{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,29]],"date-time":"2025-03-29T10:40:14Z","timestamp":1743244814593,"version":"3.40.3"},"reference-count":36,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100001843","name":"Science and Engineering Research Board","doi-asserted-by":"publisher","award":["CRG\/2021\/00827"],"award-info":[{"award-number":["CRG\/2021\/00827"]}],"id":[{"id":"10.13039\/501100001843","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001412","name":"Council of Scientific and Industrial Research, India","doi-asserted-by":"publisher","award":["09\/1290(0001)\/2019-EMR-I"],"award-info":[{"award-number":["09\/1290(0001)\/2019-EMR-I"]}],"id":[{"id":"10.13039\/501100001412","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations.\nThis method utilizes an upwind approach to handle the nonlinear convective terms effectively.\nWe discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations.\nFurthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi mathvariant=\"bold\">L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2023-0202_ineq_0001.png\"\/>\n                        <jats:tex-math>\\mathbf{L}^{2}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and energy norms, the temperature approximation in <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2023-0202_ineq_0002.png\"\/>\n                        <jats:tex-math>L^{2}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and energy norms and pressure approximation in <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2023-0202_ineq_0002.png\"\/>\n                        <jats:tex-math>L^{2}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-norm for <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>t<\/m:mi>\n                              <m:mo>&gt;<\/m:mo>\n                              <m:mn>0<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2023-0202_ineq_0004.png\"\/>\n                        <jats:tex-math>t&gt;0<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.\nAdditionally, under the smallness assumption on the data, we prove uniform in time error estimates.\nWe also consider a backward Euler scheme for full discretization and derive fully discrete error estimates.\nFinally, we provide numerical examples to support the theoretical conclusions.<\/jats:p>","DOI":"10.1515\/cmam-2023-0202","type":"journal-article","created":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T09:27:59Z","timestamp":1720603679000},"page":"313-348","source":"Crossref","is-referenced-by-count":0,"title":["On Error Estimates of a Discontinuous Galerkin Method of the Boussinesq System of Equations"],"prefix":"10.1515","volume":"25","author":[{"given":"Saumya","family":"Bajpai","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science , 521180 Indian Institute of Technology Goa , Ponda , Goa-403401 , India"}]},{"given":"Debendra Kumar","family":"Swain","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science , 521180 Indian Institute of Technology Goa , Ponda , Goa-403401 , India"}]}],"member":"374","published-online":{"date-parts":[[2024,7,11]]},"reference":[{"unstructured":"R. A. Adams,\nSobolev Spaces,\nPure Appl. Math. 65,\nAcademic Press, New York, 1975.","key":"2025032910202826952_j_cmam-2023-0202_ref_001"},{"doi-asserted-by":"crossref","unstructured":"D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini,\nUnified analysis of discontinuous Galerkin methods for elliptic problems,\nSIAM J. Numer. Anal. 39 (2001\/02), no. 5, 1749\u20131779.","key":"2025032910202826952_j_cmam-2023-0202_ref_002","DOI":"10.1137\/S0036142901384162"},{"doi-asserted-by":"crossref","unstructured":"S. Bajpai and D. K. Swain,\nA priori error estimates of a three-step two-level finite element Galerkin method for a 2D-Boussinesq system of equations,\nComput. Math. Appl. 146 (2023), 137\u2013164.","key":"2025032910202826952_j_cmam-2023-0202_ref_003","DOI":"10.1016\/j.camwa.2023.06.025"},{"unstructured":"H. B\u00e9nard,\nLes tourbillons cellulaires dans une nappe liquide,\nRev. Gen. Sci. Pure Appl. 11 (1900), 1261\u20131271, 1309\u20131328.","key":"2025032910202826952_j_cmam-2023-0202_ref_004"},{"unstructured":"H. B\u00e9nard,\nLes tourbillons cellularies dans une nappe liquide transportant de la chaleur par convection en r\u00e9gime permanent,\nAnn. Chim. Phys. 23 (1901), 62\u2013144.","key":"2025032910202826952_j_cmam-2023-0202_ref_005"},{"doi-asserted-by":"crossref","unstructured":"J. Boland and W. Layton,\nAn analysis of the finite element method for natural convection problems,\nNumer. Methods Partial Differential Equations 6 (1990), no. 2, 115\u2013126.","key":"2025032910202826952_j_cmam-2023-0202_ref_006","DOI":"10.1002\/num.1690060202"},{"doi-asserted-by":"crossref","unstructured":"J. A. Burns, X. M. He and W. Hu,\nControl of the Boussinesq equations with implications for sensor location in energy efficient buildings,\nProceedings of the 2012 American Control Conference,\nIEEE Press, Piscataway (2012), 2232\u20132237.","key":"2025032910202826952_j_cmam-2023-0202_ref_007","DOI":"10.1109\/ACC.2012.6315623"},{"doi-asserted-by":"crossref","unstructured":"N. Chaabane, V. Girault, C. Puelz and B. Riviere,\nConvergence of IPDG for coupled time-dependent Navier\u2013Stokes and Darcy equations,\nJ. Comput. Appl. Math. 324 (2017), 25\u201348.","key":"2025032910202826952_j_cmam-2023-0202_ref_008","DOI":"10.1016\/j.cam.2017.04.002"},{"unstructured":"B. Cockburn and C.-W. Shu,\nRunge\u2013Kutta discontinuous Galerkin methods for convection-dominated problems,\nJ. Sci. Comput. 16 (2001), no. 3, 173\u2013261.","key":"2025032910202826952_j_cmam-2023-0202_ref_009"},{"doi-asserted-by":"crossref","unstructured":"E. Colmenares, R. Oyarz\u00faa and F. 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Paquet,\nA refined mixed finite element method for the Boussinesq equations in polygonal domains,\nIMA J. Numer. Anal. 21 (2001), no. 2, 525\u2013551.","key":"2025032910202826952_j_cmam-2023-0202_ref_013","DOI":"10.1093\/imanum\/21.2.525"},{"doi-asserted-by":"crossref","unstructured":"V. Girault and B. Rivi\u00e8re,\nDG approximation of coupled Navier\u2013Stokes and Darcy equations by Beaver\u2013Joseph\u2013Saffman interface condition,\nSIAM J. Numer. Anal. 47 (2009), no. 3, 2052\u20132089.","key":"2025032910202826952_j_cmam-2023-0202_ref_014","DOI":"10.1137\/070686081"},{"doi-asserted-by":"crossref","unstructured":"V. Girault, B. Rivi\u00e8re and M. F. Wheeler,\nA discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier\u2013Stokes problems,\nMath. Comp. 74 (2005), no. 249, 53\u201384.","key":"2025032910202826952_j_cmam-2023-0202_ref_015","DOI":"10.1090\/S0025-5718-04-01652-7"},{"doi-asserted-by":"crossref","unstructured":"V. Girault, B. 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Huang,\nDecoupled two level finite element methods for the steady natural convection problem,\nNumer. Algorithms 68 (2015), no. 4, 837\u2013866.","key":"2025032910202826952_j_cmam-2023-0202_ref_036","DOI":"10.1007\/s11075-014-9874-4"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0202\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0202\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,29]],"date-time":"2025-03-29T10:24:49Z","timestamp":1743243889000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0202\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,11]]},"references-count":36,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,11,14]]},"published-print":{"date-parts":[[2025,4,1]]}},"alternative-id":["10.1515\/cmam-2023-0202"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2023-0202","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2024,7,11]]}}}