{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,1,4]],"date-time":"2025-01-04T05:35:38Z","timestamp":1735968938051,"version":"3.32.0"},"reference-count":29,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This paper develops and analyses a semi-discrete numerical method for the two-dimensional Vlasov\u2013Stokes system with periodic boundary condition.\nThe method is based on the coupling of the semi-discrete discontinuous Galerkin method for the Vlasov equation with discontinuous Galerkin scheme for the stationary incompressible Stokes equation.\nThe proposed method is both mass and momentum conservative.\nSince it is difficult to establish non-negativity of the discrete local density, the generalized discrete Stokes operator become non-coercive and indefinite, and under the smallness condition on the discretization parameter, optimal error estimates are established with help of a modified the Stokes projection to deal with the Stokes part and, with the help of a special projection, to tackle the Vlasov part.\nFinally, numerical experiments based on the dG method combined with a splitting algorithm are performed.<\/jats:p>","DOI":"10.1515\/cmam-2023-0243","type":"journal-article","created":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T11:24:57Z","timestamp":1706613897000},"page":"93-113","source":"Crossref","is-referenced-by-count":0,"title":["Discontinuous Galerkin Methods for the Vlasov\u2013Stokes System"],"prefix":"10.1515","volume":"25","author":[{"given":"Harsha","family":"Hutridurga","sequence":"first","affiliation":[{"name":"Department of Mathematics , 29491 Indian Institute of Technology Bombay , Powai , Mumbai , India"}]},{"given":"Krishan","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics , 29491 Indian Institute of Technology Bombay , Powai , Mumbai , India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0477-9272","authenticated-orcid":false,"given":"Amiya K.","family":"Pani","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Birla Institute of Technology and Science, Pilani , K.K. Birla Goa Campus, NH 17 B, Zuarinagar , Goa , India"}]}],"member":"374","published-online":{"date-parts":[[2024,1,30]]},"reference":[{"key":"2025010318340231985_j_cmam-2023-0243_ref_001","doi-asserted-by":"crossref","unstructured":"S. Agmon,\nLectures on Elliptic Boundary Value Problems,\nAmerican Mathematical Society, Providence, 2010.","DOI":"10.1090\/chel\/369"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_002","doi-asserted-by":"crossref","unstructured":"C. Amrouche and V. Girault,\nOn the existence and regularity of the solution of Stokes problem in arbitrary dimension,\nProc. Japan Acad. Ser. A Math. Sci. 67 (1991), no. 5, 171\u2013175.","DOI":"10.3792\/pjaa.67.171"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_003","doi-asserted-by":"crossref","unstructured":"D. N. Arnold,\nAn interior penalty finite element method with discontinuous elements,\nSIAM J. Numer. Anal. 19 (1982), no. 4, 742\u2013760.","DOI":"10.1137\/0719052"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_004","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.","DOI":"10.1137\/S0036142901384162"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_005","doi-asserted-by":"crossref","unstructured":"B. Ayuso, J. A. Carrillo and C.-W. Shu,\nDiscontinuous Galerkin methods for the one-dimensional Vlasov\u2013Poisson system,\nKinet. Relat. Models 4 (2011), no. 4, 955\u2013989.","DOI":"10.3934\/krm.2011.4.955"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_006","doi-asserted-by":"crossref","unstructured":"C. Baranger, L. Boudin, P.-E. Jabin and S. Mancini,\nA modeling of biospray for the upper airways,\nCEMRACS 2004\u2014Mathematics and Applications to Biology and Medicine,\nESAIM Proc. 14,\nEDP Sciences, Les Ulis (2005), 41\u201347.","DOI":"10.1051\/proc:2005004"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_007","doi-asserted-by":"crossref","unstructured":"F. Brezzi, L. D. Marini and E. S\u00fcli,\nDiscontinuous Galerkin methods for first-order hyperbolic problems,\nMath. Models Methods Appl. Sci. 14 (2004), no. 12, 1893\u20131903.","DOI":"10.1142\/S0218202504003866"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_008","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nClassics Appl. Math. 40,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2002.","DOI":"10.1137\/1.9780898719208"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_009","doi-asserted-by":"crossref","unstructured":"B. Cockburn, G. Kanschat, I. Perugia and D. Sch\u00f6tzau,\nSuperconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids,\nSIAM J. Numer. Anal. 39 (2001), no. 1, 264\u2013285.","DOI":"10.1137\/S0036142900371544"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_010","doi-asserted-by":"crossref","unstructured":"M. Crouzeix and V. 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Comp. 48 (1987), no. 178, 521\u2013532.","DOI":"10.1090\/S0025-5718-1987-0878688-2"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_011","doi-asserted-by":"crossref","unstructured":"B. A. de Dios, J. A. Carrillo and C.-W. Shu,\nDiscontinuous Galerkin methods for the multi-dimensional Vlasov\u2013Poisson problem,\nMath. Models Methods Appl. Sci. 22 (2012), no. 12, Article ID 1250042.","DOI":"10.1142\/S021820251250042X"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_012","doi-asserted-by":"crossref","unstructured":"D. A. Di Pietro and A. Ern,\nMathematical Aspects of Discontinuous Galerkin Methods,\nMath. Appl. (Berlin) 69,\nSpringer, Heidelberg, 2012.","DOI":"10.1007\/978-3-642-22980-0"},{"key":"2025010318340231985_j_cmam-2023-0243_ref_013","doi-asserted-by":"crossref","unstructured":"I. Gasser, P.-E. Jabin and B. Perthame,\nRegularity and propagation of moments in some nonlinear Vlasov systems,\nProc. Roy. Soc. Edinburgh Sect. 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