{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T14:12:00Z","timestamp":1778595120668,"version":"3.51.4"},"reference-count":49,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We investigate the discretization of <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>H<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:mi>curl<\/m:mi>\n                                 <m:mo stretchy=\"false\">)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0053_ineq_0001.png\"\/>\n                        <jats:tex-math>H(\\mathrm{curl})<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>H<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:mi>div<\/m:mi>\n                                 <m:mo stretchy=\"false\">)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0053_ineq_0002.png\"\/>\n                        <jats:tex-math>H(\\mathrm{div})<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> in two and three space dimensions by partially discontinuous nodal finite elements, i.e., vector-valued Lagrange finite elements with discontinuity in certain directions.\nThese spaces can be implemented as a combination of continuous and discontinuous Lagrange elements and fit in de Rham complexes.\nWe construct well-conditioned nodal bases.<\/jats:p>","DOI":"10.1515\/cmam-2022-0053","type":"journal-article","created":{"date-parts":[[2022,6,8]],"date-time":"2022-06-08T10:20:54Z","timestamp":1654683654000},"page":"613-629","source":"Crossref","is-referenced-by-count":3,"title":["Partially Discontinuous Nodal Finite Elements for \ud835\udc3b(curl) and \ud835\udc3b(div)"],"prefix":"10.1515","volume":"22","author":[{"given":"Jun","family":"Hu","sequence":"first","affiliation":[{"name":"LMAM and School of Mathematical Sciences , Peking University , Beijing 100871 , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kaibo","family":"Hu","sequence":"additional","affiliation":[{"name":"Mathematical Institute , University of Oxford , Andrew Wiles Building, Radcliffe Observatory Quarter , Oxford , OX2 6GG , United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1503-193X","authenticated-orcid":false,"given":"Qian","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences , Michigan Technological University , Houghton , MI 49931 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2022,6,9]]},"reference":[{"key":"2023033112440969372_j_cmam-2022-0053_ref_001","doi-asserted-by":"crossref","unstructured":"M. Ainsworth and J. Coyle,\nHierarchic \n                  \n                     \n                        \n                           h\n                           \u2062\n                           p\n                        \n                     \n                     \n                     hp\n                  \n               -edge element families for Maxwell\u2019s equations on hybrid quadrilateral\/triangular meshes,\nComput. Methods Appl. Mech. Engrg. 190 (2001), no. 49\u201350, 6709\u20136733.","DOI":"10.1016\/S0045-7825(01)00259-6"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_002","doi-asserted-by":"crossref","unstructured":"M. Ainsworth and S. Jiang,\nPreconditioning the mass matrix for high order finite element approximation on tetrahedra,\nSIAM J. Sci. Comput. 43 (2021), no. 1, A384\u2013A414.","DOI":"10.1137\/20M1333018"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_003","doi-asserted-by":"crossref","unstructured":"D. N. Arnold,\nFinite Element Exterior Calculus,\nCBMS-NSF Regional Conf. Ser. in Appl. Math. 93,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2018.","DOI":"10.1137\/1.9781611975543"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_004","doi-asserted-by":"crossref","unstructured":"D. N. Arnold, R. S. Falk and R. Winther,\nDifferential complexes and stability of finite element methods. II. The elasticity complex,\nCompatible Spatial Discretizations,\nIMA Vol. Math. Appl. 142,\nSpringer, New York (2006), 47\u201367.","DOI":"10.1007\/0-387-38034-5_3"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_005","doi-asserted-by":"crossref","unstructured":"D. N. Arnold, R. S. Falk and R. Winther,\nFinite element exterior calculus, homological techniques, and applications,\nActa Numer. 15 (2006), 1\u2013155.","DOI":"10.1017\/S0962492906210018"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_006","doi-asserted-by":"crossref","unstructured":"D. N. Arnold and K. Hu,\nComplexes from complexes,\nFound. Comput. Math. 21 (2021), no. 6, 1739\u20131774.","DOI":"10.1007\/s10208-021-09498-9"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_007","unstructured":"D. N. Arnold and J. Qin,\nQuadratic velocity\/linear pressure Stokes elements,\nAdv. Comput. Methods Partial Differ. Equ. 7 (1992), 28\u201334."},{"key":"2023033112440969372_j_cmam-2022-0053_ref_008","doi-asserted-by":"crossref","unstructured":"S. Badia and R. Codina,\nA nodal-based finite element approximation of the Maxwell problem suitable for singular solutions,\nSIAM J. Numer. Anal. 50 (2012), no. 2, 398\u2013417.","DOI":"10.1137\/110835360"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_009","doi-asserted-by":"crossref","unstructured":"D. Boffi, F. Brezzi and M. Fortin,\nMixed Finite Element Methods and Applications,\nSpringer Ser. Comput. Math. 44,\nSpringer, Heidelberg, 2013.","DOI":"10.1007\/978-3-642-36519-5"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_010","doi-asserted-by":"crossref","unstructured":"D. Boffi, J. Guzman and M. Neilan,\nConvergence of Lagrange finite elements for the Maxwell eigenvalue problem in two dimensions,\nIMA J. Numer. Anal. (2022), 10.1093\/imanum\/drab104.","DOI":"10.1093\/imanum\/drab104"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_011","doi-asserted-by":"crossref","unstructured":"A. Bossavit,\nWhitney forms: A class of finite elements for three-dimensional computations in electromagnetism,\nIEE Proc. A 135 (1988), no. 8, 493\u2013500.","DOI":"10.1049\/ip-a-1.1988.0077"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_012","doi-asserted-by":"crossref","unstructured":"A. Bossavit,\nSolving Maxwell equations in a closed cavity, and the question of \u201cspurious modes\u201d,\nIEEE Trans. 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Hu,\nFinite element approximations of symmetric tensors on simplicial grids in \n                  \n                     \n                        \n                           R\n                           n\n                        \n                     \n                     \n                     \\mathbb{R}^{n}\n                  \n               : The higher order case,\nJ. Comput. Math. 33 (2015), no. 3, 283\u2013296.","DOI":"10.4208\/jcm.1412-m2014-0071"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_027","doi-asserted-by":"crossref","unstructured":"J. Hu and S. Zhang,\nA family of symmetric mixed finite elements for linear elasticity on tetrahedral grids,\nSci. China Math. 58 (2015), no. 2, 297\u2013307.","DOI":"10.1007\/s11425-014-4953-5"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_028","doi-asserted-by":"crossref","unstructured":"J. Hu and S. 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Breinbjerg,\nHigher order hierarchical Legendre basis functions for electromagnetic modeling,\nIEEE Trans. Antennas Propagation 52 (2004), no. 11, 2985\u20132995.","DOI":"10.1109\/TAP.2004.835279"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_031","unstructured":"G. E. Karniadakis and S. J. Sherwin,\nSpectral\/\n                  \n                     \n                        \n                           \n                              h\n                              \u2062\n                              p\n                           \n                        \n                        \n                        hp\n                     \n                  \n                Element Methods for CFD,\nOxford University, Oxford, 2013."},{"key":"2023033112440969372_j_cmam-2022-0053_ref_032","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":"2023033112440969372_j_cmam-2022-0053_ref_033","doi-asserted-by":"crossref","unstructured":"P. Le Tallec,\nA mixed finite element approximation of the Navier\u2013Stokes equations,\nNumer. Math. 35 (1980), no. 4, 381\u2013404.","DOI":"10.1007\/BF01399007"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_034","doi-asserted-by":"crossref","unstructured":"J.-C. N\u00e9d\u00e9lec,\nA new family of mixed finite elements in \n                  \n                     \n                        \n                           R\n                           3\n                        \n                     \n                     \n                     {\\mathbf{R}}^{3}\n                  \n               ,\nNumer. Math. 50 (1986), no. 1, 57\u201381.","DOI":"10.1007\/BF01389668"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_035","doi-asserted-by":"crossref","unstructured":"M. Neilan,\nDiscrete and conforming smooth de Rham complexes in three dimensions,\nMath. Comp. 84 (2015), no. 295, 2059\u20132081.","DOI":"10.1090\/S0025-5718-2015-02958-5"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_036","doi-asserted-by":"crossref","unstructured":"M. Neilan,\nThe Stokes complex: a review of exactly divergence-free finite element pairs for incompressible flows,\n75 years of Mathematics of Computation,\nContemp. Math. 754,\nAmerican Mathematical Society, Providence (2020), 141\u2013158.","DOI":"10.1090\/conm\/754\/15142"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_037","doi-asserted-by":"crossref","unstructured":"H. Schenck and T. Sorokina,\nSubdivision and spline spaces,\nConstr. Approx. 47 (2018), no. 2, 237\u2013247.","DOI":"10.1007\/s00365-017-9367-5"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_038","doi-asserted-by":"crossref","unstructured":"J. Sch\u00f6berl and S. Zaglmayr,\nHigh order N\u00e9d\u00e9lec elements with local complete sequence properties,\nCOMPEL 24 (2005), no. 2, 374\u2013384.","DOI":"10.1108\/03321640510586015"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_039","doi-asserted-by":"crossref","unstructured":"L. R. Scott and M. Vogelius,\nNorm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials,\nRAIRO Mod\u00e9l. Math. Anal. Num\u00e9r. 19 (1985), no. 1, 111\u2013143.","DOI":"10.1051\/m2an\/1985190101111"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_040","doi-asserted-by":"crossref","unstructured":"R. Stenberg,\nA nonstandard mixed finite element family,\nNumer. Math. 115 (2010), no. 1, 131\u2013139.","DOI":"10.1007\/s00211-009-0272-0"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_041","doi-asserted-by":"crossref","unstructured":"D.-K. Sun, J.-F. Lee and Z. Cendes,\nConstruction of nearly orthogonal Nedelec bases for rapid convergence with multilevel preconditioned solvers,\nSIAM J. Sci. Comput. 23 (2001), no. 4, 1053\u20131076.","DOI":"10.1137\/S1064827500367531"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_042","doi-asserted-by":"crossref","unstructured":"D. Sun, J. Manges, X. Yuan and Z. Cendes,\nSpurious modes in finite-element methods,\nIEEE Antennas Propagation Mag. 37 (1995), no. 5, 12\u201324.","DOI":"10.1109\/74.475860"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_043","unstructured":"W. Tonnon,\nSemi-Lagrangian discretization of the incompressible Euler equation,\nMaster\u2019s thesis, ETH Z\u00fcrich, 2021."},{"key":"2023033112440969372_j_cmam-2022-0053_ref_044","doi-asserted-by":"crossref","unstructured":"J. P. Webb,\nHierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements,\nIEEE Trans. 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Comput. Phys. 9 (2011), no. 3, 780\u2013806.","DOI":"10.4208\/cicp.220310.030610s"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_047","doi-asserted-by":"crossref","unstructured":"J. Xin and W. Cai,\nWell-conditioned orthonormal hierarchical \n                  \n                     \n                        \n                           L\n                           2\n                        \n                     \n                     \n                     \\mathcal{L}_{2}\n                  \n                Bases on \n                  \n                     \n                        \n                           R\n                           n\n                        \n                     \n                     \n                     {\\mathbb{R}}^{n}\n                  \n                simplicial elements,\nJ. Sci. Comput. 50 (2012), no. 2, 446\u2013461.","DOI":"10.1007\/s10915-011-9491-5"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_048","doi-asserted-by":"crossref","unstructured":"J. Xin, W. Cai and N. Guo,\nOn the construction of well-conditioned hierarchical bases for \n                  \n                     \n                        \n                           H\n                           \u2062\n                           \n                              (\n                              div\n                              )\n                           \n                        \n                     \n                     \n                     H(\\mathrm{div})\n                  \n               -conforming \n                  \n                     \n                        \n                           R\n                           n\n                        \n                     \n                     \n                     \\mathbb{R}^{n}\n                  \n                simplicial elements,\nCommun. Comput. Phys. 14 (2013), no. 3, 621\u2013638.","DOI":"10.4208\/cicp.100412.041112a"},{"key":"2023033112440969372_j_cmam-2022-0053_ref_049","unstructured":"S. Zaglmayr,\nHigh order finite element methods for electromagnetic field computation,\nPhD thesis, JKU, Linz, 2006."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0053\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0053\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T17:04:56Z","timestamp":1680282296000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0053\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,9]]},"references-count":49,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3,26]]},"published-print":{"date-parts":[[2022,7,1]]}},"alternative-id":["10.1515\/cmam-2022-0053"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0053","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,9]]}}}