{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T00:38:35Z","timestamp":1767919115504,"version":"3.49.0"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,8,1]],"date-time":"2020-08-01T00:00:00Z","timestamp":1596240000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,8,1]],"date-time":"2020-08-01T00:00:00Z","timestamp":1596240000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11601119"],"award-info":[{"award-number":["11601119"]}],"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"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2020,8]]},"DOI":"10.1007\/s10915-020-01290-1","type":"journal-article","created":{"date-parts":[[2020,8,3]],"date-time":"2020-08-03T20:04:43Z","timestamp":1596485083000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Superconvergence Error Estimate of a Finite Element Method on Nonuniform Time Meshes for Reaction\u2013Subdiffusion Equations"],"prefix":"10.1007","volume":"84","author":[{"given":"Jincheng","family":"Ren","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong-lin","family":"Liao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhimin","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,8,3]]},"reference":[{"key":"1290_CR1","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1016\/j.jcp.2014.09.031","volume":"280","author":"AA Alikhanov","year":"2015","unstructured":"Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424\u2013438 (2015)","journal-title":"J. Comput. Phys."},{"key":"1290_CR2","doi-asserted-by":"publisher","first-page":"624","DOI":"10.1007\/s10915-018-0863-y","volume":"79","author":"H Chen","year":"2019","unstructured":"Chen, H., Stynes, M.: Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. J. Sci. Comput. 79, 624\u2013647 (2019)","journal-title":"J. Sci. Comput."},{"key":"1290_CR3","series-title":"Finite Element Methods (Part 1)","volume-title":"Handbook of Numerical Analysis","author":"PG Ciarlet","year":"1991","unstructured":"Ciarlet, P.G., Lions, J.L.: Handbook of Numerical Analysis. Finite Element Methods (Part 1), vol. II. North-Holland, Amsterdam (1991)"},{"key":"1290_CR4","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1137\/080714130","volume":"47","author":"WH Deng","year":"2008","unstructured":"Deng, W.H.: Finite element method for the space and time fractional Fokker\u2013Planck equation. SIAM J. Numer. Anal. 47, 204\u2013226 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"1290_CR5","series-title":"Lecture Notes in Mathematics","volume-title":"The Analysis of Fractional Differential Equations","author":"K Diethelm","year":"2004","unstructured":"Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (2004)"},{"key":"1290_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-020-09782-2","author":"BQ Ji","year":"2020","unstructured":"Ji, B.Q., Liao, H.-L., Zhang, L.M.: Simple maximum-principle preserving time-stepping methods for time-fractional Allen\u2013Cahn equation. Adv. Comput. Math. (2020). https:\/\/doi.org\/10.1007\/s10444-020-09782-2","journal-title":"Adv. Comput. Math."},{"key":"1290_CR7","doi-asserted-by":"publisher","first-page":"3285","DOI":"10.1016\/j.cam.2011.01.011","volume":"235","author":"Y Jiang","year":"2011","unstructured":"Jiang, Y., Ma, J.T.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235, 3285\u20133290 (2011)","journal-title":"J. Comput. Appl. Math."},{"key":"1290_CR8","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1093\/imanum\/dru018","volume":"35","author":"B Jin","year":"2015","unstructured":"Jin, B., Lazarov, R., Pascal, J., Zhou, Z.: Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion. IMA J. Numer. Anal. 35, 561\u2013582 (2015)","journal-title":"IMA J. Numer. Anal."},{"key":"1290_CR9","first-page":"197","volume":"36","author":"B Jin","year":"2016","unstructured":"Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 197\u2013221 (2016)","journal-title":"IMA J. Numer. Anal."},{"key":"1290_CR10","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1137\/120873984","volume":"51","author":"B Jin","year":"2013","unstructured":"Jin, B., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51, 445\u2013466 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"1290_CR11","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1016\/j.jcp.2014.10.051","volume":"281","author":"B Jin","year":"2015","unstructured":"Jin, B., Lazarov, R., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825\u2013843 (2015)","journal-title":"J. Comput. Phys."},{"key":"1290_CR12","doi-asserted-by":"publisher","first-page":"A146","DOI":"10.1137\/140979563","volume":"38","author":"B Jin","year":"2016","unstructured":"Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations. SIAM J. Sci. Comput. 38, A146\u2013A170 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"1290_CR13","first-page":"86","volume":"24","author":"DF Li","year":"2018","unstructured":"Li, D.F., Liao, H.-L., Sun, W., Wang, J., Zhang, J.W.: Analysis of L1-Galerkin FEMs for time-fractional nonlinear parabolic problems. Commun. Comput. Phys. 24, 86\u2013103 (2018)","journal-title":"Commun. Comput. Phys."},{"key":"1290_CR14","doi-asserted-by":"publisher","first-page":"A3067","DOI":"10.1137\/16M1105700","volume":"39","author":"DF Li","year":"2017","unstructured":"Li, D.F., Wang, J., Zhang, J.W.: Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional Schr\u00f6dinger equations. SIAM J. Sci. Comput. 39, A3067\u2013A3088 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"1290_CR15","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1007\/s10915-019-00943-0","volume":"80","author":"DF Li","year":"2019","unstructured":"Li, D.F., Wu, C., Zhang, Z.M.: Linearized Galerkin FEMs for nonlinear time fractional Parabolic problems with non-smooth solutions in time direction. J. Sci. Comput. 80, 403\u2013419 (2019)","journal-title":"J. Sci. Comput."},{"key":"1290_CR16","doi-asserted-by":"publisher","first-page":"1223","DOI":"10.1007\/s11075-019-00722-w","volume":"83","author":"X Li","year":"2020","unstructured":"Li, X., Zhang, L., Liao, H.-L.: Sharp $$H^1$$-norm error estimate of a cosine pseudo-spectral scheme for 2D reaction-subdiffusion equations. Numer. Algor. 83, 1223\u20131248 (2020)","journal-title":"Numer. Algor."},{"key":"1290_CR17","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.jcp.2017.06.036","volume":"347","author":"Z Li","year":"2017","unstructured":"Li, Z., Wang, H., Yang, D.P.: A space-time fractional phase-field model with tunable sharpness and decay behavior and its efficient numerical simulation. J. Comput. Phys. 347, 20\u201338 (2017)","journal-title":"J. Comput. Phys."},{"key":"1290_CR18","doi-asserted-by":"publisher","first-page":"1112","DOI":"10.1137\/17M1131829","volume":"56","author":"H-L Liao","year":"2018","unstructured":"Liao, H.-L., Li, D.F., Zhang, J.W.: Sharp error estimate of nonuniform L1 formula for linear reaction-subdiffusion equations. SIAM J. Numer. Anal. 56, 1112\u20131133 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"1290_CR19","doi-asserted-by":"publisher","first-page":"218","DOI":"10.1137\/16M1175742","volume":"57","author":"H-L Liao","year":"2019","unstructured":"Liao, H.-L., McLean, W., Zhang, J.W.: A discrete Gr\u00f6nwall inequality with application to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57, 218\u2013237 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"1290_CR20","unstructured":"Liao, H.-L., McLean, W., Zhang, J.W.: A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion equation (2018). arXiv: 1803.09873v2"},{"key":"1290_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-019-00927-0","volume":"80","author":"H-L Liao","year":"2019","unstructured":"Liao, H.-L., Yan, Y., Zhang, J.W.: Unconditional convergence of a two-level linearized fast algorithm for semilinear subdiffusion equations. J. Sci. Comput. 80, 1\u201325 (2019)","journal-title":"J. Sci. Comput."},{"key":"1290_CR22","volume-title":"The Construction and Analysis of High Efficient Elements","author":"Q Lin","year":"1996","unstructured":"Lin, Q., Yan, N.N.: The Construction and Analysis of High Efficient Elements. Hebei University Press, Hebei (1996)"},{"key":"1290_CR23","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.1016\/j.jcp.2007.02.001","volume":"225","author":"YM Lin","year":"2007","unstructured":"Lin, Y.M., Xu, C.J.: Finite difference\/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533\u20131552 (2007)","journal-title":"J. Comput. Phys."},{"key":"1290_CR24","doi-asserted-by":"publisher","first-page":"1876","DOI":"10.1016\/j.camwa.2018.07.036","volume":"76","author":"H Liu","year":"2018","unstructured":"Liu, H., Cheng, A.J., Wang, H., Zhao, J.: Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation. Comput. Math. Appl. 76, 1876\u20131892 (2018)","journal-title":"Comput. Math. Appl."},{"key":"1290_CR25","volume-title":"Fractals and Fractional Calculus Continuum Mechanics","author":"F Mainardi","year":"1997","unstructured":"Mainardi, F.: Fractals and Fractional Calculus Continuum Mechanics. Springer, Berlin (1997)"},{"key":"1290_CR26","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":"1290_CR27","volume-title":"The Fractional Calculus","author":"K Oldham","year":"1974","unstructured":"Oldham, K., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)"},{"key":"1290_CR28","volume-title":"Fractional Differential Equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)"},{"key":"1290_CR29","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1007\/s10915-017-0385-z","volume":"72","author":"JC Ren","year":"2017","unstructured":"Ren, J.C., Long, X.N., Mao, S.P., Zhang, J.W.: Superconvergence of finite element approximations for the fractional diffusion-wave equation. J. Sci. Comput. 72, 917\u2013935 (2017)","journal-title":"J. Sci. Comput."},{"key":"1290_CR30","unstructured":"Ren, J.C., Liao, H.-L., Zhang, J.W., Zhang, Z.M.: Sharp $$H^1$$-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems (2018). arXiv: 1811.08059v1"},{"key":"1290_CR31","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1002\/num.22428","volume":"36","author":"JC Ren","year":"2020","unstructured":"Ren, J.C., Shi, D.Y., Vong, S.W.: High accuracy error estimates of a Galerkin FEM for nonlinear time fractional diffusion equation. Numer. Methods Part. Differ. Equ. 36, 284\u2013301 (2020)","journal-title":"Numer. Methods Part. Differ. Equ."},{"key":"1290_CR32","doi-asserted-by":"publisher","first-page":"456","DOI":"10.1016\/j.jcp.2012.08.026","volume":"232","author":"JC Ren","year":"2013","unstructured":"Ren, J.C., Sun, Z.Z., Zhao, X.: Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 232, 456\u2013467 (2013)","journal-title":"J. Comput. Phys."},{"key":"1290_CR33","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/s10915-012-9681-9","volume":"56","author":"JC Ren","year":"2013","unstructured":"Ren, J.C., Sun, Z.Z.: Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with Neumann boundary conditions. J. Sci. Comput. 56, 381\u2013408 (2013)","journal-title":"J. Sci. Comput."},{"key":"1290_CR34","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":"1290_CR35","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1016\/j.aml.2014.07.019","volume":"38","author":"DY Shi","year":"2014","unstructured":"Shi, D.Y., Wang, P.L., Zhao, Y.M.: Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schr\u00f6dinger equation. Appl. Math. Lett. 38, 129\u2013134 (2014)","journal-title":"Appl. Math. Lett."},{"key":"1290_CR36","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":"1290_CR37","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.apnum.2005.03.003","volume":"56","author":"ZZ Sun","year":"2006","unstructured":"Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193\u2013209 (2006)","journal-title":"Appl. Numer. Math."},{"key":"1290_CR38","series-title":"Springer Series in Computational Mathematics","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 Series in Computational Mathematics, vol. 25. Springer, Berlin (1997)"},{"key":"1290_CR39","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.: 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":"1290_CR40","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1007\/s10915-015-0152-y","volume":"70","author":"YM Zhao","year":"2017","unstructured":"Zhao, Y.M., Chen, P., Bu, W.P., Liu, X.T., Tang, Y.F.: Two mixed finite element methods for time-fractional diffusion equations. J. Sci. Comput. 70, 407\u2013428 (2017)","journal-title":"J. Sci. Comput."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01290-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-020-01290-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01290-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,2]],"date-time":"2021-08-02T23:20:00Z","timestamp":1627946400000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-020-01290-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8]]},"references-count":40,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,8]]}},"alternative-id":["1290"],"URL":"https:\/\/doi.org\/10.1007\/s10915-020-01290-1","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8]]},"assertion":[{"value":"17 December 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 July 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 July 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 August 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"38"}}