{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T16:44:06Z","timestamp":1771605846941,"version":"3.50.1"},"reference-count":35,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100001412","name":"Council of Scientific and Industrial Research, India","doi-asserted-by":"publisher","award":["09\/796(0095)\/2019-EMR-I"],"award-info":[{"award-number":["09\/796(0095)\/2019-EMR-I"]}],"id":[{"id":"10.13039\/501100001412","id-type":"DOI","asserted-by":"publisher"}]},{"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"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier\u2013Stokes model.\nThe two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size \ud835\udc3b, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size \u210e and solving the resulting system to produce an approximate solution with desired accuracy.\nWe establish optimal error estimates of the two-grid DG approximations for the velocity and pressure in energy and <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-2023-0035_ineq_0001.png\"\/>\n                        <jats:tex-math>L^{2}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-norms, respectively, for an appropriate choice of coarse and fine mesh parameters.\nWe further discretize the two-grid DG model in time, using the backward Euler method, and derive the fully discrete error estimates.\nFinally, numerical results are presented to confirm the efficiency of the proposed scheme.<\/jats:p>","DOI":"10.1515\/cmam-2023-0035","type":"journal-article","created":{"date-parts":[[2023,11,24]],"date-time":"2023-11-24T18:37:25Z","timestamp":1700851045000},"page":"935-966","source":"Crossref","is-referenced-by-count":1,"title":["Discontinuous Galerkin Two-Grid Method for the Transient Navier\u2013Stokes Equations"],"prefix":"10.1515","volume":"24","author":[{"given":"Kallol","family":"Ray","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences , Tezpur University , Tezpur , Sonitpur, Assam-784028 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Deepjyoti","family":"Goswami","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences , Tezpur University , Tezpur , Sonitpur, Assam-784028 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saumya","family":"Bajpai","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science , Indian Institute of Technology Goa , Ponda , Goa - 403401 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2023,11,25]]},"reference":[{"key":"2024100217254161564_j_cmam-2023-0035_ref_001","doi-asserted-by":"crossref","unstructured":"H. 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Math. 25 (2017), no. 4, 199\u2013228.","DOI":"10.1515\/jnma-2016-1055"},{"key":"2024100217254161564_j_cmam-2023-0035_ref_005","doi-asserted-by":"crossref","unstructured":"C. Bi and V. Ginting,\nTwo-grid discontinuous Galerkin method for quasi-linear elliptic problems,\nJ. Sci. Comput. 49 (2011), no. 3, 311\u2013331.","DOI":"10.1007\/s10915-011-9463-9"},{"key":"2024100217254161564_j_cmam-2023-0035_ref_006","doi-asserted-by":"crossref","unstructured":"C. Bi, C. Wang and Y. Lin,\nPointwise error estimates and two-grid algorithms of discontinuous Galerkin method for strongly nonlinear elliptic problems,\nJ. Sci. Comput. 67 (2016), no. 1, 153\u2013175.","DOI":"10.1007\/s10915-015-0072-x"},{"key":"2024100217254161564_j_cmam-2023-0035_ref_007","doi-asserted-by":"crossref","unstructured":"A. Cesmelioglu, B. Cockburn and W. Qiu,\nAnalysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier\u2013Stokes equations,\nMath. 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