{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T08:12:32Z","timestamp":1777623152078,"version":"3.51.4"},"reference-count":33,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["91430215"],"award-info":[{"award-number":["91430215"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11101415"],"award-info":[{"award-number":["11101415"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11471026"],"award-info":[{"award-number":["11471026"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,10,1]]},"abstract":"Abstract<\/jats:title>\n This paper studies the mixed element method for the boundary value problem of the biharmonic equation \n \n \n \n \n \n \u0394<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n \u2062<\/m:mo>\n u<\/m:mi>\n <\/m:mrow>\n =<\/m:mo>\n f<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n {\\Delta^{2}u=f}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in two dimensions.\nWe start from a \n \n \n \n u<\/m:mi>\n \u223c<\/m:mo>\n \n \u2207<\/m:mo>\n \u2061<\/m:mo>\n u<\/m:mi>\n <\/m:mrow>\n \u223c<\/m:mo>\n \n \n \u2207<\/m:mo>\n 2<\/m:mn>\n <\/m:msup>\n \u2061<\/m:mo>\n u<\/m:mi>\n <\/m:mrow>\n \u223c<\/m:mo>\n \n div<\/m:mi>\n \u2062<\/m:mo>\n \n \n \u2207<\/m:mo>\n 2<\/m:mn>\n <\/m:msup>\n \u2061<\/m:mo>\n u<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {u\\sim\\nabla u\\sim\\nabla^{2}u\\sim\\operatorname{div}\\nabla^{2}u}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> formulation that is discussed in [4] and construct its stability on \n \n \n \n \n \n \n \n \n \n \n H<\/m:mi>\n 0<\/m:mn>\n 1<\/m:mn>\n <\/m:msubsup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \u00d7<\/m:mo>\n \n \n H<\/m:mi>\n ~<\/m:mo>\n <\/m:mover>\n 0<\/m:mn>\n 1<\/m:mn>\n <\/m:msubsup>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \u00d7<\/m:mo>\n \n \n L<\/m:mi>\n \u00af<\/m:mo>\n <\/m:mover>\n sym<\/m:mi>\n 2<\/m:mn>\n <\/m:msubsup>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n \u00d7<\/m:mo>\n \n H<\/m:mi>\n \n -<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n <\/m:mrow>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n div<\/m:mi>\n ,<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H^{1}_{0}(\\Omega)\\times\\tilde{H}^{1}_{0}(\\Omega)\\times\\bar{L}_{\\mathrm{sym}}^%\n{2}(\\Omega)\\times H^{-1}(\\operatorname{div},\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nThen we utilise the Helmholtz decomposition of \n \n \n \n \n H<\/m:mi>\n \n -<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n div<\/m:mi>\n ,<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H^{-1}(\\operatorname{div},\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and construct a new formulation stable on first-order and zero-order Sobolev spaces.\nFinite element discretisations are then given with respect to the new formulation, and both theoretical analysis and numerical verification are given.<\/jats:p>","DOI":"10.1515\/cmam-2017-0002","type":"journal-article","created":{"date-parts":[[2017,4,13]],"date-time":"2017-04-13T10:00:53Z","timestamp":1492077653000},"page":"601-616","source":"Crossref","is-referenced-by-count":5,"title":["A Stable Mixed Element Method for the Biharmonic Equation with First-Order Function Spaces"],"prefix":"10.1515","volume":"17","author":[{"given":"Zheng","family":"Li","sequence":"first","affiliation":[{"name":"Kunming University of Science and Technology , Kunming , P. R. China"}]},{"given":"Shuo","family":"Zhang","sequence":"additional","affiliation":[{"name":"LSEC, Institute of Computational Mathematics and Scientific\/Engineering Computing, Academy of Mathematics and System Sciences, NCMIS , Chinese Academy of Sciences , Beijing 100190 , P. R. China"}]}],"member":"374","published-online":{"date-parts":[[2017,4,13]]},"reference":[{"key":"2023033116270860654_j_cmam-2017-0002_ref_001_w2aab3b7b2b1b6b1ab1b7b1Aa","doi-asserted-by":"crossref","unstructured":"M. Amara and F. Dabaghi,\nAn optimal C0{C^{0}} finite element algorithm for the 2D biharmonic problem: Theoretical analysis and numerical results,\nNumer. Math. 90 (2001), no. 1, 19\u201346.","DOI":"10.1007\/s002110100284"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_002_w2aab3b7b2b1b6b1ab1b7b2Aa","doi-asserted-by":"crossref","unstructured":"D. N. Arnold and R. S. Falk,\nA uniformly accurate finite element method for the Reissner\u2013Mindlin plate,\nSIAM J. Numer. Anal. 26 (1989), no. 6, 1276\u20131290.\n10.1137\/0726074","DOI":"10.1137\/0726074"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_003_w2aab3b7b2b1b6b1ab1b7b3Aa","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka,\nThe finite element method with Lagrangian multipliers,\nNumer. Math. 20 (1973), no. 3, 179\u2013192.\n10.1007\/BF01436561","DOI":"10.1007\/BF01436561"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_004_w2aab3b7b2b1b6b1ab1b7b4Aa","doi-asserted-by":"crossref","unstructured":"E. M. Behrens and J. Guzm\u00e1n,\nA mixed method for the biharmonic problem based on a system of first-order equations,\nSIAM J. Numer. Anal. 49 (2011), no. 2, 789\u2013817.\n10.1137\/090775245","DOI":"10.1137\/090775245"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_005_w2aab3b7b2b1b6b1ab1b7b5Aa","doi-asserted-by":"crossref","unstructured":"C. Bernardi, V. Girault and Y. Maday,\nMixed spectral element approximation of the Navier\u2013Stokes equations in the stream-function and vorticity formulation,\nIMA J. Numer. Anal. 12 (1992), no. 4, 565\u2013608.\n10.1093\/imanum\/12.4.565","DOI":"10.1093\/imanum\/12.4.565"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_006_w2aab3b7b2b1b6b1ab1b7b6Aa","doi-asserted-by":"crossref","unstructured":"D. Boffi, F. Brezzi and M. Fortin,\nMixed Finite Element Methods and Applications,\nSpringer Ser. Comput. Math. 44,\nSpringer, Berlin, 2013.","DOI":"10.1007\/978-3-642-36519-5"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_007_w2aab3b7b2b1b6b1ab1b7b7Aa","doi-asserted-by":"crossref","unstructured":"J. H. Bramble and R. S. Falk,\nA mixed-Lagrange multiplier finite element method for the polyharmonic equation,\nRAIRO Mod\u00e9lisation Math. Anal. Num\u00e9r. 19 (1985), no. 4, 519\u2013557.\n10.1051\/m2an\/1985190405191","DOI":"10.1051\/m2an\/1985190405191"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_008_w2aab3b7b2b1b6b1ab1b7b8Aa","doi-asserted-by":"crossref","unstructured":"J. H. Bramble and J. Xu,\nA local post-processing technique for improving the accuracy in mixed finite-element approximations,\nSIAM J. Numer. Anal. 26 (1989), no. 6, 1267\u20131275.\n10.1137\/0726073","DOI":"10.1137\/0726073"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_009_w2aab3b7b2b1b6b1ab1b7b9Aa","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L.-Y. Sung,\nC0{C^{0}} interior penalty methods for fourth order elliptic boundary value problems on polygonal domains,\nJ. Sci. Comput. 22 (2005), no. 1\u20133, 83\u2013118.","DOI":"10.1007\/s10915-004-4135-7"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_010_w2aab3b7b2b1b6b1ab1b7c10Aa","doi-asserted-by":"crossref","unstructured":"F. Brezzi,\nOn the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers,\nRev. Franc. Automat. Inform. Rech. Operat. R 8 (1974), no. 2, 129\u2013151.","DOI":"10.1051\/m2an\/197408R201291"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_011_w2aab3b7b2b1b6b1ab1b7c11Aa","doi-asserted-by":"crossref","unstructured":"F. Brezzi and M. Fortin,\nNumerical approximation of Mindlin\u2013Reissner plates,\nMath. Comp. 47 (1986), no. 175, 151\u2013158.\n10.1090\/S0025-5718-1986-0842127-7","DOI":"10.1090\/S0025-5718-1986-0842127-7"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_012_w2aab3b7b2b1b6b1ab1b7c12Aa","unstructured":"F. Brezzi and M. Fortin,\nMixed and Hybrid Finite Element Methods,\nSpringer Ser. Comput. Math. 15,\nSpringer, New York, 2012."},{"key":"2023033116270860654_j_cmam-2017-0002_ref_013_w2aab3b7b2b1b6b1ab1b7c13Aa","doi-asserted-by":"crossref","unstructured":"X.-L. Cheng, W. Han and H.-C. Huang,\nSome mixed finite element methods for biharmonic equation,\nJ. Comput. Appl. Math. 126 (2000), no. 1, 91\u2013109.\n10.1016\/S0377-0427(99)00342-8","DOI":"10.1016\/S0377-0427(99)00342-8"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_014_w2aab3b7b2b1b6b1ab1b7c14Aa","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet and P.-A. Raviart,\nA mixed finite element method for the biharmonic equation,\nMathematical Aspects of Finite Elements in Partial Differential Equations (Madison 1974),\nAcademic Press, New York (1974), 125\u2013145.","DOI":"10.1016\/B978-0-12-208350-1.50009-1"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_015_w2aab3b7b2b1b6b1ab1b7c15Aa","doi-asserted-by":"crossref","unstructured":"M. Comodi,\nThe Hellan\u2013Herrmann\u2013Johnson method: Some new error estimates and postprocessing,\nMath. Comp. 52 (1989), no. 185, 17\u201329.\n10.1090\/S0025-5718-1989-0946601-7","DOI":"10.1090\/S0025-5718-1989-0946601-7"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_016_w2aab3b7b2b1b6b1ab1b7c16Aa","doi-asserted-by":"crossref","unstructured":"R. S. Falk,\nApproximation of the biharmonic equation by a mixed finite element method,\nSIAM J. Numer. Anal. 15 (1978), no. 3, 556\u2013567.\n10.1137\/0715036","DOI":"10.1137\/0715036"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_017_w2aab3b7b2b1b6b1ab1b7c17Aa","doi-asserted-by":"crossref","unstructured":"R. S. Falk,\nFinite elements for the Reissner\u2013Mindlin plate,\nMixed Finite Elements, Compatibility Conditions, and Applications (Cetraro 2006),\nLecture Notes in Math. 1939,\nSpringer, Berlin (2008), 195\u2013232.","DOI":"10.1007\/978-3-540-78319-0_5"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_018_w2aab3b7b2b1b6b1ab1b7c18Aa","doi-asserted-by":"crossref","unstructured":"C. Feng and S. Zhang,\nOptimal solver for Morley element discretization of biharmonic equation on shape-regular grids,\nJ. Comput. Math. 34 (2016), no. 2, 159\u2013173.\n10.4208\/jcm.1510-m2014-0085","DOI":"10.4208\/jcm.1510-m2014-0085"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_019_w2aab3b7b2b1b6b1ab1b7c19Aa","doi-asserted-by":"crossref","unstructured":"V. Girault and P.-A. Raviart,\nFinite Element Methods for Navier\u2013Stokes Equations,\nSpring, Berlin, 1986.","DOI":"10.1007\/978-3-642-61623-5"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_020_w2aab3b7b2b1b6b1ab1b7c20Aa","unstructured":"K. Hellan,\nAnalysis of elastic plates in flexure by a simplified finite element method,\nActa Polytechn. Scand. Civil Eng. Building Construct. Ser. (46):1\u201329, 1967."},{"key":"2023033116270860654_j_cmam-2017-0002_ref_021_w2aab3b7b2b1b6b1ab1b7c21Aa","doi-asserted-by":"crossref","unstructured":"L. R. Herrmann,\nFinite-element bending analysis for plates,\nAmer. Soc. Civil Eng. Eng. Mech. Divis. J. 93 (1967), 13\u201326.","DOI":"10.1061\/JMCEA3.0000891"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_022_w2aab3b7b2b1b6b1ab1b7c22Aa","doi-asserted-by":"crossref","unstructured":"J. Hu and Z.-C. Shi,\nThe best L2{L^{2}} norm error estimate of lower order finite element methods for the fourth order problem,\nJ. Comput. Math. 30 (2012), no. 5, 449\u2013460.","DOI":"10.4208\/jcm.1203-m3855"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_023_w2aab3b7b2b1b6b1ab1b7c23Aa","doi-asserted-by":"crossref","unstructured":"C. Johnson,\nOn the convergence of a mixed finite-element method for plate bending problems,\nNumer. Math. 21 (1973), no. 1, 43\u201362.\n10.1007\/BF01436186","DOI":"10.1007\/BF01436186"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_024_w2aab3b7b2b1b6b1ab1b7c24Aa","unstructured":"W. Krendl and W. Zulehner,\nA decomposition result for biharmonic problems and the Hellan\u2013Herrmann\u2013Johnson method,\nElectron. Trans. Numer. Anal. 45 (2016), 257\u2013282."},{"key":"2023033116270860654_j_cmam-2017-0002_ref_025_w2aab3b7b2b1b6b1ab1b7c25Aa","unstructured":"T. Miyoshi,\nA finite element method for the solutions of fourth order partial differential equations,\nKumamoto J. Sci. Math. 9 (1973), no. 2, 87\u2013116."},{"key":"2023033116270860654_j_cmam-2017-0002_ref_026_w2aab3b7b2b1b6b1ab1b7c26Aa","doi-asserted-by":"crossref","unstructured":"P. Monk,\nA mixed finite element method for the biharmonic equation,\nSIAM J. Numer. Anal. 24 (1987), no. 4, 737\u2013749.\n10.1137\/0724048","DOI":"10.1137\/0724048"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_027_w2aab3b7b2b1b6b1ab1b7c27Aa","doi-asserted-by":"crossref","unstructured":"A. Oukit and R. Pierre,\nMixed finite element for the linear plate problem: The Hermann\u2013Miyoshi model revisited,\nNumer. Math. 74 (1996), no. 4, 453\u2013477.\n10.1007\/s002110050225","DOI":"10.1007\/s002110050225"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_028_w2aab3b7b2b1b6b1ab1b7c28Aa","doi-asserted-by":"crossref","unstructured":"J. Pitk\u00e4ranta,\nAnalysis of some low-order finite element schemes for Mindlin\u2013Reissner and Kirchhoff plates,\nNumer. Math. 53 (1988), no. 1\u20132, 237\u2013254.\n10.1007\/BF01395887","DOI":"10.1007\/BF01395887"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_029_w2aab3b7b2b1b6b1ab1b7c29Aa","doi-asserted-by":"crossref","unstructured":"R. Rannacher,\nOn nonconforming an mixed finite element methods for plate bending problems. The linear case,\nRAIRO Anal. Numer. 13 (1979), no. 4, 369\u2013387.\n10.1051\/m2an\/1979130403691","DOI":"10.1051\/m2an\/1979130403691"},{"key":"2023033116270860654_j_cmam-2017-0002_ref_030_w2aab3b7b2b1b6b1ab1b7c30Aa","unstructured":"Z.-C. Shi and M. Wang,\nFinite Element Methods,\nScience Press, Beijing, 2013."},{"key":"2023033116270860654_j_cmam-2017-0002_ref_031_w2aab3b7b2b1b6b1ab1b7c31Aa","unstructured":"S. Zhang,\nStable mixed element schemes for plate models on multiply-connected domains,\npreprint (2017), https:\/\/arxiv.org\/abs\/1702.06401."},{"key":"2023033116270860654_j_cmam-2017-0002_ref_032_w2aab3b7b2b1b6b1ab1b7c32Aa","unstructured":"S. Zhang, Y. Xi and X. Ji,\nA multi-level mixed element method for the eigenvalue problem of biharmonic equation,\npreprint (2016), https:\/\/arxiv.org\/abs\/1606.05419."},{"key":"2023033116270860654_j_cmam-2017-0002_ref_033_w2aab3b7b2b1b6b1ab1b7c33Aa","doi-asserted-by":"crossref","unstructured":"S. Zhang and J. Xu,\nOptimal solvers for fourth-order PDEs discretized on unstructured grids,\nSIAM J. Numer. Anal. 52 (2014), no. 1, 282\u2013307.\n10.1137\/120878148","DOI":"10.1137\/120878148"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/17\/4\/article-p601.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0002\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0002\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T23:06:03Z","timestamp":1680303963000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0002\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,4,13]]},"references-count":33,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,9,9]]},"published-print":{"date-parts":[[2017,10,1]]}},"alternative-id":["10.1515\/cmam-2017-0002"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0002","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2017,4,13]]}}}