{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T00:34:26Z","timestamp":1781656466471,"version":"3.54.5"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,10,23]],"date-time":"2020-10-23T00:00:00Z","timestamp":1603411200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,10,23]],"date-time":"2020-10-23T00:00:00Z","timestamp":1603411200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004731","name":"Natural Science Foundation of Zhejiang Province","doi-asserted-by":"publisher","award":["LY19A010008"],"award-info":[{"award-number":["LY19A010008"]}],"id":[{"id":"10.13039\/501100004731","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":[[2020,11]]},"DOI":"10.1007\/s10915-020-01345-3","type":"journal-article","created":{"date-parts":[[2020,10,23]],"date-time":"2020-10-23T03:02:47Z","timestamp":1603422167000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":38,"title":["A Uniformly Convergent Weak Galerkin Finite Element Method on Shishkin Mesh for 1d Convection\u2013Diffusion Problem"],"prefix":"10.1007","volume":"85","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0452-6926","authenticated-orcid":false,"given":"Peng","family":"Zhu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shenglan","family":"Xie","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,10,23]]},"reference":[{"key":"1345_CR1","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.: A $$P_0$$-$$P_0$$ weak Galerkin finite element method for solving singularly perturbed reaction\u2013diffusion problems. Numer. Methods Partial Differ. Eq. 36, 213\u2013227 (2020)","journal-title":"Numer. Methods Partial Differ. Eq."},{"key":"1345_CR2","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/s10915-019-01120-z","volume":"82","author":"M Cui","year":"2020","unstructured":"Cui, M., Zhang, S.: On the uniform convergence of the weak Galerkin finite element method for a singularly-perturbed biharmonic equation. J. Sci. Comput. 82, 5 (2020). https:\/\/doi.org\/10.1007\/s10915-019-01120-z","journal-title":"J. Sci. Comput."},{"key":"1345_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-22980-0","volume-title":"Mathematical aspects of discontinuous Galerkrin methods","author":"DA Di Pietro","year":"2012","unstructured":"Di Pietro, D.A., Ern, A.: Mathematical aspects of discontinuous Galerkrin methods. Springer-Verlag, Berlin (2012)"},{"issue":"1","key":"1345_CR4","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1137\/090757344","volume":"53","author":"S Franz","year":"2011","unstructured":"Franz, S., Roos, H.-G.: The capriciousness of numerical methods for singular perturbations. SIAM Rev. 53(1), 157\u2013173 (2011)","journal-title":"SIAM Rev."},{"key":"1345_CR5","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1137\/070700267","volume":"47","author":"R Lin","year":"2008","unstructured":"Lin, R.: Discontinuous discretization for least-squares formulation of singularly perturbed reaction\u2013diffusion problems in one and two dimensions. SIAM J. Numer. Anal. 47, 89\u2013108 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"1345_CR6","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/s00211-008-0208-0","volume":"112","author":"R Lin","year":"2009","unstructured":"Lin, R.: Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction\u2013diffusion problems with discontinuous coefficients and boundary singularities. Numer. Math. 112, 295\u2013318 (2009)","journal-title":"Numer. Math."},{"issue":"3","key":"1345_CR7","doi-asserted-by":"publisher","first-page":"1482","DOI":"10.1137\/17M1152528","volume":"56","author":"R Lin","year":"2018","unstructured":"Lin, R., Ye, X., Zhang, S., Zhu, P.: A weak Galerkin finite element method for singularly perturbed convection\u2013diffusion\u2013reaction problems. SIAM J. Numer. Anal. 56(3), 1482\u20131497 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"1345_CR8","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1016\/S0893-9659(01)00061-1","volume":"14","author":"T Lin\u00df","year":"2001","unstructured":"Lin\u00df, T.: The necessity of Shishkin decompositions. Appl. Math. Lett. 14, 891\u2013896 (2001)","journal-title":"Appl. Math. Lett."},{"key":"1345_CR9","doi-asserted-by":"publisher","first-page":"1061","DOI":"10.1016\/S0045-7825(02)00630-8","volume":"192","author":"T Lin\u00df","year":"2003","unstructured":"Lin\u00df, T.: Layer-adapted meshes for convection\u2013diffusion problems. Comput. Methods Appl. Mech. Eng. 192, 1061\u20131105 (2003)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1345_CR10","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1007\/PL00005420","volume":"87","author":"T Lin\u00df","year":"2001","unstructured":"Lin\u00df, T., Stynes, M.: The SDFEM on Shishkin meshes for linear convection\u2013diffusion problems. Numer. Math. 87, 457\u2013484 (2001)","journal-title":"Numer. Math."},{"key":"1345_CR11","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/s10092-018-0265-4","volume":"55","author":"L Liu","year":"2018","unstructured":"Liu, L., Leng, H., Long, G.: Analysis of the SDFEM for singularly perturbed differential\u2013difference equations. Calcolo 55, 23 (2018). https:\/\/doi.org\/10.1007\/s10092-018-0265-4","journal-title":"Calcolo"},{"key":"1345_CR12","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1007\/s10915-014-9964-4","volume":"65","author":"L Mu","year":"2015","unstructured":"Mu, L., Wang, J., Ye, X., Zhang, S.: A weak Galerkin finite element method for the Maxwell equations. J. Sci. Comput. 65, 363\u2013386 (2015)","journal-title":"J. Sci. Comput."},{"key":"1345_CR13","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1002\/(SICI)1521-4001(199805)78:5<291::AID-ZAMM291>3.0.CO;2-R","volume":"78","author":"H-G Roos","year":"1998","unstructured":"Roos, H.-G.: Layer-adapted grids for singular perturbation problem. ZAMM Z. Angew. Math. Mech. 78, 291\u2013309 (1998)","journal-title":"ZAMM Z. Angew. Math. Mech."},{"issue":"6","key":"1345_CR14","doi-asserted-by":"publisher","first-page":"1560","DOI":"10.1002\/num.20241","volume":"23","author":"H-G Roos","year":"2007","unstructured":"Roos, H.-G., Zarin, H.: A supercloseness result for the discontinuous Galerkin stabilization of convection\u2013diffusion problems on Shishkin meshes. Numer. Methods Partial Differ. Equ. 23(6), 1560\u20131576 (2007)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"1345_CR15","volume-title":"Robust numerical methods for singularly perturbed differential equations, Springer Series in Computational Mathematics","author":"H-G Roos","year":"2008","unstructured":"Roos, H.-G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations, Springer Series in Computational Mathematics, vol. 24. Springer-Verlag, Berlin Heidelberg (2008)"},{"key":"1345_CR16","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1006\/jmaa.1997.5581","volume":"214","author":"M Stynes","year":"1997","unstructured":"Stynes, M., O\u2019Riorddan, E.: A uniformly convergent Galerkin method on a Shishkin mesh for a convection\u2013diffusion problem. J. Math. Anal. Appl. 214, 36\u201354 (1997)","journal-title":"J. Math. Anal. Appl."},{"key":"1345_CR17","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1515\/JNMA.2001.59","volume":"9","author":"M Stynes","year":"2001","unstructured":"Stynes, M., Tobiska, L.: Analysis of the streamline-diffusion finite element method on a Shishkin mesh for a convection\u2013diffusion problem with exponential layers. J. Numer. Math. 9, 59\u201376 (2001)","journal-title":"J. Numer. Math."},{"key":"1345_CR18","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1007\/s10092-018-0297-9","volume":"55","author":"G Singh","year":"2018","unstructured":"Singh, G., Natesan, S.: Superconvergence of discontinuous Galerkin method with interior penalties for singularly perturbed two-point boundary-value problems. Calcolo 55, 54 (2018). https:\/\/doi.org\/10.1007\/s10092-018-0297-9","journal-title":"Calcolo"},{"key":"1345_CR19","doi-asserted-by":"publisher","first-page":"538","DOI":"10.1016\/j.cma.2006.05.009","volume":"196","author":"L Tobiska","year":"2006","unstructured":"Tobiska, L.: Analysis of a new stabilized higher order finite element method for advection\u2013diffusion equations. Comput. Methods Appl. Mech. Eng. 196, 538\u2013550 (2006)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1345_CR20","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":"1345_CR21","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1007\/s10444-015-9415-2","volume":"42","author":"J Wang","year":"2016","unstructured":"Wang, J., Ye, X.: A weak Galerkin finite element method for the Stokes equations. Adv. Comput. Math. 42, 155\u2013174 (2016)","journal-title":"Adv. Comput. Math."},{"key":"1345_CR22","doi-asserted-by":"publisher","first-page":"2314","DOI":"10.1016\/j.camwa.2014.03.021","volume":"68","author":"C Wang","year":"2014","unstructured":"Wang, C., Wang, J.: An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes. Comput. Math. Appl. 68, 2314\u20132330 (2014)","journal-title":"Comput. Math. Appl."},{"key":"1345_CR23","first-page":"185","volume":"25","author":"ZQ Xie","year":"2007","unstructured":"Xie, Z.Q., Zhang, Z.: Superconvergence of DG method for one-dimensional singularly perturbed problems. J. Comput. Math. 25, 185\u2013200 (2007)","journal-title":"J. Comput. Math."},{"key":"1345_CR24","first-page":"280","volume":"27","author":"ZQ Xie","year":"2009","unstructured":"Xie, Z.Q., Zhang, Z.Z., Zhang, Z.: A numerical study of uniform superconvergence for solving singularly perturbed problems. J. Comput. Math. 27, 280\u2013298 (2009)","journal-title":"J. Comput. Math."},{"key":"1345_CR25","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1007\/s00211-005-0598-1","volume":"100","author":"H Zarin","year":"2005","unstructured":"Zarin, H., Roos, H.-G.: Interior penalty discontinuous approximations of convection\u2013diffusion problems with parabolic layers. Numer. Math. 100, 735\u2013759 (2005)","journal-title":"Numer. Math."},{"key":"1345_CR26","doi-asserted-by":"publisher","first-page":"1147","DOI":"10.1090\/S0025-5718-03-01486-8","volume":"245","author":"Z Zhang","year":"2003","unstructured":"Zhang, Z.: Finite element superconvergence on Shishkin mesh for 2-D convection\u2013diffusion problems. Math. Comput. 245, 1147\u20131177 (2003)","journal-title":"Math. Comput."},{"key":"1345_CR27","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1007\/s10915-009-9288-y","volume":"41","author":"ZZ Zhang","year":"2009","unstructured":"Zhang, Z.Z., Xie, Z.Q., Zhang, Z.: Superconvergence of discontinuous Galerkin methods for convection diffusion problems. J. Sci. Comput. 41, 70\u201393 (2009)","journal-title":"J. Sci. Comput."},{"key":"1345_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cam.2016.03.025","volume":"280","author":"T Zhang","year":"2016","unstructured":"Zhang, T., Tang, L.: A weak finite element method for elliptic problems in one space dimension. Appl. Math. Comput. 280, 1\u201310 (2016)","journal-title":"Appl. Math. Comput."},{"key":"1345_CR29","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1007\/s11464-014-0358-6","volume":"9","author":"R Zhang","year":"2014","unstructured":"Zhang, R., Song, H., Luan, N.: Weak Galerkin finite element method for valuation of American options. Front. Math. China 9, 455\u2013476 (2014)","journal-title":"Front. Math. China"},{"key":"1345_CR30","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1090\/S0025-5718-2013-02736-6","volume":"83","author":"H Zhu","year":"2014","unstructured":"Zhu, H., Zhang, Z.: Uniform convergence of the LDG method for a singularly perturbed problem with the exponential boundary layer. Math. Comput. 83, 635\u2013663 (2014)","journal-title":"Math. Comput."},{"key":"1345_CR31","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1016\/j.apnum.2013.10.001","volume":"76","author":"P Zhu","year":"2014","unstructured":"Zhu, P., Xie, S.: Higher order uniformly convergent continuous\/discontinuous Galerkin methods for singularly perturbed problems of convection-diffusion type. Appl. Numer. Math. 76, 48\u201359 (2014)","journal-title":"Appl. Numer. Math."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01345-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-020-01345-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01345-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,22]],"date-time":"2021-10-22T19:54:39Z","timestamp":1634932479000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-020-01345-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,23]]},"references-count":31,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,11]]}},"alternative-id":["1345"],"URL":"https:\/\/doi.org\/10.1007\/s10915-020-01345-3","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,23]]},"assertion":[{"value":"1 June 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 September 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 October 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 October 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"34"}}