{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:44:43Z","timestamp":1776843883090,"version":"3.51.2"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2011,9,17]],"date-time":"2011-09-17T00:00:00Z","timestamp":1316217600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2012,10]]},"DOI":"10.1007\/s10444-011-9215-2","type":"journal-article","created":{"date-parts":[[2011,9,16]],"date-time":"2011-09-16T14:03:27Z","timestamp":1316181807000},"page":"393-416","source":"Crossref","is-referenced-by-count":30,"title":["L 2 error estimates and superconvergence of the finite volume element methods on quadrilateral meshes"],"prefix":"10.1007","volume":"37","author":[{"given":"Junliang","family":"Lv","sequence":"first","affiliation":[]},{"given":"Yonghai","family":"Li","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2011,9,17]]},"reference":[{"key":"9215_CR1","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1137\/0724050","volume":"24","author":"RE Bank","year":"1987","unstructured":"Bank, R.E., Rose, D.J.: Some error estimates for the box method. SIAM J. Numer. Anal. 24, 777\u2013787 (1987)","journal-title":"SIAM J. Numer. Anal."},{"key":"9215_CR2","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3658-8","volume-title":"The Mathematical Theory of Finite Element Methods","author":"SC Brenner","year":"2002","unstructured":"Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 2nd edn. Springer, New York (2002)","edition":"2"},{"key":"9215_CR3","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1007\/BF01385651","volume":"58","author":"ZQ Cai","year":"1991","unstructured":"Cai, Z.Q.: On the finite volume element method. Numer. Math. 58, 713\u2013735 (1991)","journal-title":"Numer. Math."},{"key":"9215_CR4","first-page":"22","volume":"33","author":"ZY Chen","year":"1994","unstructured":"Chen, Z.Y.: L 2 estimates of linear element generalized difference schemes. Acta Sci. Natur. Univ. Sunyatseni 33, 22\u201328 (1994)","journal-title":"Acta Sci. Natur. Univ. Sunyatseni"},{"key":"9215_CR5","first-page":"163","volume":"3","author":"ZY Chen","year":"1994","unstructured":"Chen, Z.Y.: Superconvergence of generalized difference methods for elliptic boundary value problem. Numer. Math., A Journal of Chinese Univ. 3, 163\u2013171 (1994)","journal-title":"A Journal of Chinese Univ."},{"key":"9215_CR6","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1023\/A:1014577215948","volume":"16","author":"ZY Chen","year":"2002","unstructured":"Chen, Z.Y., Li, R.H., Zhou, A.H.: A note on the optimal L 2-estimate of the finite volume element method. Adv. Comput. Math. 16, 291\u2013303 (2002)","journal-title":"Adv. Comput. Math."},{"key":"9215_CR7","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1007\/s11425-008-0120-1","volume":"51","author":"ZY Chen","year":"2008","unstructured":"Chen, Z.Y., He C.N., Wu B.: High order finite volume methods for singular perturbation problems. Sci. China Ser. A 51, 1391\u20131400 (2008)","journal-title":"Sci. China Ser. A"},{"key":"9215_CR8","doi-asserted-by":"crossref","first-page":"991","DOI":"10.1090\/S0025-5718-99-01090-X","volume":"68","author":"SH Chou","year":"1999","unstructured":"Chou, S.H., Vassilevski, P.S.: A general mixed covolume framework for constructing conservative schemes for elliptic problems. Math. Comput. 68, 991\u20131011 (1999)","journal-title":"Math. Comput."},{"key":"9215_CR9","doi-asserted-by":"crossref","first-page":"3706","DOI":"10.1016\/j.cma.2006.10.025","volume":"196","author":"SH Chou","year":"2007","unstructured":"Chou, S.H., Ye, X.: Superconvergence of finite volume methods for the second order elliptic problem. Comput. Methods Appl. Mech. Eng. 196, 3706\u20133712 (2007)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"9215_CR10","doi-asserted-by":"crossref","first-page":"1850","DOI":"10.1137\/S0036142997321285","volume":"35","author":"SH Chou","year":"1998","unstructured":"Chou, S.H., Kwak, D.Y., Vassilevski, P.S.: Mixed covolume methods for elliptic problems on triangular grids. SIAM J. Numer. Anal. 35, 1850\u20131861 (1998)","journal-title":"SIAM J. Numer. Anal."},{"key":"9215_CR11","volume-title":"The Finite Element Methods for Elliptic Problems","author":"PG Ciarlet","year":"1978","unstructured":"Ciarlet, P.G.: The Finite Element Methods for Elliptic Problems. North-Holland, Amesterdam (1978)"},{"key":"9215_CR12","doi-asserted-by":"crossref","first-page":"772","DOI":"10.1137\/S0036142997322801","volume":"36","author":"RE Ewing","year":"1999","unstructured":"Ewing, R.E., Liu, M.M., Wang, J.P.: Superconvergence of mixed finite element aproximations over quadrialterals. SIAM J. Numer. Anal. 36, 772\u2013787 (1999)","journal-title":"SIAM J. Numer. Anal."},{"key":"9215_CR13","doi-asserted-by":"crossref","first-page":"1865","DOI":"10.1137\/S0036142900368873","volume":"39","author":"RE Ewing","year":"2002","unstructured":"Ewing, R.E., Lin, T., Lin, Y.P.: On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39, 1865\u20131888 (2002)","journal-title":"SIAM J. Numer. Anal."},{"key":"9215_CR14","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1051\/m2an\/1979130201391","volume":"3","author":"P Lesaint","year":"1979","unstructured":"Lesaint, P., Zl\u00e1mal, M.: Superconvergence of the gradient of finite element solutions. RAIRO Numer. Anal. 3, 139\u2013166 (1979)","journal-title":"RAIRO Numer. Anal."},{"key":"9215_CR15","first-page":"653","volume":"17","author":"YH Li","year":"1999","unstructured":"Li, Y.H., Li, R.H.: Generalized difference methods on arbitrary quadrilateral networks. J. Comput. Math. 17, 653\u2013672 (1999)","journal-title":"J. Comput. Math."},{"key":"9215_CR16","volume-title":"Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods","author":"RH Li","year":"2000","unstructured":"Li, R.H., Chen, Z.Y., Wu, W.: Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000)"},{"key":"9215_CR17","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s10444-009-9121-z","volume":"33","author":"JL Lv","year":"2010","unstructured":"Lv, J.L., Li, Y.H.: L 2 error estimate of the finite volume element methods on quadrilateral meshes. Adv. Comput. Math. 33, 129\u2013148 (2010)","journal-title":"Adv. Comput. Math."},{"key":"9215_CR18","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/BF02238536","volume":"51","author":"T Schmidt","year":"1993","unstructured":"Schmidt, T.: Box schemes on quadrilateral meshes. Compting 51, 271\u2013292 (1993)","journal-title":"Compting"},{"key":"9215_CR19","volume-title":"Finite Difference Schemes and Partial Differential Equations","author":"JC Strikwerda","year":"1989","unstructured":"Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations. Wadsworth Brooks\/Cole, Pacific Grove (1989)"},{"key":"9215_CR20","doi-asserted-by":"crossref","first-page":"1419","DOI":"10.1137\/0728073","volume":"28","author":"E S\u00fcli","year":"1991","unstructured":"S\u00fcli, E.: Convergence of finite volume schemes for Poisson\u2019s equation on nonuniform meshes. SIAM J. Numer. Anal. 28, 1419\u20131430 (1991)","journal-title":"SIAM J. Numer. Anal."},{"key":"9215_CR21","first-page":"229","volume":"33","author":"JP Wang","year":"2000","unstructured":"Wang, J.P.: A superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems. J. Math. Study 33, 229\u2013243 (2000)","journal-title":"J. Math. Study"},{"key":"9215_CR22","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1007\/s00211-008-0189-z","volume":"111","author":"JC Xu","year":"2009","unstructured":"Xu, J.C., Zou, Q.S.: Analysis of linear and quadratic simplical finite volume methods for elliptic equations. Numer. Math. 111, 469\u2013492 (2009)","journal-title":"Numer. Math."},{"key":"9215_CR23","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1051\/m2an:2007002","volume":"40","author":"M Yang","year":"2007","unstructured":"Yang, M.: A second-order finite volume element method on quadrilateral meshes for elliptic equations. M2AN, 40, 1053\u20131067 (2007)","journal-title":"M2AN"},{"key":"9215_CR24","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1002\/(SICI)1099-0887(199612)12:12<849::AID-CNM25>3.0.CO;2-N","volume":"12","author":"L Zhang","year":"1996","unstructured":"Zhang, L., Li, L.K.: On the superconvergence of isoparametric bilinear element. Commun. Numer. Methods Eng. 12, 849\u2013862 (1996)","journal-title":"Commun. Numer. Methods Eng."},{"key":"9215_CR25","first-page":"360","volume":"4","author":"PQ Zhu","year":"1982","unstructured":"Zhu, P.Q., Li, R.H.: Generalized difference methods for second order elliptic partial differential equations (II)-quadrilateral grids. Numer. Math., A Journal of Chinese Univ. 4, 360\u2013375 (1982)","journal-title":"Numer. Math., A Journal of Chinese Univ."},{"key":"9215_CR26","unstructured":"Zhu, Q.D., Lin, Q.: The Superconvergence Theory of Finite Elements. Hunan Science and Technology Publishing House, Changsha (1989). In Chinese"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-011-9215-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-011-9215-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-011-9215-2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T17:08:15Z","timestamp":1559149695000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-011-9215-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,9,17]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,10]]}},"alternative-id":["9215"],"URL":"https:\/\/doi.org\/10.1007\/s10444-011-9215-2","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,9,17]]}}}