{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:45:46Z","timestamp":1759837546640},"reference-count":36,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In finite element approximation of the Oseen problem, one needs to handle two major difficulties, namely, the lack of stability due to convection dominance and the incompatibility between the approximating finite element spaces for the velocity and the pressure. These difficulties are addressed in this article by using an edge patch-wise local projection (EPLP) stabilization technique. The article analyses the EPLP stabilized nonconforming finite element methods for the Oseen problem. For approximating the velocity, the lowest-order Crouzeix\u2013Raviart (CR) nonconforming finite element space is considered; whereas for approximating the pressure, two discrete spaces are considered, namely, the piecewise constant polynomial space and the lowest-order CR finite element space. The proposed discrete weak formulation is a combination of the standard Galerkin method, EPLP stabilization and weakly imposed boundary condition by using Nitsche\u2019s technique. The resulting bilinear form satisfies an inf-sup condition with respect to EPLP norm, which leads to the well-posedness of the discrete problem. A priori error analysis assures the optimal order of convergence in both the cases, that is, order one in the case of piecewise constant approximation and <jats:inline-formula id=\"j_cmam-2018-0020_ineq_9999_w2aab3b7e2560b1b6b1aab1c14b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mfrac>\n                              <m:mn>3<\/m:mn>\n                              <m:mn>2<\/m:mn>\n                           <\/m:mfrac>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0020_eq_0374.png\" \/>\n                        <jats:tex-math>\\frac{3}{2}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> in the case of CR-finite element approximation for pressure. The numerical experiments illustrate the theoretical findings.<\/jats:p>","DOI":"10.1515\/cmam-2018-0020","type":"journal-article","created":{"date-parts":[[2019,3,31]],"date-time":"2019-03-31T12:04:02Z","timestamp":1554033842000},"page":"189-214","source":"Crossref","is-referenced-by-count":6,"title":["Edge Patch-Wise Local Projection Stabilized Nonconforming FEM for the Oseen Problem"],"prefix":"10.1515","volume":"19","author":[{"given":"Rahul","family":"Biswas","sequence":"first","affiliation":[{"name":"Department of Mathematics , Indian Institute of Science , Bangalore 560012 , India"}]},{"given":"Asha K.","family":"Dond","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Indian Institute of Science , Bangalore 560012 , India"}]},{"given":"Thirupathi","family":"Gudi","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Indian Institute of Science , Bangalore 560012 , India"}]}],"member":"374","published-online":{"date-parts":[[2018,6,21]]},"reference":[{"key":"2023033110021985249_j_cmam-2018-0020_ref_001_w2aab3b7e2560b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"D. 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Braack,\nA finite element pressure gradient stabilization for the Stokes equations based on local projections,\nCalcolo 38 (2001), no. 4, 173\u2013199.","DOI":"10.1007\/s10092-001-8180-4"},{"key":"2023033110021985249_j_cmam-2018-0020_ref_004_w2aab3b7e2560b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"M.  Braack and E.  Burman,\nLocal projection stabilization for the Oseen problem and its interpretation as a variational multiscale method,\nSIAM J. Numer. Anal. 43 (2006), no. 6, 2544\u20132566.","DOI":"10.1137\/050631227"},{"key":"2023033110021985249_j_cmam-2018-0020_ref_005_w2aab3b7e2560b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"M.  Braack, E.  Burman, V.  John and G.  Lube,\nStabilized finite element methods for the generalized Oseen problem,\nComput. Methods Appl. Mech. Engrg. 196 (2007), no. 4\u20136, 853\u2013866.","DOI":"10.1016\/j.cma.2006.07.011"},{"key":"2023033110021985249_j_cmam-2018-0020_ref_006_w2aab3b7e2560b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"J. H.  Bramble, J. E.  Pasciak and O.  Steinbach,\nOn the stability of the L2L^{2} projection in H1\u2062(\u03a9)H^{1}(\\Omega),\nMath. Comp. 71 (2002), no. 237, 147\u2013156.","DOI":"10.1090\/S0025-5718-01-01314-X"},{"key":"2023033110021985249_j_cmam-2018-0020_ref_007_w2aab3b7e2560b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L. R.  Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023033110021985249_j_cmam-2018-0020_ref_008_w2aab3b7e2560b1b6b1ab2b1b8Aa","doi-asserted-by":"crossref","unstructured":"E.  Burman,\nA unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty,\nSIAM J. 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J\u00e1nos Bolyai 59,\nNorth-Holland, Amsterdam (1991), 259\u2013266."},{"key":"2023033110021985249_j_cmam-2018-0020_ref_034_w2aab3b7e2560b1b6b1ab2b1c34Aa","doi-asserted-by":"crossref","unstructured":"L.  Tobiska and R.  Verf\u00fcrth,\nAnalysis of a streamline diffusion finite element method for the Stokes and Navier\u2013Stokes equations,\nSIAM J. Numer. Anal. 33 (1996), no. 1, 107\u2013127.","DOI":"10.1137\/0733007"},{"key":"2023033110021985249_j_cmam-2018-0020_ref_035_w2aab3b7e2560b1b6b1ab2b1c35Aa","doi-asserted-by":"crossref","unstructured":"S.  Turek and A.  Ouazzi,\nUnified edge-oriented stabilization of nonconforming FEM for incompressible flow problems: Numerical investigations,\nJ. Numer. Math. 15 (2007), no. 4, 299\u2013322.","DOI":"10.1515\/jnum.2007.014"},{"key":"2023033110021985249_j_cmam-2018-0020_ref_036_w2aab3b7e2560b1b6b1ab2b1c36Aa","unstructured":"J.  Volker,\nFinite Element Methods for Incompressible Flow Problems,\nSpringer Ser. Comput. 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