{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T09:33:00Z","timestamp":1787563980657,"version":"build-2736575974"},"reference-count":54,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11771338"],"award-info":[{"award-number":["11771338"]}],"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":["11671304"],"award-info":[{"award-number":["11671304"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1418934"],"award-info":[{"award-number":["DMS-1418934"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004731","name":"Natural Science Foundation of Zhejiang Province","doi-asserted-by":"publisher","award":["LY17A010010"],"award-info":[{"award-number":["LY17A010010"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004731","name":"Natural Science Foundation of Zhejiang Province","doi-asserted-by":"publisher","award":["LY15A010015"],"award-info":[{"award-number":["LY15A010015"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007194","name":"Wenzhou Municipal Science and Technology Bureau","doi-asserted-by":"publisher","award":["G20160019"],"award-info":[{"award-number":["G20160019"]}],"id":[{"id":"10.13039\/501100007194","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>A framework to systematically decouple high order elliptic equations into a combination of Poisson-type and Stokes-type equations is developed. The key is to systematically construct the underling commutative diagrams involving the complexes and Helmholtz decompositions in a general way. Discretizing the decoupled formulation leads to a natural superconvergence between the Galerkin projection and the decoupled approximation. Examples include but are not limited to the primal formulations and mixed formulations of the biharmonic equation, fourth order curl equation, and triharmonic equation. As a byproduct, Helmholtz decompositions for many dual spaces are obtained.<\/jats:p>","DOI":"10.1137\/17m1145872","type":"journal-article","created":{"date-parts":[[2018,9,11]],"date-time":"2018-09-11T16:39:22Z","timestamp":1536683962000},"page":"2796-2825","source":"Crossref","is-referenced-by-count":27,"title":["Decoupling of Mixed Methods Based on Generalized Helmholtz Decompositions"],"prefix":"10.1137","volume":"56","author":[{"given":"Long","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2966-7426","authenticated-orcid":true,"given":"Xuehai","family":"Huang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,9,11]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050222"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1142\/S0219530510001497"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/PL00005386"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492906210018"},{"key":"atypb5","first-page":"281","volume":"47","author":"Arnold D. N.","year":"2010","journal-title":"S.)"},{"key":"atypb6","first-page":"1","volume-title":"The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Univ. Maryland","author":"Babu\u0161ka I.","year":"1972"},{"key":"atypb7","series-title":"Lecture Notes in Math.","volume-title":"Compatibility Conditions, and Applications","author":"Boffi D.","year":"1939"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-36519-5"},{"key":"atypb9","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511618635","volume-title":"Fast Solvers, and Applications in Elasticity Theory","author":"Braess D.","year":"2007","edition":"3"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-017-0449-0"},{"key":"atypb12","volume-title":"Universitext","author":"Brezis H.","year":"2011"},{"key":"atypb13","first-page":"129","volume":"8","author":"Brezzi F.","year":"1974","journal-title":"Rev. Fran\u00e7aise Automat. Informat. Recherche Op\u00e9rationnelle S\u00e9r. Rouge"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1986-0842127-7"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202591000083"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-004-0577-y"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-02-01402-3"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100378"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-007-0068-z"},{"key":"atypb20","volume-title":"Multigrid Methods for Constrained Minimization Problems and Application to Saddle Point Problems, preprint, https:\/\/arxiv.org\/abs\/1601.04091","author":"Chen L.","year":"2016"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-017-0636-z"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s11425-017-9181-2"},{"key":"atypb23","first-page":"1","author":"Chen L.","year":"2018","journal-title":"J. Sci. Comput., ("},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396415"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s00209-009-0517-8"},{"key":"atypb26","unstructured":"R. de la Mbyadrid Modino,\n                      Quantum Mechanics in Rigged Hilbert Space Language\n                      , Ph.D. thesis, Universidad de Valladolid, Valladolid, Spain, 2001."},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3208"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/16M1080495"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61623-5"},{"key":"atypb30","volume-title":"Cambridge University Press","author":"Hatcher A.","year":"2002"},{"key":"atypb31","volume-title":"Trondheim","author":"Hellan K.","year":"1967"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1061\/JMCEA3.0000891"},{"key":"atypb33","first-page":"133","volume":"6","author":"Hiptmair R.","year":"1997","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492902000041"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/060660588"},{"key":"atypb36","volume-title":"Tech. report","author":"Huang J.","year":"2012"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1137\/090773374"},{"key":"atypb38","unstructured":"X. Huang,\n                      New Finite Element Methods and Efficient Algorithms for Fourth Order Elliptic Equations\n                      , Ph.D. thesis, Shanghai Jiao Tong University, Shanghai Shi, China, 2010."},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1007\/BF01436186"},{"key":"atypb40","first-page":"257","volume":"45","author":"Krendl W.","year":"2016","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb41","volume-title":"A Uniform inf--sup Condition with Applications to Preconditioning, preprint, https:\/\/arxiv.org\/abs\/1201.1513","author":"Mardal K.-A.","year":"2012"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901383910"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-004-0529-6"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198508885.001.0001"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-006-0031-4"},{"key":"atypb46","volume-title":"Corresponding Compact Embeddings for Tensor Rotations, and a Related Decomposition Result for Biharmonic Problems in 3D, https:\/\/arxiv.org\/abs\/1609.05873","author":"Pauly D.","year":"2016"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1137\/17M1118427"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1137\/15M1013651"},{"key":"atypb49","first-page":"25","volume-title":"Journal f\u00fcr die reine und angewandte Mathematik, 55","author":"von Helmholtz H.","year":"1858"},{"key":"atypb50","first-page":"437","volume":"26","author":"Xie X.","year":"2008","journal-title":"J. Comput. Math."},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61859-8"},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2018005"},{"key":"atypb53","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-017-0902-x"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-017-0592-7"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1145872","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:40:40Z","timestamp":1787326840000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1145872"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":54,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1145872"],"URL":"https:\/\/doi.org\/10.1137\/17m1145872","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}