{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:42:15Z","timestamp":1740123735507,"version":"3.37.3"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,3,28]],"date-time":"2023-03-28T00:00:00Z","timestamp":1679961600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,3,28]],"date-time":"2023-03-28T00:00:00Z","timestamp":1679961600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100004735","name":"Natural Science Foundation of Hunan Province","doi-asserted-by":"publisher","award":["2020JJ4242, 2021JJ30178"],"award-info":[{"award-number":["2020JJ4242, 2021JJ30178"]}],"id":[{"id":"10.13039\/501100004735","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2023,10]]},"DOI":"10.1007\/s11075-023-01518-9","type":"journal-article","created":{"date-parts":[[2023,3,28]],"date-time":"2023-03-28T15:02:56Z","timestamp":1680015776000},"page":"733-763","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A two-grid combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem"],"prefix":"10.1007","volume":"94","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1161-6663","authenticated-orcid":false,"given":"Jiming","family":"Yang","sequence":"first","affiliation":[]},{"given":"Jing","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,3,28]]},"reference":[{"key":"1518_CR1","doi-asserted-by":"crossref","unstructured":"F. Brezzi, J. Douglas JR and L. D. Marini, Two families of mixed finite elements for second order elliptic problems. Appl. Numer. Math. 47 (1985), 217\u2013235","DOI":"10.1007\/BF01389710"},{"key":"1518_CR2","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1016\/j.apnum.2009.08.004","volume":"60","author":"C Chen","year":"2010","unstructured":"Chen, C., Liu, W.: Two-grid finite volume element methods for semilinear parabolic problems. Appl. Numer. Math. 60, 10\u201318 (2010)","journal-title":"Appl. Numer. Math."},{"key":"1518_CR3","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1016\/j.cam.2008.09.001","volume":"228","author":"C Chen","year":"2009","unstructured":"Chen, C., Yang, M., Bi, C.: Two-grid methods for finite volume element approximations of nonlinear parabolic equations. J. Comp. Appl. Math. 228, 123\u2013132 (2009)","journal-title":"J. Comp. Appl. Math."},{"issue":"2","key":"1518_CR4","doi-asserted-by":"publisher","first-page":"203","DOI":"10.4208\/aamm.12-m12130","volume":"6","author":"L Chen","year":"2014","unstructured":"Chen, L., Chen, Y.: Two-grid discretization scheme for nonlinear reaction diffusion equations by mixed finite element methods. Adv. Appl. Math. Mech. 6(2), 203\u2013219 (2014)","journal-title":"Adv. Appl. Math. Mech."},{"issue":"2","key":"1518_CR5","doi-asserted-by":"publisher","first-page":"408","DOI":"10.1002\/nme.1775","volume":"69","author":"Y Chen","year":"2007","unstructured":"Chen, Y., Liu, H., Liu, S.: Analysis of two-grid methods for reaction diffusion equations by expanded mixed finite element methods. Int. J. Numer. Meth. Eng. 69(2), 408\u2013422 (2007)","journal-title":"Int. J. Numer. Meth. Eng."},{"issue":"5","key":"1518_CR6","doi-asserted-by":"publisher","first-page":"1503","DOI":"10.4208\/cicp.scpde14.46s","volume":"19","author":"Y Chen","year":"2016","unstructured":"Chen, Y., Hu, H.: Two-grid method for miscible displacement problem by mixed finite element methods and mixed finite element method of characteristics. Commun. Comput. Phys. 19(5), 1503\u20131528 (2016)","journal-title":"Commun. Comput. Phys."},{"issue":"2","key":"1518_CR7","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1002\/nme.668","volume":"57","author":"Y Chen","year":"2003","unstructured":"Chen, Y., Huang, Y., Yu, D.: A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations. Int. J. Numer. Meth. Eng. 57(2), 193\u2013209 (2003)","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"1518_CR8","first-page":"1","volume":"2012","author":"C Chen","year":"2012","unstructured":"Chen, C., Liu, W.: A two-grid method for finite element solutions of nonlinear parabolic equations. Abs. Appl. Anal. 2012, 1\u201311 (2012)","journal-title":"Abs. Appl. Anal."},{"issue":"2","key":"1518_CR9","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1007\/s10444-018-9628-2","volume":"45","author":"C Chen","year":"2019","unstructured":"Chen, C., Li, K., Chen, Y., Huang, Y.: Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations. Adv. Comp. Math. 45(2), 611\u2013630 (2019)","journal-title":"Adv. Comp. Math."},{"issue":"2","key":"1518_CR10","first-page":"222","volume":"4","author":"A Cheng","year":"1995","unstructured":"Cheng, A.: Optimal error estimate in $$L^\\infty (J;L^2 (\\Omega ))$$ norm of FEM for a model for compressible miscible displacement in porous media. Numer. Math. 4(2), 222\u2013236 (1995)","journal-title":"Numer. Math."},{"issue":"2","key":"1518_CR11","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1137\/S0036142995293493","volume":"35","author":"CN Dawson","year":"1998","unstructured":"Dawson, C.N., Wheeler, M.F., Woodward, C.S.: A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35(2), 435\u2013452 (1998)","journal-title":"SIAM J. Numer. Anal."},{"key":"1518_CR12","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1090\/conm\/180\/01971","volume":"180","author":"CN Dawson","year":"1994","unstructured":"Dawson, C.N., Wheeler, M.F.: Two-grid methods for mixed finite element approximations of nonlinear parabolic equations. Contemp. Math. 180, 191\u2013203 (1994)","journal-title":"Contemp. Math."},{"issue":"11","key":"1518_CR13","doi-asserted-by":"publisher","first-page":"2694","DOI":"10.1016\/j.camwa.2016.09.002","volume":"72","author":"H Hu","year":"2016","unstructured":"Hu, H., Chen, Y., Zhou, J.: Two-grid method for miscible displacement problem by mixed finite element methods and finite element method of characteristics. Comp. Math. Appl. 72(11), 2694\u20132715 (2016)","journal-title":"Comp. Math. Appl."},{"key":"1518_CR14","doi-asserted-by":"publisher","first-page":"879","DOI":"10.1007\/s11075-018-0575-2","volume":"81","author":"H Hu","year":"2019","unstructured":"Hu, H., Fu, Y., Zhou, J.: Numerical solution of a miscible displacement problem with dispersion term using a two-grid mixed finite element approach. Numer. Algor. 81, 879\u2013914 (2019)","journal-title":"Numer. Algor."},{"key":"1518_CR15","unstructured":"Q. Lin and N. Yan, Structure and analysis for efficient finite element methods, Publishers of Hebei university (in Chinese), 2002"},{"issue":"8","key":"1518_CR16","doi-asserted-by":"publisher","first-page":"1453","DOI":"10.1080\/00207160.2017.1322689","volume":"95","author":"S Liu","year":"2018","unstructured":"Liu, S., Chen, Y., Huang, Y., Zhou, J.: Two-grid methods for miscible displacement problem by Galerkin methods and mixed finite-element methods. Int. J. Comp. Math. 95(8), 1453\u20131477 (2018)","journal-title":"Int. J. Comp. Math."},{"key":"1518_CR17","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1006\/jcph.1998.6032","volume":"146","author":"JT Oden","year":"1998","unstructured":"Oden, J.T., Babu\u0161ka, I., Baumann, C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146, 491\u2013516 (1998)","journal-title":"J. Comput. Phys."},{"key":"1518_CR18","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1016\/j.jmaa.2005.07.027","volume":"315","author":"MR Ohm","year":"2000","unstructured":"Ohm, M.R., Lee, H.Y., Shin, J.Y.: Error estimates for discontinuous Galerkin method for nonlinear parabolic equations. J. Math. Anal. Appl. 315, 132\u2013143 (2000)","journal-title":"J. Math. Anal. Appl."},{"key":"1518_CR19","first-page":"292","volume-title":"A mixed finite element method for second order elliptic problems","author":"RA Raviart","year":"1977","unstructured":"Raviart, R.A., Thomas, J.M.: A mixed finite element method for second order elliptic problems, pp. 292\u2013315. Lecture notes in Mathematics, Springer, New York, Mathematics Aspects of the Finite Element Method (1977)"},{"key":"1518_CR20","doi-asserted-by":"crossref","unstructured":"B. Rivi\u00e8re, Discontinuous Galerkin methods for solving elliptic and parabolic equations: Theory and implementation, SIAM, 2008","DOI":"10.1137\/1.9780898717440"},{"key":"1518_CR21","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1002\/cnm.464","volume":"18","author":"B Rivi\u00e8re","year":"2002","unstructured":"Rivi\u00e8re, B., Wheeler, M.F.: Discontinuous Galerkin methods for coupled flow and transport problems. Comm. Numer. Methods Eng. 18, 63\u201368 (2002)","journal-title":"Comm. Numer. Methods Eng."},{"key":"1518_CR22","series-title":"Lecture Notes in Comput","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1007\/978-3-642-59721-3_17","volume-title":"Discontinuous Galerkin Methods: Theory, Computation and Applications","author":"B Rivi\u00e8re","year":"2000","unstructured":"Rivi\u00e8re, B., Wheeler, M.F.: A discontinuous Galerkin method applied to nonlinear parabolic equations. In: Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Comput, pp. 231\u2013244. Springer-Verlag, Sci. and Engrg. (2000)"},{"issue":"3\u20134","key":"1518_CR23","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1023\/A:1011591328604","volume":"3","author":"B Rivi\u00e8re","year":"1999","unstructured":"Rivi\u00e8re, B., Wheeler, M.F., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems (Part I). Comput. Geosci. 3(3\u20134), 337\u2013360 (1999)","journal-title":"Comput. Geosci."},{"key":"1518_CR24","doi-asserted-by":"publisher","first-page":"1289","DOI":"10.1016\/S0898-1221(03)90220-3","volume":"46","author":"A Romkes","year":"2003","unstructured":"Romkes, A., Prudhomme, S., Oden, J.T.: A priori error analysis of a stabilized discontinuous Galerkin method. Comp. Math. Appl. 46, 1289\u20131311 (2003)","journal-title":"Comp. Math. Appl."},{"key":"1518_CR25","unstructured":"S. Sun, Discontinuous Galerkin Methods for Reactive Transport in Porous Media, Ph. D. thesis, The university of Texas at Austin, (2003)"},{"key":"1518_CR26","doi-asserted-by":"crossref","unstructured":"S. Sun, B. Rivi\u00e8re and M. F. Wheeler, A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media,Recent Progress in Computational and Applied PDEs , Kluwer Academic Publishers, Plenum Press, Dordrecht, New York, (2002), 323-351","DOI":"10.1007\/978-1-4615-0113-8_23"},{"key":"1518_CR27","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1016\/j.apnum.2004.08.035","volume":"52","author":"S Sun","year":"2005","unstructured":"Sun, S., Wheeler, M.F.: Discontinuous Galerkin methods for coupled flow and reactive transport problems. Appl. Numer. Math. 52, 273\u2013298 (2005)","journal-title":"Appl. Numer. Math."},{"key":"1518_CR28","first-page":"511","volume":"22","author":"S Sun","year":"2005","unstructured":"Sun, S., Wheeler, M.F.: $$L^{2}(H^{1})$$ norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems. J. Sci. Comp. 22, 511\u2013540 (2005)","journal-title":"J. Sci. Comp."},{"issue":"5","key":"1518_CR29","doi-asserted-by":"publisher","first-page":"1759","DOI":"10.1137\/S0036142992232949","volume":"33","author":"J Xu","year":"1996","unstructured":"Xu, J.: Two-grid discretization techniques for linear and nonlinear PDE. SIAM J. Numer. Anal. 33(5), 1759\u20131777 (1996)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"1518_CR30","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1137\/0915016","volume":"15","author":"J Xu","year":"1994","unstructured":"Xu, J.: A novel two-grid method for semilinear equations. SIAM J. Sci. Comp. 15(1), 231\u2013237 (1994)","journal-title":"SIAM J. Sci. Comp."},{"issue":"3","key":"1518_CR31","first-page":"425","volume":"24","author":"J Yang","year":"2006","unstructured":"Yang, J., Chen, Y.: A unified a posteriori error analysis for discontinuous Galerkin approximations of reactive transport equations. J. Comput. Math. 24(3), 425\u2013434 (2006)","journal-title":"J. Comput. Math."},{"issue":"4","key":"1518_CR32","doi-asserted-by":"publisher","first-page":"1523","DOI":"10.1007\/s11075-020-00943-4","volume":"86","author":"J Yang","year":"2021","unstructured":"Yang, J., Zhou, J.: A two-grid method for discontinuous Galerkin approximations to nonlinear Sobolev equations. Numer. Algo. 86(4), 1523\u20131541 (2021)","journal-title":"Numer. Algo."},{"key":"1518_CR33","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1016\/j.amc.2018.09.067","volume":"346","author":"J Yang","year":"2019","unstructured":"Yang, J., Xing, X.: A two-grid discontinuous Galerkin method for a kind of nonlinear parabolic problems. Appl. Math. Comp. 346, 96\u2013108 (2019)","journal-title":"Appl. Math. Comp."},{"issue":"6","key":"1518_CR34","doi-asserted-by":"publisher","first-page":"1801","DOI":"10.1002\/num.21777","volume":"29","author":"J Yang","year":"2013","unstructured":"Yang, J., Chen, Y., Xiong, Z.: Superconvergence of a full-discrete combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem. Numer. Methods PDEs 29(6), 1801\u20131820 (2013)","journal-title":"Numer. Methods PDEs"},{"issue":"1","key":"1518_CR35","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s10440-015-0017-2","volume":"142","author":"J Yang","year":"2016","unstructured":"Yang, J., Xiong, Z.: Superconvergence analysis of a full-discrete combined mixed finite element and discontinuous Galerkin approximation for an incompressible miscible displacement problem. Acta Appl. Math. 142(1), 107\u2013121 (2016)","journal-title":"Acta Appl. Math."},{"issue":"3","key":"1518_CR36","doi-asserted-by":"publisher","first-page":"1106","DOI":"10.1016\/j.apm.2011.07.054","volume":"36","author":"J Yang","year":"2012","unstructured":"Yang, J., Chen, Y.: Superconvergence of a combined mixed finite element and discontinuous Galerkin approximation for an incompressible miscible displacement problem. Appl. Math. Modell. 36(3), 1106\u20131113 (2012)","journal-title":"Appl. Math. Modell."},{"issue":"1","key":"1518_CR37","doi-asserted-by":"publisher","first-page":"91","DOI":"10.4208\/jcm.1006-m2991","volume":"29","author":"J Yang","year":"2011","unstructured":"Yang, J., Chen, Y.: A priori error estimates of a combined mixed finite element and discontinuous Galerkin method for compressible miscible displacement with molecular diffusion and dispersion. J. Comput. Math. 29(1), 91\u2013107 (2011)","journal-title":"J. Comput. Math."},{"key":"1518_CR38","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/j.cam.2017.12.041","volume":"337","author":"J Zeng","year":"2018","unstructured":"Zeng, J., Chen, Y., Hu, H.: Two-grid method for compressible miscible displacement problem by CFEM-MFEM. J. Comp. Appl. Math. 337, 175\u2013189 (2018)","journal-title":"J. Comp. Appl. Math."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01518-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-023-01518-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01518-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,5]],"date-time":"2023-09-05T09:15:25Z","timestamp":1693905325000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-023-01518-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,28]]},"references-count":38,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["1518"],"URL":"https:\/\/doi.org\/10.1007\/s11075-023-01518-9","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2023,3,28]]},"assertion":[{"value":"4 November 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 February 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 March 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"No conflict of interest exists. We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}