{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T04:18:25Z","timestamp":1768969105794,"version":"3.49.0"},"reference-count":40,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071350"],"award-info":[{"award-number":["12071350"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003399","name":"Science and Technology Commission of Shanghai Municipality","doi-asserted-by":"publisher","award":["2021SHZDZX0100"],"award-info":[{"award-number":["2021SHZDZX0100"]}],"id":[{"id":"10.13039\/501100003399","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,10,1]]},"abstract":"Abstract<\/jats:title>\n We construct a \n \n \n \n C<\/m:mi>\n 1<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n C^{1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-\n \n \n \n P<\/m:mi>\n 7<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n P_{7}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> Bell finite element by restricting its normal derivative from a \n \n \n \n P<\/m:mi>\n 6<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n P_{6}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> polynomial to a \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> polynomial, and its second normal derivative from a \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> polynomial to a \n \n \n \n P<\/m:mi>\n 4<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n P_{4}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> polynomial, on the three edges of every triangle.\nOn one triangle, the finite element space contains the \n \n \n \n P<\/m:mi>\n 6<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n P_{6}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> polynomial space.\nWe show the method converges at order 7 in \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n L^{2}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-norm.\nBy eliminating all degrees of freedom on edges of \n \n \n \n C<\/m:mi>\n 1<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n C^{1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-\n \n \n \n P<\/m:mi>\n 7<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n P_{7}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> Argyris finite element, the global degrees of freedom of the new element are reduced substantially from \n \n \n \n 27<\/m:mn>\n \u2062<\/m:mo>\n V<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n 27V<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> to \n \n \n \n 12<\/m:mn>\n \u2062<\/m:mo>\n V<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n 12V<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> asymptotically, where \ud835\udc49 is the number of vertices in the triangular mesh.\nWhile the global degrees of freedom of the \n \n \n \n C<\/m:mi>\n 1<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n C^{1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-\n \n \n \n P<\/m:mi>\n 6<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n P_{6}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> Argyris finite element is \n \n \n \n 19<\/m:mn>\n \u2062<\/m:mo>\n V<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n 19V<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, the new element is equally accurate but more economic.\nNumerical tests are presented, showing the new element is more accurate than the existing element while having less global unknowns.<\/jats:p>","DOI":"10.1515\/cmam-2023-0068","type":"journal-article","created":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T19:12:09Z","timestamp":1700680329000},"page":"995-1000","source":"Crossref","is-referenced-by-count":2,"title":["A \ud835\udc361<\/sup>-\ud835\udc437<\/sub> Bell Finite Element on Triangle"],"prefix":"10.1515","volume":"24","author":[{"given":"Xuejun","family":"Xu","sequence":"first","affiliation":[{"name":"School of Mathematical Science , Tongji University , Shanghai , 200092; and Institute of Computational Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100190 , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2023,11,23]]},"reference":[{"key":"2024100217254070369_j_cmam-2023-0068_ref_001","doi-asserted-by":"crossref","unstructured":"J. H. Argyris, I. Fried and D. W. Scharpf,\nThe TUBA family of plate elements for the matrix displacement method,\nAeronautical J. Roy. Aeronautical Soc. 72 (1968), 514\u2013517.","DOI":"10.1017\/S0001924000084396"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_002","doi-asserted-by":"crossref","unstructured":"K. Bell,\nA refined triangular plate bending element,\nInternat. J. Numer. Methods Engrg. 1 (1969), 101\u2013122.","DOI":"10.1002\/nme.1620010108"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_003","unstructured":"F. K. Bogner, R. L. Fox and L. A. Schmit,\nThe generation of interelement compatible stiffness and mass matrices by the use of interpolation formulas,\nProceedings of the Conference on Matrix Methods in Structural Mechanics,\nAir Force Institute of Technology, Ohio (1965), 397\u2013444."},{"key":"2024100217254070369_j_cmam-2023-0068_ref_004","doi-asserted-by":"crossref","unstructured":"H. Chen, S. Chen and Z. Qiao,\n\n \n \n \n C\n 0\n \n \n \n C^{0}\n \n -nonconforming tetrahedral and cuboid elements for the three-dimensional fourth order elliptic problem,\nNumer. Math. 124 (2013), no. 1, 99\u2013119.","DOI":"10.1007\/s00211-012-0508-2"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_005","doi-asserted-by":"crossref","unstructured":"H.-R. Chen, S.-C. Chen and Z.-H. Qiao,\n\n \n \n \n C\n 0\n \n \n \n C^{0}\n \n -nonconforming triangular prism elements for the three-dimensional fourth order elliptic problem,\nJ. Sci. Comput. 55 (2013), no. 3, 645\u2013658.","DOI":"10.1007\/s10915-012-9652-1"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_006","unstructured":"R. W. Clough and J. L. Tocher,\nFinite element stiffness matrices for analysis of plates in bending,\nProceedings of the Conference on Matrix Methods in Structural Mechanics,\nAir Force Institute of Technology, Ohio (1965), 515\u2013545."},{"key":"2024100217254070369_j_cmam-2023-0068_ref_007","doi-asserted-by":"crossref","unstructured":"M. Cui and S. Zhang,\nOn the uniform convergence of the weak Galerkin finite element method for a singularly-perturbed biharmonic equation,\nJ. Sci. Comput. 82 (2020), no. 1, Paper No. 5","DOI":"10.1007\/s10915-019-01120-z"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_008","doi-asserted-by":"crossref","unstructured":"J. Douglas, Jr., T. Dupont, P. Percell and R. Scott,\nA family of \n \n \n \n C\n 1\n \n \n \n C^{1}\n \n finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems,\nRAIRO Anal. Num\u00e9r. 13 (1979), no. 3, 227\u2013255.","DOI":"10.1051\/m2an\/1979130302271"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_009","unstructured":"B. Fraeijs de Veubeke,\nA conforming finite element for plate bending,\nStress Analysis,\nWiley, New York (1965), 145\u2013197."},{"key":"2024100217254070369_j_cmam-2023-0068_ref_010","doi-asserted-by":"crossref","unstructured":"V. Girault and L. R. Scott,\nHermite interpolation of nonsmooth functions preserving boundary conditions,\nMath. Comp. 71 (2002), no. 239, 1043\u20131074.","DOI":"10.1090\/S0025-5718-02-01446-1"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_011","doi-asserted-by":"crossref","unstructured":"H. Han, Z. Huang and S. Zhang,\nAn iterative method based on equation decomposition for the fourth-order singular perturbation problem,\nNumer. Methods Partial Differential Equations 29 (2013), no. 3, 961\u2013978.","DOI":"10.1002\/num.21740"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_012","doi-asserted-by":"crossref","unstructured":"J. Hu, Y. Huang and S. Zhang,\nThe lowest order differentiable finite element on rectangular grids,\nSIAM J. Numer. Anal. 49 (2011), no. 4, 1350\u20131368.","DOI":"10.1137\/100806497"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_013","doi-asserted-by":"crossref","unstructured":"J. Hu and Z. Shi,\nA new a posteriori error estimate for the Morley element,\nNumer. Math. 112 (2009), no. 1, 25\u201340.","DOI":"10.1007\/s00211-008-0205-3"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_014","doi-asserted-by":"crossref","unstructured":"J. Hu and Z. Shi,\nA lower bound of the \n \n \n \n L\n 2\n \n \n \n L^{2}\n \n norm error estimate for the Adini element of the biharmonic equation,\nSIAM J. Numer. Anal. 51 (2013), no. 5, 2651\u20132659.","DOI":"10.1137\/130907136"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_015","doi-asserted-by":"crossref","unstructured":"J. Hu, Z. Shi and J. Xu,\nConvergence and optimality of the adaptive Morley element method,\nNumer. Math. 121 (2012), no. 4, 731\u2013752.","DOI":"10.1007\/s00211-012-0445-0"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_016","doi-asserted-by":"crossref","unstructured":"J. Hu, S. Tian and S. Zhang,\nA family of 3D \n \n \n \n H\n 2\n \n \n \n H^{2}\n \n -nonconforming tetrahedral finite elements for the biharmonic equation,\nSci. China Math. 63 (2020), no. 8, 1505\u20131522.","DOI":"10.1007\/s11425-019-1661-8"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_017","doi-asserted-by":"crossref","unstructured":"J. Hu and S. Zhang,\nThe minimal conforming \n \n \n \n H\n k\n \n \n \n H^{k}\n \n finite element spaces on \n \n \n \n R\n n\n \n \n \n R^{n}\n \n rectangular grids,\nMath. Comp. 84 (2015), no. 292, 563\u2013579.","DOI":"10.1090\/S0025-5718-2014-02871-8"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_018","doi-asserted-by":"crossref","unstructured":"J. Hu and S. Zhang,\nA cubic \n \n \n \n H\n 3\n \n \n \n H^{3}\n \n -nonconforming finite element,\nCommun. Appl. Math. Comput. 1 (2019), no. 1, 81\u2013100.","DOI":"10.1007\/s42967-019-0009-8"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_019","doi-asserted-by":"crossref","unstructured":"J. Hu and S. Zhang,\nAn error analysis method SPP-BEAM and a construction guideline of nonconforming finite elements for fourth order elliptic problems,\nJ. Comput. Math. 38 (2020), no. 1, 195\u2013222.","DOI":"10.4208\/jcm.1811-m2018-0162"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_020","doi-asserted-by":"crossref","unstructured":"J. Huang, X. Huang and S. Zhang,\nA superconvergence of the Morley element via postprocessing,\nRecent Advances in Scientific Computing and Applications,\nContemp. Math. 586,\nAmerican Mathematical Society, Providence (2013), 189\u2013196.","DOI":"10.1090\/conm\/586\/11640"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_021","doi-asserted-by":"crossref","unstructured":"M.-J. Lai and L. L. Schumaker,\nSpline Functions on Triangulations,\nEncyclopedia Math. Appl. 110,\nCambridge University, Cambridge, 2007.","DOI":"10.1017\/CBO9780511721588"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_022","doi-asserted-by":"crossref","unstructured":"P. Lascaux and P. Lesaint,\nSome nonconforming finite elements for the plate bending problem,\nRev. Fran\u00e7aise Automat. Inform. Rech. Op\u00e9r. S\u00e9r. Rouge Anal. Num\u00e9r. 9 (1975), 9\u201353.","DOI":"10.1051\/m2an\/197509R100091"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_023","doi-asserted-by":"crossref","unstructured":"J. Morgan and R. Scott,\nA nodal basis for \n \n \n \n C\n 1\n \n \n \n C^{1}\n \n piecewise polynomials of degree \n \n \n \n n\n \u2265\n 5\n \n \n \n n\\geq 5\n \n ,\nMath. Comput. 29 (1975), 736\u2013740.","DOI":"10.1090\/S0025-5718-1975-0375740-7"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_024","doi-asserted-by":"crossref","unstructured":"L. Morley,\nThe triangular equilibrium element in the solution of plate bending problems,\nAero. Quart. 19 (1968), 149\u2013169.","DOI":"10.1017\/S0001925900004546"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_025","doi-asserted-by":"crossref","unstructured":"L. Mu, J. Wang, X. Ye and S. Zhang,\nA \n \n \n \n C\n 0\n \n \n \n C^{0}\n \n -weak Galerkin finite element method for the biharmonic equation,\nJ. Sci. Comput. 59 (2014), no. 2, 473\u2013495.","DOI":"10.1007\/s10915-013-9770-4"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_026","doi-asserted-by":"crossref","unstructured":"L. Mu, X. Ye and S. Zhang,\nDevelopment of a \n \n \n \n P\n 2\n \n \n \n P_{2}\n \n element with optimal \n \n \n \n L\n 2\n \n \n \n L^{2}\n \n convergence for biharmonic equation,\nNumer. Methods Partial Differential Equations 35 (2019), no. 4, 1497\u20131508.","DOI":"10.1002\/num.22361"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_027","doi-asserted-by":"crossref","unstructured":"P. Percell,\nOn cubic and quartic Clough\u2013Tocher finite elements,\nSIAM J. Numer. Anal. 13 (1976), no. 1, 100\u2013103.","DOI":"10.1137\/0713011"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_028","doi-asserted-by":"crossref","unstructured":"M. J. D. Powell and M. A. Sabin,\nPiecewise quadratic approximations on triangles,\nACM Trans. Math. Software 3 (1977), no. 4, 316\u2013325.","DOI":"10.1145\/355759.355761"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_029","unstructured":"G. Sander,\nBornes sup\u00e9rieures et inf\u00e9rieures dans l\u2019analyse matricielle des plaques en flexion-torsion,\nBull. Soc. Roy. Sci. Li\u00e8ge 33 (1964), 456\u2013494."},{"key":"2024100217254070369_j_cmam-2023-0068_ref_030","doi-asserted-by":"crossref","unstructured":"L. R. Scott and S. Zhang,\nFinite element interpolation of nonsmooth functions satisfying boundary conditions,\nMath. Comp. 54 (1990), no. 190, 483\u2013493.","DOI":"10.1090\/S0025-5718-1990-1011446-7"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_031","doi-asserted-by":"crossref","unstructured":"M. Wang and J. Xu,\nThe Morley element for fourth order elliptic equations in any dimensions,\nNumer. Math. 103 (2006), no. 1, 155\u2013169.","DOI":"10.1007\/s00211-005-0662-x"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_032","doi-asserted-by":"crossref","unstructured":"M. Wang and J. Xu,\nMinimal finite element spaces for \n \n \n \n 2\n \u2062\n m\n \n \n \n 2m\n \n -th-order partial differential equations in \n \n \n \n R\n n\n \n \n \n R^{n}\n \n ,\nMath. Comp. 82 (2013), no. 281, 25\u201343.","DOI":"10.1090\/S0025-5718-2012-02611-1"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_033","doi-asserted-by":"crossref","unstructured":"X. Ye and S. Zhang,\nA stabilizer free weak Galerkin method for the biharmonic equation on polytopal meshes,\nSIAM J. Numer. Anal. 58 (2020), no. 5, 2572\u20132588.","DOI":"10.1137\/19M1276601"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_034","unstructured":"X. Ye, S. Zhang and Z. Zhang,\nA new \n \n \n \n P\n 1\n \n \n \n P_{1}\n \n weak Galerkin method for the biharmonic equation,\nJ. Comput. Appl. Math. 364 (2020), Article ID 12337."},{"key":"2024100217254070369_j_cmam-2023-0068_ref_035","doi-asserted-by":"crossref","unstructured":"A. \u017den\u00ed\u0161ek,\nPolynomial approximation on tetrahedrons in the finite element method,\nJ. Approximation Theory 7 (1973), 334\u2013351.","DOI":"10.1016\/0021-9045(73)90036-1"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_036","doi-asserted-by":"crossref","unstructured":"A. \u017den\u00ed\u0161ek,\nA general theorem on triangular finite \n \n \n \n C\n \n (\n m\n )\n \n \n \n \n C^{(m)}\n \n -elements,\nRev. Fran\u00e7aise Automat. Inform. Rech. Op\u00e9r. S\u00e9r. Rouge Anal. Num\u00e9r. 8 (1974), 119\u2013127.","DOI":"10.1051\/m2an\/197408R201191"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_037","doi-asserted-by":"crossref","unstructured":"S. Zhang,\nAn optimal order multigrid method for biharmonic, \n \n \n \n C\n 1\n \n \n \n C^{1}\n \n finite element equations,\nNumer. Math. 56 (1989), no. 6, 613\u2013624.","DOI":"10.1007\/BF01396347"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_038","doi-asserted-by":"crossref","unstructured":"S. Zhang,\nA \n \n \n \n C\n 1\n \n \n \n C_{1}\n \n -\n \n \n \n P\n 2\n \n \n \n P_{2}\n \n finite element without nodal basis,\nM2AN Math. Model. Numer. Anal. 42 (2008), no. 2, 175\u2013192.","DOI":"10.1051\/m2an:2008002"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_039","doi-asserted-by":"crossref","unstructured":"S. Zhang,\nA family of 3D continuously differentiable finite elements on tetrahedral grids,\nAppl. Numer. Math. 59 (2009), no. 1, 219\u2013233.","DOI":"10.1016\/j.apnum.2008.02.002"},{"key":"2024100217254070369_j_cmam-2023-0068_ref_040","unstructured":"S. Y. Zhang,\nA family of differentiable finite elements on simplicial grids in four space dimensions,\nMath. Numer. Sin. 38 (2016), no. 3, 309\u2013324."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0068\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0068\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,2]],"date-time":"2024-10-02T17:26:12Z","timestamp":1727889972000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0068\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,23]]},"references-count":40,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,1,2]]},"published-print":{"date-parts":[[2024,10,1]]}},"alternative-id":["10.1515\/cmam-2023-0068"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2023-0068","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,23]]}}}