{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T03:53:48Z","timestamp":1784260428810,"version":"3.55.0"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2019,3,22]],"date-time":"2019-03-22T00:00:00Z","timestamp":1553212800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11771162"],"award-info":[{"award-number":["11771162"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11726603"],"award-info":[{"award-number":["11726603"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871092"],"award-info":[{"award-number":["11871092"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["91430216"],"award-info":[{"award-number":["91430216"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U1530401"],"award-info":[{"award-number":["U1530401"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11471031"],"award-info":[{"award-number":["11471031"]}],"id":[{"id":"10.13039\/501100001809","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":[[2019,7]]},"DOI":"10.1007\/s10915-019-00943-0","type":"journal-article","created":{"date-parts":[[2019,3,22]],"date-time":"2019-03-22T13:02:51Z","timestamp":1553259771000},"page":"403-419","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":102,"title":["Linearized Galerkin FEMs for Nonlinear Time Fractional Parabolic Problems with Non-smooth Solutions in Time Direction"],"prefix":"10.1007","volume":"80","author":[{"given":"Dongfang","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3490-9912","authenticated-orcid":false,"given":"Chengda","family":"Wu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhimin","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2019,3,22]]},"reference":[{"key":"943_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-3455-9","volume-title":"Nonoscillation Theory of Functional Differential Equations with Applications","author":"PR Agarwal","year":"2012","unstructured":"Agarwal, P.R., Berezansky, L., Braverman, E., Domoshnitsky, A.: Nonoscillation Theory of Functional Differential Equations with Applications. Springer, New York (2012)"},{"key":"943_CR2","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/0370-1573(90)90099-N","volume":"195","author":"J Bouchaud","year":"1990","unstructured":"Bouchaud, J., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127\u2013293 (1990)","journal-title":"Phys. Rep."},{"key":"943_CR3","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1090\/S0025-5718-1985-0804933-3","volume":"45","author":"H Brunner","year":"1985","unstructured":"Brunner, H.: The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes. Math. Comput. 45, 417\u2013437 (1985)","journal-title":"Math. Comput."},{"key":"943_CR4","doi-asserted-by":"publisher","first-page":"6613","DOI":"10.1016\/j.jcp.2010.05.015","volume":"229","author":"H Brunner","year":"2010","unstructured":"Brunner, H., Ling, L., Yamamoto, M.: Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys. 229, 6613\u20136622 (2010)","journal-title":"J. Comput. Phys."},{"key":"943_CR5","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1090\/S0025-5718-06-01788-1","volume":"75","author":"E Cuesta","year":"2006","unstructured":"Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75, 673\u2013696 (2006)","journal-title":"Math. Comput."},{"key":"943_CR6","doi-asserted-by":"publisher","first-page":"A3070","DOI":"10.1137\/16M1070323","volume":"38","author":"W Cao","year":"2016","unstructured":"Cao, W., Zeng, F., Zhang, Z., Karniadakis, G.E.: Implicit\u2013explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions. SIAM J. Sci. Comput. 38, A3070\u2013A3093 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"943_CR7","doi-asserted-by":"publisher","first-page":"160","DOI":"10.1016\/j.aml.2018.05.007","volume":"84","author":"X Chen","year":"2018","unstructured":"Chen, X., Di, Y., Duan, J., Li, D.: Linearized compact ADI schemes for nonlinear time-fractional Schrodinger equations. Appl. Math. Lett. 84, 160\u2013167 (2018)","journal-title":"Appl. Math. Lett."},{"key":"943_CR8","doi-asserted-by":"crossref","unstructured":"Chen, Y., Wu, L.: Second Order Elliptic Equations and Elliptic Systems, Translations of Mathematical Monographs 174. AMS, USA (1998)","DOI":"10.1090\/mmono\/174"},{"key":"943_CR9","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1016\/j.jcp.2014.10.016","volume":"293","author":"F Chen","year":"2015","unstructured":"Chen, F., Xu, Q., Hesthaven, J.S.: A multi-domain spectral method for time-fractional differential equations. J. Comput. Phys. 293, 157\u2013172 (2015)","journal-title":"J. Comput. Phys."},{"key":"943_CR10","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1515\/cmam-2017-0027","volume":"18","author":"D Hou","year":"2018","unstructured":"Hou, D., Hasan, M.T., Xu, X.: M\u00fcntz spectral methods for the time-fractional diffusion equation. Comput. Methods Appl. Math. 18, 43\u201362 (2018)","journal-title":"Comput. Methods Appl. Math."},{"key":"943_CR11","doi-asserted-by":"publisher","first-page":"A3129","DOI":"10.1137\/17M1118816","volume":"39","author":"B Jin","year":"2017","unstructured":"Jin, B., Li, B., Zhou, Z.: Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM. J. Sci. Comput. 39, A3129\u2013A3152 (2017)","journal-title":"SIAM. J. Sci. Comput."},{"key":"943_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/16M1089320","volume":"56","author":"B Jin","year":"2018","unstructured":"Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM. J. Numer. Anal. 56, 1\u201323 (2018)","journal-title":"SIAM. J. Numer. Anal."},{"key":"943_CR13","volume-title":"Theory and Application of Fractional Differential Equations","author":"AA Kilbas","year":"2006","unstructured":"Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. Elsevier, Amsterdam (2006)"},{"key":"943_CR14","doi-asserted-by":"publisher","first-page":"1763","DOI":"10.1137\/15M1031734","volume":"54","author":"KN Le","year":"2016","unstructured":"Le, K.N., Mclean, W., Mustapha, K.: Numerical solution of the time-fractional Fokker\u2013Planck equation with general forcing. SIAM. J. Numer. Anal. 54, 1763\u20131784 (2016)","journal-title":"SIAM. J. Numer. Anal."},{"key":"943_CR15","doi-asserted-by":"publisher","first-page":"933","DOI":"10.1137\/120892465","volume":"52","author":"B Li","year":"2014","unstructured":"Li, B., Gao, H., Sun, W.: Unconditionally optimal error estimate of a Crank\u2013Nicolson Galerkin method for nonlinear thermistor equations. SIAM J. Numer. Anal. 52, 933\u2013954 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"943_CR16","first-page":"622","volume":"10","author":"B Li","year":"2013","unstructured":"Li, B., Sun, W.: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. Model. 10, 622\u2013633 (2013)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"943_CR17","doi-asserted-by":"publisher","first-page":"1959","DOI":"10.1137\/120871821","volume":"51","author":"B Li","year":"2013","unstructured":"Li, B., Sun, W.: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51, 1959\u20131977 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"943_CR18","first-page":"86","volume":"24","author":"D Li","year":"2018","unstructured":"Li, D., Liao, H., Sun, W., Wang, J., Zhang, J.: Analysis of L1-Galerkin FEMs for time fractional nonlinear parabolic problems. Commun. Comput. Phys. 24, 86\u2013103 (2018)","journal-title":"Commun. Comput. Phys."},{"key":"943_CR19","doi-asserted-by":"publisher","first-page":"892","DOI":"10.1007\/s10915-017-0381-3","volume":"72","author":"D Li","year":"2017","unstructured":"Li, D., Wang, J.: Unconditionally optimal error analysis of Crank\u2013Nicolson Galerkin FEMs for a strongly nonlinear parabolic system. J. Sci. Comput. 72, 892\u2013915 (2017)","journal-title":"J. Sci. Comput."},{"key":"943_CR20","doi-asserted-by":"publisher","first-page":"A3067","DOI":"10.1137\/16M1105700","volume":"39","author":"D Li","year":"2017","unstructured":"Li, D., Wang, J., Zhang, J.: Unconditionally convergent \n                    \n                      \n                    \n                    $$L1$$\n                    \n                      \n                        \n                          L\n                          1\n                        \n                      \n                    \n                  -Galerkin FEMs for nonlinear time-fractional Schr\u00f6dinger equations. SIAM. J. Sci. Comput. 39, A3067\u2013A3088 (2017)","journal-title":"SIAM. J. Sci. Comput."},{"key":"943_CR21","doi-asserted-by":"publisher","first-page":"848","DOI":"10.1007\/s10915-018-0642-9","volume":"76","author":"D Li","year":"2018","unstructured":"Li, D., Zhang, J., Zhang, Z.: Unconditionally optimal error estimates of a linearized Galerkin method for nonlinear time fractional reaction\u2013subdiffusion equations. J. Sci. Comput. 76, 848\u2013866 (2018)","journal-title":"J. Sci. Comput."},{"key":"943_CR22","doi-asserted-by":"publisher","first-page":"439","DOI":"10.4208\/eajam.031116.080317a","volume":"7","author":"H Li","year":"2017","unstructured":"Li, H., Wu, X., Zhang, J.: Numerical solution of the time-fractional sub-diffusion equation on an unbounded domain in two-dimensional space. East Asia J. Appl. Math. 7, 439\u2013454 (2017)","journal-title":"East Asia J. Appl. Math."},{"key":"943_CR23","doi-asserted-by":"publisher","first-page":"1112","DOI":"10.1137\/17M1131829","volume":"56","author":"H Liao","year":"2018","unstructured":"Liao, H., Li, D., Zhang, J.: Sharp error estimate of the nonuniform L1 formula for linear reaction\u2013subdiffusion equations. SIAM J. Numer. Anal. 56, 1112\u20131133 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"943_CR24","volume-title":"Automated Solution of Differential Equations by the Finite Element Method","year":"2012","unstructured":"Logg, A., Mardal, K., Wells, G. (eds.): Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)"},{"key":"943_CR25","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1017\/S1446181111000617","volume":"52","author":"W McLean","year":"2010","unstructured":"McLean, W.: Regularity of solutions to a time-fractional diffusion equation. ANZIAM J. 52, 123\u2013138 (2010)","journal-title":"ANZIAM J."},{"key":"943_CR26","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1007\/s00211-014-0669-2","volume":"130","author":"K Mustapha","year":"2015","unstructured":"Mustapha, K.: Time-stepping discontinuous Galerkin methods for fractional diffusion problems. Numer. Math. 130, 497\u2013516 (2015)","journal-title":"Numer. Math."},{"key":"943_CR27","doi-asserted-by":"publisher","first-page":"2259","DOI":"10.1090\/mcom\/3304","volume":"87","author":"K Mustapha","year":"2018","unstructured":"Mustapha, K.: FEM for time-fractional diffusion equations, novel optimal error analysis. Math. Comput. 87, 2259\u20132272 (2018)","journal-title":"Math. Comput."},{"key":"943_CR28","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1093\/imanum\/drn075","volume":"30","author":"K Mustapha","year":"2010","unstructured":"Mustapha, K., Mustapha, H.: A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel. IMA. J. Numer. Anal. 30, 555\u2013578 (2010)","journal-title":"IMA. J. Numer. Anal."},{"key":"943_CR29","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1016\/j.jmaa.2011.04.058","volume":"382","author":"K Sakamoto","year":"2011","unstructured":"Sakamoto, K., Yamamoto, M.: Initial value\/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426\u2013447 (2011)","journal-title":"J. Math. Anal. Appl."},{"key":"943_CR30","doi-asserted-by":"publisher","first-page":"854","DOI":"10.4208\/nmtma.2018.s10","volume":"11","author":"C Shen","year":"2018","unstructured":"Shen, C., Shen, J.: A space-time Petrov\u2013Galerkin spectral method for time fractional diffusion. Numer. Math. Theor. Methods Appl. 11, 854\u2013876 (2018)","journal-title":"Numer. Math. Theor. Methods Appl."},{"key":"943_CR31","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/s00211-015-0767-9","volume":"134","author":"Z Si","year":"2016","unstructured":"Si, Z., Wang, J., Sun, W.: Unconditional stability and error estimates of modified characteristics FEMs for the Navier\u2013Stokes equations. Numer. Math. 134, 139\u2013161 (2016)","journal-title":"Numer. Math."},{"key":"943_CR32","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1016\/j.aml.2017.05.016","volume":"74","author":"M Stynes","year":"2017","unstructured":"Stynes, M., Gracia, J.L.: Preprocessing schemes for fractional-derivative problems to improve their convergence rates. Appl. Math. Lett. 74, 187\u2013192 (2017)","journal-title":"Appl. Math. Lett."},{"key":"943_CR33","doi-asserted-by":"publisher","first-page":"1057","DOI":"10.1137\/16M1082329","volume":"55","author":"M Stynes","year":"2017","unstructured":"Stynes, M., O\u2019Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057\u20131079 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"943_CR34","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03359-3","volume-title":"Galerkin Finite Element Methods for Parabolic Problems","author":"V Thom\u00e9e","year":"1997","unstructured":"Thom\u00e9e, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (1997)"},{"key":"943_CR35","doi-asserted-by":"publisher","first-page":"1291","DOI":"10.1137\/17M113544X","volume":"56","author":"C Wu","year":"2018","unstructured":"Wu, C., Sun, W.: Analysis of Galerkin FEMs for mixed formulation of time-dependent Ginzburg\u2013Landau equations under temporal gauge. SIAM. J. Numer. Anal. 56, 1291\u20131312 (2018)","journal-title":"SIAM. J. Numer. Anal."},{"key":"943_CR36","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1016\/j.jcp.2017.12.035","volume":"357","author":"Y Xing","year":"2018","unstructured":"Xing, Y., Yan, Y.: A higher order numerical method for time fractional partial differential equations with nonsmooth data. J. Comput. Phys. 357, 305\u2013323 (2018)","journal-title":"J. Comput. Phys."},{"key":"943_CR37","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1137\/16M1094257","volume":"56","author":"Y Yan","year":"2018","unstructured":"Yan, Y., Khan, M., Ford, N.J.: An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data. SIAM. J. Numer. Anal. 56, 210\u2013227 (2018)","journal-title":"SIAM. J. Numer. Anal."},{"key":"943_CR38","doi-asserted-by":"publisher","first-page":"036126","DOI":"10.1103\/PhysRevE.69.036126","volume":"69","author":"SB Yuste","year":"2004","unstructured":"Yuste, S.B., Acedo, L., Lindenberg, K.: Reaction front in an \n                    \n                      \n                    \n                    $$A+B\\rightarrow C$$\n                    \n                      \n                        \n                          A\n                          +\n                          B\n                          \u2192\n                          C\n                        \n                      \n                    \n                   reaction\u2013subdiffusion process. Phys. Rev. E. 69, 036126 (2004)","journal-title":"Phys. Rev. E."},{"key":"943_CR39","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.jcp.2014.02.008","volume":"265","author":"Y Zhang","year":"2014","unstructured":"Zhang, Y., Sun, Z., Liao, H.: Finite difference methods for time fractional diffusion equations on no-uniform meshes. J. Comput. Phys. 265, 195\u2013210 (2014)","journal-title":"J. Comput. Phys."},{"key":"943_CR40","doi-asserted-by":"publisher","first-page":"1209","DOI":"10.1007\/s11425-015-5113-2","volume":"59","author":"X Zhou","year":"2016","unstructured":"Zhou, X., Xu, C.: Well-posedness of a kind of nonlinear coupled system of fractional differential equations. Sci. China Math. 59, 1209\u20131220 (2016)","journal-title":"Sci. China Math."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-00943-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-019-00943-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-00943-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,21]],"date-time":"2020-03-21T00:18:21Z","timestamp":1584749901000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-019-00943-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,22]]},"references-count":40,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,7]]}},"alternative-id":["943"],"URL":"https:\/\/doi.org\/10.1007\/s10915-019-00943-0","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,22]]},"assertion":[{"value":"24 October 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 January 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 March 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 March 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}