{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T15:22:24Z","timestamp":1790004144926,"version":"4.0.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T00:00:00Z","timestamp":1774310400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T00:00:00Z","timestamp":1774310400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12471385"],"award-info":[{"award-number":["12471385"]}],"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":["11971276"],"award-info":[{"award-number":["11971276"]}],"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":[[2026,5]]},"DOI":"10.1007\/s10915-026-03263-2","type":"journal-article","created":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T06:57:45Z","timestamp":1774335465000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Adaptive Finite Element Approximation of Semilinear Fractional Optimal Control Problem"],"prefix":"10.1007","volume":"107","author":[{"given":"Tongxin","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2944-1809","authenticated-orcid":false,"given":"Zhaojie","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,3,24]]},"reference":[{"key":"3263_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cma.2017.08.019","volume":"327","author":"M Ainsworth","year":"2017","unstructured":"Ainsworth, M., Glusa, C.: Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver. Comput. Methods Appl. Mech. Eng. 327, 1\u201348 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"3","key":"3263_CR2","doi-asserted-by":"publisher","first-page":"2106","DOI":"10.1093\/imanum\/drz025","volume":"40","author":"A Allendes","year":"2020","unstructured":"Allendes, A., Fuica, F., Ot\u00e1rola, E.: Adaptive finite element methods for sparse PDE-constrained optimization. IMA J. Numer. Anal. 40(3), 2106\u20132142 (2020)","journal-title":"IMA J. Numer. Anal."},{"key":"3263_CR3","doi-asserted-by":"crossref","unstructured":"Allendes, A., Fuica, F., Ot\u00e1rola, E., et\u00a0al.: A posteriori error estimates for semilinear optimal control problems. ESAIM: Math. Model. Numer. Anal. 55(5), 2293\u20132322 (2021)","DOI":"10.1051\/m2an\/2021033"},{"issue":"6","key":"3263_CR4","doi-asserted-by":"publisher","first-page":"3432","DOI":"10.1137\/140975061","volume":"53","author":"H Antil","year":"2015","unstructured":"Antil, H., Ot\u00e1rola, E.: A FEM for an optimal control problem of fractional powers of elliptic operators. SIAM J. Control. Optim. 53(6), 3432\u20133456 (2015)","journal-title":"SIAM J. Control. Optim."},{"key":"3263_CR5","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1023\/A:1020576801966","volume":"23","author":"N Arada","year":"2002","unstructured":"Arada, N., Casas, E., Tr\u00f6ltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23, 201\u2013229 (2002)","journal-title":"Comput. Optim. Appl."},{"issue":"5\u20136","key":"3263_CR6","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/s00791-018-0289-y","volume":"19","author":"A Bonito","year":"2018","unstructured":"Bonito, A., Borthagaray, J.P., Nochetto, R.H., Ot\u00e1rola, E., Salgado, A.J.: Numerical methods for fractional diffusion. Comput. Vis. Sci. 19(5\u20136), 19\u201346 (2018)","journal-title":"Comput. Vis. Sci."},{"issue":"4","key":"3263_CR7","doi-asserted-by":"publisher","first-page":"2775","DOI":"10.1137\/18M1219989","volume":"57","author":"M D\u2019Elia","year":"2019","unstructured":"D\u2019Elia, M., Glusa, C., Ot\u00e1rola, E.: A priori error estimates for the optimal control of the integral fractional Laplacian. SIAM J. Control. Optim. 57(4), 2775\u20132798 (2019)","journal-title":"SIAM J. Control. Optim."},{"key":"3263_CR8","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s10092-019-0334-3","volume":"56","author":"S Dohr","year":"2019","unstructured":"Dohr, S., Kahle, C., Rogovs, S., Swierczynski, P.: A FEM for an optimal control problem subject to the fractional Laplace equation. Calcolo 56, 37 (2019)","journal-title":"Calcolo"},{"key":"3263_CR9","doi-asserted-by":"publisher","first-page":"259","DOI":"10.4208\/jcm.1612-m2016-0696","volume":"36","author":"N Du","year":"2018","unstructured":"Du, N., Shen, W.: A fast stochastic method for a constrained optimal control problem governed by a random fractional diffusion equation. J. Math. Comput. 36, 259\u2013275 (2018)","journal-title":"J. Math. Comput."},{"key":"3263_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-015-0125-1","volume":"68","author":"N Du","year":"2016","unstructured":"Du, N., Wang, H., Liu, W.: A fast gradient projection method for a constrained fractional optimal control. J. Sci. Comput. 68, 1\u201320 (2016)","journal-title":"J. Sci. Comput."},{"issue":"330","key":"3263_CR11","doi-asserted-by":"publisher","first-page":"1557","DOI":"10.1090\/mcom\/3603","volume":"90","author":"M Faustmann","year":"2021","unstructured":"Faustmann, M., Melenk, J.M., Praetorius, D.: Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian. Math. Comp. 90(330), 1557\u20131587 (2021)","journal-title":"Math. Comp."},{"key":"3263_CR12","doi-asserted-by":"publisher","first-page":"1121","DOI":"10.1007\/s00211-016-0827-9","volume":"135","author":"W Gong","year":"2017","unstructured":"Gong, W., Yan, N.: Adaptive finite element method for elliptic optimal control problems: convergence and optimality. Numer. Math. 135, 1121\u20131170 (2017)","journal-title":"Numer. Math."},{"key":"3263_CR13","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1137\/17M1155636","volume":"57","author":"M Gunzburger","year":"2019","unstructured":"Gunzburger, M., Wang, J.: Error analysis of fully discrete finite element approximations to an optimal control problem governed by a time-fractional PDE. SIAM J. Control. Optim. 57, 241\u2013263 (2019)","journal-title":"SIAM J. Control. Optim."},{"key":"3263_CR14","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1007\/s10589-005-4559-5","volume":"30","author":"M Hinze","year":"2005","unstructured":"Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30, 45\u201363 (2005)","journal-title":"Comput. Optim. Appl."},{"issue":"1","key":"3263_CR15","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1093\/imanum\/dry064","volume":"40","author":"B Jin","year":"2020","unstructured":"Jin, B., Li, B., Zhou, Z.: Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint. IMA J. Numer. Anal. 40(1), 377\u2013404 (2020)","journal-title":"IMA J. Numer. Anal."},{"key":"3263_CR16","doi-asserted-by":"publisher","first-page":"1832","DOI":"10.1137\/130909251","volume":"52","author":"K Kohls","year":"2014","unstructured":"Kohls, K., R\u00f6sch, A., Siebert, K.G.: A posteriori error analysis of optimal control problems with control constraints. SIAM J. Control. Optim. 52, 1832\u20131861 (2014)","journal-title":"SIAM J. Control. Optim."},{"issue":"5","key":"3263_CR17","doi-asserted-by":"publisher","first-page":"1321","DOI":"10.1137\/S0363012901389342","volume":"41","author":"R Li","year":"2002","unstructured":"Li, R., Liu, W., Ma, H., Tang, T.: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control. Optim. 41(5), 1321\u20131349 (2002)","journal-title":"SIAM J. Control. Optim."},{"issue":"3","key":"3263_CR18","doi-asserted-by":"publisher","first-page":"1852","DOI":"10.1137\/24M1688771","volume":"63","author":"S Li","year":"2025","unstructured":"Li, S., Cao, W.: Spectral Galerkin method for optimal control of stochastic fractional Laplacian equations with white noise on a disk. SIAM J. Control. Optim. 63(3), 1852\u20131877 (2025)","journal-title":"SIAM J. Control. Optim."},{"key":"3263_CR19","doi-asserted-by":"crossref","unstructured":"Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971)","DOI":"10.1007\/978-3-642-65024-6"},{"issue":"1\u20134","key":"3263_CR20","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1023\/A:1014239012739","volume":"15","author":"W Liu","year":"2001","unstructured":"Liu, W., Yan, N.: A posteriori error analysis for convex distributed optimal control problems. Adv. Comput. Math. 15(1\u20134), 285\u2013309 (2001)","journal-title":"Adv. Comput. Math."},{"key":"3263_CR21","volume-title":"Adaptive Finite Element Methods for Optimal Control Governed by PDEs","author":"WB Liu","year":"2008","unstructured":"Liu, W.B., Yan, N.N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Science Press, Beijing (2008)"},{"issue":"2","key":"3263_CR22","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1002\/num.22429","volume":"36","author":"E Ot\u00e1rola","year":"2019","unstructured":"Ot\u00e1rola, E.: An adaptive finite element method for the sparse optimal control of fractional diffusion. Numer. Methods Partial Differential Equations 36(2), 302\u2013328 (2019)","journal-title":"Numer. Methods Partial Differential Equations"},{"issue":"1","key":"3263_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/20M1356294","volume":"60","author":"E Ot\u00e1rola","year":"2022","unstructured":"Ot\u00e1rola, E.: Fractional semilinear optimal control: optimality conditions, convergence, and error analysis. SIAM J. Numer. Anal. 60(1), 1\u201327 (2022)","journal-title":"SIAM J. Numer. Anal."},{"key":"3263_CR24","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1007\/s00245-023-10009-1","volume":"88","author":"E Ot\u00e1rola","year":"2023","unstructured":"Ot\u00e1rola, E.: Error estimates for fractional semilinear optimal control on Lipschitz polytopes. Appl. Math. Optim. 88, 40 (2023)","journal-title":"Appl. Math. Optim."},{"issue":"1","key":"3263_CR25","doi-asserted-by":"publisher","first-page":"198","DOI":"10.1093\/imanum\/drx005","volume":"38","author":"H Antil","year":"2017","unstructured":"Antil, H., Ot\u00e1rola, E.: An a posteriori error analysis for an optimal control problem involving the fractional Laplacian. IMA J. Numer. Anal. 38(1), 198\u2013226 (2017)","journal-title":"IMA J. Numer. Anal."},{"issue":"1","key":"3263_CR26","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1515\/cmam-2017-0030","volume":"18","author":"E Ot\u00e1rola","year":"2018","unstructured":"Ot\u00e1rola, E., Salgado, A.J.: Sparse Optimal Control for Fractional Diffusion. Comput. Methods Appl. Math. 18(1), 95\u2013110 (2018)","journal-title":"Comput. Methods Appl. Math."},{"issue":"2","key":"3263_CR27","doi-asserted-by":"publisher","first-page":"1140","DOI":"10.1137\/19M1267581","volume":"59","author":"C Glusa","year":"2021","unstructured":"Glusa, C., Ot\u00e1rola, E.: Error Estimates for the Optimal Control of a Parabolic Fractional PDE. SIAM J. Numer. Anal. 59(2), 1140\u20131165 (2021)","journal-title":"SIAM J. Numer. Anal."},{"key":"3263_CR28","volume-title":"Strongly Elliptic Systems and Boundary Integral Equations","author":"W McLean","year":"2000","unstructured":"McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)"},{"key":"3263_CR29","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/112","volume-title":"Optimal Control of Partial Differential Equations: Theory","author":"F Tr\u00f6ltzsch","year":"2010","unstructured":"Tr\u00f6ltzsch, F., Sprekels, J.: Optimal Control of Partial Differential Equations: Theory. American Mathematical Society, Methods and Applications (2010)"},{"issue":"1","key":"3263_CR30","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/s10915-024-02739-3","volume":"102","author":"F Wang","year":"2025","unstructured":"Wang, F., Wang, Q., Zhou, Z.: Adaptive finite element approximation of sparse optimal control problem with integral fractional Laplacian. J. Sci. Comput. 102(1), 17 (2025)","journal-title":"J. Sci. Comput."},{"key":"3263_CR31","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.113233","volume":"386","author":"FY Wang","year":"2021","unstructured":"Wang, F.Y., Zhang, Z.Q., Zhou, Z.J.: A spectral Galerkin approximation of optimal control problem governed by fractional advection-diffusion-reaction equations. J. Comput. Appl. Math. 386, 113233 (2021)","journal-title":"J. Comput. Appl. Math."},{"key":"3263_CR32","first-page":"1053","volume":"46","author":"X Ye","year":"2016","unstructured":"Ye, X., Xu, C.: A spectral method for optimal control problem governed by the abnormal diffusion equation with integral constraint on the state. Sci China Math 46, 1053\u20131070 (2016)","journal-title":"Sci China Math"},{"key":"3263_CR33","doi-asserted-by":"publisher","first-page":"993","DOI":"10.1007\/s10915-019-00964-9","volume":"80","author":"C Zhang","year":"2019","unstructured":"Zhang, C., Liu, H., Zhou, Z.: A priori error analysis for time-stepping discontinuous Galerkin finite element approximation of time fractional optimal control problem. J. Sci. Comput. 80, 993\u20131018 (2019)","journal-title":"J. Sci. Comput."},{"issue":"3","key":"3263_CR34","doi-asserted-by":"publisher","first-page":"1851","DOI":"10.1137\/20M1344962","volume":"59","author":"X Zheng","year":"2021","unstructured":"Zheng, X., Wang, H.: A hidden-memory variable-order time-fractional optimal control model: analysis and approximation. SIAM J. Control. Optim. 59(3), 1851\u20131880 (2021)","journal-title":"SIAM J. Control. Optim."},{"issue":"4","key":"3263_CR35","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1007\/s10444-023-10064-w","volume":"49","author":"Z Zhou","year":"2023","unstructured":"Zhou, Z., Wang, Q.: Adaptive finite element approximation of optimal control problems with the integral fractional Laplacian. Adv. Comput. Math. 49(4), 59 (2023)","journal-title":"Adv. Comput. Math."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03263-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03263-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03263-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T16:38:42Z","timestamp":1778690322000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-026-03263-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,24]]},"references-count":35,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,5]]}},"alternative-id":["3263"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03263-2","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,24]]},"assertion":[{"value":"27 August 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 January 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 March 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 March 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"48"}}