{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:59:48Z","timestamp":1759838388265,"version":"3.37.3"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,5,8]],"date-time":"2021-05-08T00:00:00Z","timestamp":1620432000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,5,8]],"date-time":"2021-05-08T00:00:00Z","timestamp":1620432000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,6]]},"DOI":"10.1007\/s40314-021-01521-7","type":"journal-article","created":{"date-parts":[[2021,5,8]],"date-time":"2021-05-08T16:03:28Z","timestamp":1620489808000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A new $$P_0$$ weak Galerkin finite element scheme for second-order problems"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3352-5829","authenticated-orcid":false,"given":"AllahBakhsh Yazdani","family":"Charati","sequence":"first","affiliation":[]},{"given":"Hamid","family":"Momeni","sequence":"additional","affiliation":[]},{"given":"Mohammed S.","family":"Cheichan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,8]]},"reference":[{"issue":"1\u20133","key":"1521_CR1","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/s10915-005-9044-x","volume":"27","author":"M Ainsworth","year":"2006","unstructured":"Ainsworth M, Monk P, Muniz W (2006) Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. J Sci Comput 27(1\u20133):5\u201340","journal-title":"J Sci Comput"},{"issue":"2","key":"1521_CR2","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1002\/num.22415","volume":"36","author":"A Al-Taweel","year":"2020","unstructured":"Al-Taweel A, Hussain S, Wang X, Jones B (2020) A $$P_0-P_0$$ weak Galerkin finite element method for solving singularly perturbed reaction-diffusion problems. Numer Methods Partial Differ Equ 36(2):213\u2013227","journal-title":"Numer Methods Partial Differ Equ"},{"key":"1521_CR3","doi-asserted-by":"publisher","unstructured":"Arnold DN, Brezzi F, Cockburn B, MariniD (2000) Discontinuous galerkin methods for elliptic problems. In: Cockburn B,Karniadakis GE, Shu CW (eds) Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering, vol 11. Springer, Berlin, Heidelberg. https:\/\/doi.org\/10.1007\/978-3-642-59721-3_5","DOI":"10.1007\/978-3-642-59721-3_5"},{"issue":"5","key":"1521_CR4","doi-asserted-by":"publisher","first-page":"1749","DOI":"10.1137\/S0036142901384162","volume":"39","author":"DN Arnold","year":"2002","unstructured":"Arnold DN, Brezzi F, Cockburn B, Marini LD (2002) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal 39(5):1749\u20131779","journal-title":"SIAM J Numer Anal"},{"issue":"2","key":"1521_CR5","doi-asserted-by":"publisher","first-page":"524","DOI":"10.1137\/S1064827501388339","volume":"24","author":"P Castillo","year":"2002","unstructured":"Castillo P (2002) Performance of discontinuous Galerkin methods for elliptic PDEs. SIAM J Sci Comput 24(2):524\u2013547","journal-title":"SIAM J Sci Comput"},{"issue":"5","key":"1521_CR6","doi-asserted-by":"publisher","first-page":"1676","DOI":"10.1137\/S0036142900371003","volume":"38","author":"P Castillo","year":"2000","unstructured":"Castillo P, Cockburn B, Perugia I, Sch\u00f6tzau D (2000) An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J Numer Anal 38(5):1676\u20131706","journal-title":"SIAM J Numer Anal"},{"key":"1521_CR7","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.amc.2019.02.043","volume":"354","author":"MS Cheichan","year":"2019","unstructured":"Cheichan MS, Kashkool HA, Gao F (2019) A weak Galerkin finite element method for solving nonlinear convection-diffusion problems in two dimensions. Appl Math Comput 354:149\u2013163","journal-title":"Appl Math Comput"},{"key":"1521_CR8","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/j.cam.2018.08.044","volume":"348","author":"Y Chen","year":"2019","unstructured":"Chen Y, Zhang T (2019) A weak Galerkin finite element method for Burgers\u2019 equation. J Comput Appl Math 348:103\u2013119","journal-title":"J Comput Appl Math"},{"key":"1521_CR9","first-page":"199","volume":"141(2)","author":"B Cockburn","year":"1997","unstructured":"Cockburn B, Shu CW (1997) The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J Comput Phys 141(2):199\u2013224","journal-title":"J Comput Phys"},{"issue":"6","key":"1521_CR10","doi-asserted-by":"publisher","first-page":"2440","DOI":"10.1137\/S0036142997316712","volume":"35","author":"B Cockburn","year":"1998","unstructured":"Cockburn B, Shu CW (1998) The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J Numer Anal 35(6):2440\u20132463","journal-title":"SIAM J Numer Anal"},{"issue":"3","key":"1521_CR11","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1023\/A:1012873910884","volume":"16","author":"B Cockburn","year":"2001","unstructured":"Cockburn B, Shu CW (2001) Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J Sci Comput 16(3):173\u2013261","journal-title":"J Sci Comput"},{"issue":"246","key":"1521_CR12","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1090\/S0025-5718-03-01552-7","volume":"73","author":"B Cockburn","year":"2004","unstructured":"Cockburn B, Kanschat G, Sch\u00c3k\u0327tzau D (2004) The local discontinuous Galerkin method for the Oseen equations. Math Comput 73(246):569\u2013593","journal-title":"Math Comput"},{"issue":"272","key":"1521_CR13","doi-asserted-by":"publisher","first-page":"2169","DOI":"10.1090\/S0025-5718-10-02360-4","volume":"79","author":"T Gudi","year":"2010","unstructured":"Gudi T (2010) A new error analysis for discontinuous finite element methods for linear elliptic problems. Math Comput 79(272):2169\u20132189","journal-title":"Math Comput"},{"issue":"173","key":"1521_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0025-5718-1986-0815828-4","volume":"46","author":"C Johnson","year":"1986","unstructured":"Johnson C, Pitk\u00e4ranta J (1986) An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math Comput 46(173):1\u201326","journal-title":"Math Comput"},{"key":"1521_CR15","doi-asserted-by":"publisher","first-page":"442","DOI":"10.1016\/j.cam.2017.11.010","volume":"333","author":"X Liu","year":"2018","unstructured":"Liu X, Li J, Chen Z (2018) A weak Galerkin finite element method for the Navier-Stokes equations. J Comput Appl Math 333:442\u2013457","journal-title":"J Comput Appl Math"},{"issue":"3","key":"1521_CR16","doi-asserted-by":"publisher","first-page":"1003","DOI":"10.1002\/num.21855","volume":"30","author":"L Mu","year":"2014","unstructured":"Mu L, Wang J, Ye X (2014) Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes. Numer Methods Partial Differ Equ 30(3):1003\u20131029","journal-title":"Numer Methods Partial Differ Equ"},{"key":"1521_CR17","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/j.cam.2015.02.001","volume":"285","author":"L Mu","year":"2015","unstructured":"Mu L, Wang J, Ye X (2015) A weak Galerkin finite element method with polynomial reduction. J Comput Appl Math 285:45\u201358","journal-title":"J Comput Appl Math"},{"issue":"23","key":"1521_CR18","doi-asserted-by":"publisher","first-page":"8841","DOI":"10.1016\/j.jcp.2009.08.030","volume":"228","author":"NC Nguyen","year":"2009","unstructured":"Nguyen NC, Peraire J, Cockburn B (2009) An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations. J Comput Phys 228(23):8841\u20138855","journal-title":"J Comput Phys"},{"key":"1521_CR19","unstructured":"Reed WH, Hill TR (1973) Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-0479. Los Alamos Scientific Laboratory, Los Alamos, NM"},{"issue":"1\u20133","key":"1521_CR20","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1007\/s10915-004-4147-3","volume":"22","author":"B Rivi\u00e9re","year":"2005","unstructured":"Rivi\u00e9re B (2005) Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J Sci Comput 22(1\u20133):479\u2013500","journal-title":"J Sci Comput"},{"key":"1521_CR21","doi-asserted-by":"publisher","first-page":"268","DOI":"10.1016\/j.cam.2017.01.021","volume":"329","author":"T Tian","year":"2018","unstructured":"Tian T, Zhai Q, Zhang R (2018) A new modified weak Galerkin finite element scheme for solving the stationary Stokes equations. J Comput Appl Math 329:268\u2013279","journal-title":"J Comput Appl Math"},{"key":"1521_CR22","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 (2013) A weak Galerkin finite element method for second-order elliptic problems. J Comput Appl Math 241:103\u2013115","journal-title":"J Comput Appl Math"},{"issue":"289","key":"1521_CR23","doi-asserted-by":"publisher","first-page":"2101","DOI":"10.1090\/S0025-5718-2014-02852-4","volume":"83","author":"J Wang","year":"2014","unstructured":"Wang J, Ye X (2014) A weak Galerkin mixed finite element method for second order elliptic problems. Math Comput 83(289):2101\u20132126","journal-title":"Math Comput"},{"key":"1521_CR24","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/j.cam.2016.01.025","volume":"302","author":"R Wang","year":"2016","unstructured":"Wang R, Wang X, Zhai Q, Zhang R (2016) A weak Galerkin finite element scheme for solving the stationary Stokes equations. J Comput Appl Math 302:171\u2013185","journal-title":"J Comput Appl Math"},{"issue":"315","key":"1521_CR25","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1090\/mcom\/3369","volume":"88","author":"J Wang","year":"2019","unstructured":"Wang J, Zhai Q, Zhang R, Zhang S (2019) A weak Galerkin finite element scheme for the Cahn-Hilliard equation. Math Comput 88(315):211\u2013235","journal-title":"Math Comput"},{"issue":"1","key":"1521_CR26","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1016\/j.jcp.2004.11.001","volume":"205","author":"Y Xu","year":"2005","unstructured":"Xu Y, Shu CW (2005) Local discontinuous Galerkin methods for nonlinear Schr\u00f6dinger equations. J Comput Phys 205(1):72\u201397","journal-title":"J Comput Phys"},{"key":"1521_CR27","doi-asserted-by":"publisher","first-page":"112337","DOI":"10.1016\/j.cam.2019.07.002","volume":"364","author":"X Ye","year":"2020","unstructured":"Ye X, Zhang S, Zhang Z (2020) A new P1 weak Galerkin method for the Biharmonic equation. J Comput Appl Math 364:112337","journal-title":"J Comput Appl Math"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01521-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-021-01521-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01521-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T12:56:43Z","timestamp":1699016203000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-021-01521-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,8]]},"references-count":27,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["1521"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01521-7","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2021,5,8]]},"assertion":[{"value":"21 December 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 March 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 April 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 May 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"138"}}