions</topic><topic>Finite element method</topic><topic>Fluid flow</topic><topic>Linear functions</topic><topic>Linearity</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Biology</topic><topic>Mathematical Modeling and Industrial Mathematics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Mathematics, Applied</topic><topic>Navier-Stokes equations</topic><topic>Nonlinearity</topic><topic>Numerical methods</topic><topic>Operators (mathematics)</topic><topic>Physical Sciences</topic><topic>Science & Technology</topic><topic>Visualization</topic><topic>Vorticity</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Droniou, Jérome</creatorcontrib><creatorcontrib>Nataraj, Neela</creatorcontrib><creatorcontrib>Shylaja, Devika</creatorcontrib><collection>Web of Science Core Collection</collection><collection>Science Citation Index Expanded</collection><collection>Web of Science - Science Citation Index Expanded - 2021</collection><collection>CrossRef</collection><jtitle>Advances in computational mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Droniou, Jérome</au><au>Nataraj, Neela</au><au>Shylaja, Devika</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Hessian discretisation method for fourth-order semi-linear elliptic equations: applications to the von Kármán and Navier–Stokes models</atitle><jtitle>Advances in computational mathematics</jtitle><stitle>Adv Comput Math</stitle><stitle>ADV COMPUT MATH</stitle><date>2021-04-01</date><risdate>2021</risdate><volume>47</volume><issue>2</issue><artnum>20</artnum><issn>1019-7168</issn><eissn>1572-9044</eissn><abstract>This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth-order semi-linear elliptic equations with trilinear non-linearity. Four properties, namely, the coercivity, consistency, limit-conformity and compactness, enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications, namely, the Navier–Stokes equations in stream function vorticity formulation and the von Kármán equations of plate bending, are discussed. Results of numerical experiments are presented for the Morley and Adini ncFEMs, and the GR method.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10444-020-09837-4</doi><tpages>28</tpages><orcidid>https://orcid.org/0000-0002-3339-3053</orcidid></addata></record> |
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| subjects | Coercivity Computational fluid dynamics Computational mathematics Computational Mathematics and Numerical Analysis Computational Science and Engineering Convergence Discretization Elliptic functions Exact solutions Finite element method Fluid flow Linear functions Linearity Mathematical analysis Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Mathematics, Applied Navier-Stokes equations Nonlinearity Numerical methods Operators (mathematics) Physical Sciences Science & Technology Visualization Vorticity |
| title | Hessian discretisation method for fourth-order semi-linear elliptic equations: applications to the von Kármán and Navier–Stokes models |
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