{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T21:10:28Z","timestamp":1772658628008,"version":"3.50.1"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2018,9,25]],"date-time":"2018-09-25T00:00:00Z","timestamp":1537833600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2018,9,25]],"date-time":"2018-09-25T00:00:00Z","timestamp":1537833600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Nationational Science Foundation","award":["DMS-1419077"],"award-info":[{"award-number":["DMS-1419077"]}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1720473"],"award-info":[{"award-number":["DMS-1720473"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2019,3]]},"DOI":"10.1007\/s10915-018-0837-0","type":"journal-article","created":{"date-parts":[[2018,9,25]],"date-time":"2018-09-25T14:38:43Z","timestamp":1537886323000},"page":"1917-1941","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Lowest-Order Weak Galerkin Finite Element Methods for Linear Elasticity on Rectangular and Brick Meshes"],"prefix":"10.1007","volume":"78","author":[{"given":"Graham","family":"Harper","sequence":"first","affiliation":[]},{"given":"Jiangguo","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Simon","family":"Tavener","sequence":"additional","affiliation":[]},{"given":"Bin","family":"Zheng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,9,25]]},"reference":[{"key":"837_CR1","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/s00607-002-1459-8","volume":"69","author":"J Alberty","year":"2002","unstructured":"Alberty, J., Carstensen, C., Funken, S., Klose, R.: Matlab implementation of the finite element method in elasticity. Computing 69, 239\u2013263 (2002)","journal-title":"Computing"},{"key":"837_CR2","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1007\/s10444-014-9376-x","volume":"41","author":"D Arnold","year":"2015","unstructured":"Arnold, D., Awanou, G., Qiu, W.: Mixed finite elements for elasticity on quadrilateral meshes. Adv. Comput. Math. 41, 553\u2013572 (2015)","journal-title":"Adv. Comput. Math."},{"key":"837_CR3","doi-asserted-by":"publisher","first-page":"909","DOI":"10.1090\/S0025-5718-02-01439-4","volume":"71","author":"D Arnold","year":"2002","unstructured":"Arnold, D., Boffi, D., Falk, R.: Approximation by quadrilateral finite elements. Math. Comput. 71, 909\u2013922 (2002)","journal-title":"Math. Comput."},{"key":"837_CR4","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1090\/S0025-5718-1992-1140646-2","volume":"59","author":"S Brenner","year":"1992","unstructured":"Brenner, S., Sung, L.-Y.: Linear finite element methods for planar linear elasticity. Math. Comput. 59, 321\u2013338 (1992)","journal-title":"Math. Comput."},{"key":"837_CR5","series-title":"Texts in Applied Mathematics","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The Mathematical Theory of Finite Element Methods","author":"SC Brenner","year":"2008","unstructured":"Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15, Third edn. Springer, New York (2008)","edition":"Third"},{"key":"837_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and Hybrid Finite Element Methods","author":"F Brezzi","year":"1991","unstructured":"Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)"},{"key":"837_CR7","doi-asserted-by":"publisher","first-page":"1591","DOI":"10.1093\/imanum\/dru048","volume":"35","author":"C Carstensen","year":"2015","unstructured":"Carstensen, C., Schedensack, M.: Medius analysis and comparison results for first-order finite element methods in linear elasticity. IMA J. Numer. Anal. 35, 1591\u20131621 (2015)","journal-title":"IMA J. Numer. Anal."},{"key":"837_CR8","doi-asserted-by":"publisher","first-page":"1317","DOI":"10.1016\/j.camwa.2016.01.023","volume":"71","author":"J Hu","year":"2016","unstructured":"Hu, J., Man, H., Wang, J., Zhang, S.: The simplest nonconforming mixed finite element method for linear elasticity in the symmetric formulation on n-rectangular grids. Comput. Math. Appl. 71, 1317\u20131336 (2016)","journal-title":"Comput. Math. Appl."},{"key":"837_CR9","series-title":"Mathematical Surveys and Monographs","volume-title":"Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations","author":"VA Kozlov","year":"2001","unstructured":"Kozlov, V.A., Maz\u2019ya, V.G., Rossmann, J.: Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Mathematical Surveys and Monographs, vol. 85. American Mathematical Society, Providence (2001)"},{"key":"837_CR10","doi-asserted-by":"publisher","first-page":"356","DOI":"10.1016\/j.cam.2013.09.056","volume":"260","author":"BP Lamichhane","year":"2014","unstructured":"Lamichhane, B.P.: A mixed finite element method for nearly incompressible elasticity and Stokes equations using primal and dual meshes with quadrilateral and hexahedral grids. J. Comput. Appl. Math. 260, 356\u2013363 (2014)","journal-title":"J. Comput. Appl. Math."},{"key":"837_CR11","doi-asserted-by":"publisher","first-page":"1336","DOI":"10.1002\/num.20683","volume":"28","author":"BP Lamichhane","year":"2012","unstructured":"Lamichhane, B.P., Stephan, E.P.: A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems. Numer. Methods PDEs 28, 1336\u20131353 (2012)","journal-title":"Numer. Methods PDEs"},{"key":"837_CR12","unstructured":"Lawrence Livermore National Laboratory. VisIt User\u2019s Manual, 1.5 edition (2005)"},{"key":"837_CR13","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1090\/conm\/586\/11655","volume":"586","author":"H Li","year":"2013","unstructured":"Li, H., Nistor, V.: LNG FEM: graded meshes on domains of polygonal structures. Contemp. Math. 586, 239\u2013246 (2013)","journal-title":"Contemp. Math."},{"key":"837_CR14","first-page":"693","volume":"5","author":"J Liu","year":"2008","unstructured":"Liu, J., Cali, R.: A note on the approximation properties of the locally divergence-free finite elements. Int. J. Numer. Anal. Model. 5, 693\u2013703 (2008)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"837_CR15","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1016\/j.jcp.2018.01.001","volume":"359","author":"J Liu","year":"2018","unstructured":"Liu, J., Tavener, S., Wang, Z.: The lowest order weak Galerkin finite element methods for the Darcy equation on quadrilateral and hybrid meshes. J. Comput. Phys. 359, 312\u2013330 (2018)","journal-title":"J. Comput. Phys."},{"key":"837_CR16","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1007\/s10444-006-9017-0","volume":"28","author":"S Mao","year":"2008","unstructured":"Mao, S., Chen, S.: A quadrilateral nonconforming finite element for linear elasticity problem. Adv. Comput. Math. 28, 81\u2013100 (2008)","journal-title":"Adv. Comput. Math."},{"key":"837_CR17","doi-asserted-by":"publisher","first-page":"466","DOI":"10.1007\/s00466-003-0546-9","volume":"33","author":"D Mijuca","year":"2004","unstructured":"Mijuca, D.: On hexahedral finite element hc8\/27 in elasticity. Comput. Mech. 33, 466\u2013480 (2004)","journal-title":"Comput. Mech."},{"key":"837_CR18","doi-asserted-by":"publisher","first-page":"346","DOI":"10.1016\/j.cam.2015.12.015","volume":"307","author":"C Wang","year":"2016","unstructured":"Wang, C., Wang, J., Wang, R., Zhang, R.: A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation. J. Comput. Appl. Math. 307, 346\u2013366 (2016)","journal-title":"J. Comput. Appl. Math."},{"key":"837_CR19","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/j.cam.2012.10.003","volume":"241","author":"J Wang","year":"2013","unstructured":"Wang, J., Ye, X.: A weak Galerkin finite element method for second order elliptic problems. J. Comput. Appl. Math. 241, 103\u2013115 (2013)","journal-title":"J. Comput. Appl. Math."},{"key":"837_CR20","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1007\/s00211-011-0427-7","volume":"121","author":"M Wheeler","year":"2012","unstructured":"Wheeler, M., Xue, G., Yotov, I.: A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Numer. Math. 121, 165\u2013204 (2012)","journal-title":"Numer. Math."},{"key":"837_CR21","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/s10092-005-0101-5","volume":"42","author":"S-Y Yi","year":"2005","unstructured":"Yi, S.-Y.: Nonconforming mixed finite element methods for linear elasticity using rectangular elements in two and three dimensions. Calcolo 42, 115\u2013133 (2005)","journal-title":"Calcolo"},{"key":"837_CR22","doi-asserted-by":"publisher","first-page":"979","DOI":"10.1142\/S0218202506001431","volume":"16","author":"S-Y Yi","year":"2006","unstructured":"Yi, S.-Y.: A new nonconforming mixed finite element method for linear elasticity. Math. Models Methods Appl. Sci. 16, 979\u2013999 (2006)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"837_CR23","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1007\/s00211-004-0536-7","volume":"98","author":"S Zhang","year":"2004","unstructured":"Zhang, S.: On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation. Numer. Math. 98, 559\u2013579 (2004)","journal-title":"Numer. Math."},{"key":"837_CR24","doi-asserted-by":"publisher","first-page":"640","DOI":"10.1137\/S0036142995282492","volume":"34","author":"Z Zhang","year":"1997","unstructured":"Zhang, Z.: Analysis of some quadrilateral nonconforming elements for incompressible elasticity. SIAM J. Numer. Anal. 34, 640\u2013663 (1997)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0837-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-018-0837-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0837-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,17]],"date-time":"2020-05-17T09:37:30Z","timestamp":1589708250000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-018-0837-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,25]]},"references-count":24,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,3]]}},"alternative-id":["837"],"URL":"https:\/\/doi.org\/10.1007\/s10915-018-0837-0","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,9,25]]},"assertion":[{"value":"1 December 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 June 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 September 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 September 2018","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}