{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T16:48:30Z","timestamp":1764175710656,"version":"3.40.5"},"reference-count":34,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2020 YFA 0713500"],"award-info":[{"award-number":["2020 YFA 0713500"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971410","12071350","12331015"],"award-info":[{"award-number":["11971410","12071350","12331015"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,4,1]]},"abstract":"Abstract<\/jats:title>\n The P2 BDM H-div finite element is enriched by five \n \n \n \n P<\/m:mi>\n 5<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n P_{5}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> divergence-free bubbles on each triangle. The resulting finite element remains H-div and is also H-curl nonconforming. Thus the new finite element combined with the discontinuous \n \n \n \n P<\/m:mi>\n 1<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n P_{1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> element for pressure, is stabilizer-free and still divergence-free in solving the Stokes equations on triangular meshes. The stabilization can also be done by three, four, or six \n \n \n \n P<\/m:mi>\n 5<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n P_{5}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> divergence-free bubble-enrichment each triangle, shown both in theory and in computation. Numerical tests show the advantage of the divergence-free element over the \n \n \n \n P<\/m:mi>\n 2<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n P_{2}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> Taylor\u2013Hood finite element.<\/jats:p>","DOI":"10.1515\/cmam-2024-0060","type":"journal-article","created":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T17:53:13Z","timestamp":1727805193000},"page":"459-473","source":"Crossref","is-referenced-by-count":2,"title":["A P<\/i>\n 2<\/sub> H-Div-Nonconforming-H-Curl Finite Element for the Stokes Equations on Triangular Meshes"],"prefix":"10.1515","volume":"25","author":[{"given":"Yunqing","family":"Huang","sequence":"first","affiliation":[{"name":"School of Mathematics and Computational Science , Xiangtan University , Xiangtan , Hunan 411105 , P. R. China"}]},{"given":"Xuejun","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematical Science , Tongji University ; and Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University , Shanghai 200092 , P. R. China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1114-4179","authenticated-orcid":false,"given":"Shangyou","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences , University of Delaware , Newark , DE 19716 , USA"}]}],"member":"374","published-online":{"date-parts":[[2024,10,2]]},"reference":[{"key":"2025032910202893871_j_cmam-2024-0060_ref_001","unstructured":"D. N. Arnold and J. Qin,\nQuadratic velocity\/linear pressure Stokes elements,\nAdvances in Computer Methods for Partial Differential Equations VII, (1992)."},{"key":"2025032910202893871_j_cmam-2024-0060_ref_002","doi-asserted-by":"crossref","unstructured":"C. Bacuta, P. S. Vassilevski and S. Zhang,\nA new approach for solving Stokes systems arising from a distributive relaxation method,\nNumer. Methods Partial Differential Equations 27 (2011), no. 4, 898\u2013914.","DOI":"10.1002\/num.20560"},{"key":"2025032910202893871_j_cmam-2024-0060_ref_003","doi-asserted-by":"crossref","unstructured":"M. Fabien, J. Guzm\u00e1n, M. Neilan and A. Zytoon,\nLow-order divergence-free approximations for the Stokes problem on Worsey\u2013Farin and Powell\u2013Sabin splits,\nComput. Methods Appl. Mech. Engrg. 390 (2022), Paper No. 114444.","DOI":"10.1016\/j.cma.2021.114444"},{"key":"2025032910202893871_j_cmam-2024-0060_ref_004","doi-asserted-by":"crossref","unstructured":"R. S. Falk and M. Neilan,\nStokes complexes and the construction of stable finite elements with pointwise mass conservation,\nSIAM J. Numer. Anal. 51 (2013), no. 2, 1308\u20131326.","DOI":"10.1137\/120888132"},{"key":"2025032910202893871_j_cmam-2024-0060_ref_005","doi-asserted-by":"crossref","unstructured":"G. Fu, J. 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