{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T14:50:59Z","timestamp":1767970259037,"version":"3.49.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T00:00:00Z","timestamp":1764547200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T00:00:00Z","timestamp":1764547200000},"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":["No. 11971410"],"award-info":[{"award-number":["No. 11971410"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2025,12]]},"DOI":"10.1007\/s10444-025-10272-6","type":"journal-article","created":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T07:22:43Z","timestamp":1764573763000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Developing and analyzing some new finite element methods for a non-local hydrodynamic Drude model"],"prefix":"10.1007","volume":"51","author":[{"given":"Yunqing","family":"Huang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8338-8412","authenticated-orcid":false,"given":"Jichun","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,12,1]]},"reference":[{"key":"10272_CR1","doi-asserted-by":"publisher","first-page":"6709","DOI":"10.1016\/S0045-7825(01)00259-6","volume":"190","author":"M Ainsworth","year":"2001","unstructured":"Ainsworth, M., Coyle, J.: Hierarchic hp-edge element families for Maxwell\u2019s equations on hybrid quadrilateral\/triangular meshes. Comput. Methods Appl. Mech. Engrg. 190, 6709\u20136733 (2001)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"issue":"1","key":"10272_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/050639107","volume":"67","author":"D Appel\u00f6","year":"2006","unstructured":"Appel\u00f6, D., Hagstrom, T., Kreiss, G.: Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability. SIAM J. Appl. Math. 67(1), 1\u201323 (2006)","journal-title":"SIAM J. Appl. Math."},{"key":"10272_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-981-16-0061-6","volume-title":"Maxwell\u2019s equations in periodic structures, Series on Applied Mathematical Sciences","author":"G Bao","year":"2022","unstructured":"Bao, G., Li, P.: Maxwell\u2019s equations in periodic structures, Series on Applied Mathematical Sciences, vol. 208. Science Press, Springer, Singapore (2022)"},{"issue":"4","key":"10272_CR4","doi-asserted-by":"publisher","first-page":"1264","DOI":"10.1137\/S003614299731853X","volume":"36","author":"D Boffi","year":"1999","unstructured":"Boffi, D., Fernandes, P., Gastaldi, L., Perugia, I.: Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36(4), 1264\u20131290 (1999)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"10272_CR5","doi-asserted-by":"publisher","first-page":"848","DOI":"10.1007\/s10915-015-0161-x","volume":"68","author":"SC Brenner","year":"2016","unstructured":"Brenner, S.C., Gedicke, J., Sung, L.-Y.: An adaptive $$P_1$$ finite element method for two-dimensional transverse magnetic time harmonic Maxwell\u2019s equations with general material properties and general boundary conditions. J. Sci. Comput. 68(2), 848\u2013863 (2016)","journal-title":"J. Sci. Comput."},{"key":"10272_CR6","doi-asserted-by":"publisher","first-page":"2792","DOI":"10.1016\/j.camwa.2017.07.025","volume":"74","author":"M Cassier","year":"2017","unstructured":"Cassier, M., Joly, P., Kachanovska, M.: Mathematical models for dispersive electromagnetic waves: an overview. Comput. Math. Appl. 74, 2792\u20132830 (2017)","journal-title":"Comput. Math. Appl."},{"issue":"5","key":"10272_CR7","doi-asserted-by":"publisher","first-page":"1542","DOI":"10.1137\/S0036142998349977","volume":"37","author":"Z Chen","year":"2000","unstructured":"Chen, Z., Du, Q., Zou, J.: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37(5), 1542\u20131570 (2000)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"10272_CR8","doi-asserted-by":"publisher","first-page":"1745","DOI":"10.1021\/nn406153k","volume":"8","author":"T Christensen","year":"2014","unstructured":"Christensen, T., Yan, W., Raza, S., Jauho, A., Mortensen, N.A., Wubs, M.: Nonlocal response of metallic nanospheres probed by light, electrons, and atoms. ACS Nano 8(2), 1745\u20131758 (2014)","journal-title":"ACS Nano"},{"issue":"2","key":"10272_CR9","doi-asserted-by":"publisher","first-page":"023109","DOI":"10.1063\/1.4861849","volume":"104","author":"C Cirac\u00ed","year":"2014","unstructured":"Cirac\u00ed, C., Chen, X., Mock, J., McGuire, F., Liu, X., Oh, S., Smith, D.: Film-coupled nanoparticles by atomic layer deposition: comparison with organic spacing layers. Appl. Phys. Lett. 104(2), 023109 (2014)","journal-title":"Appl. Phys. Lett."},{"issue":"6098","key":"10272_CR10","doi-asserted-by":"publisher","first-page":"1072","DOI":"10.1126\/science.1224823","volume":"337","author":"C Ciraci","year":"2012","unstructured":"Ciraci, C., Hill, R., Mock, J., Urzhumov, Y., Fern\u00e1ndez-Dom\u00ednguez, A., Maier, S., Pendry, J., Chilkoti, A., Smith, D.: Probing the ultimate limits of plasmonic enhancement. Science 337(6098), 1072\u20131074 (2012)","journal-title":"Science"},{"key":"10272_CR11","doi-asserted-by":"crossref","unstructured":"Demkowicz, L., Kurtz, J., Pardo, D., Paszynski, M., Rachowicz, W., Zdunek, A.: Computing with hp-adaptive finite elements, vol. 2: Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications, CRC Press, Taylor and Francis, (2008)","DOI":"10.1201\/9781420011692"},{"issue":"1","key":"10272_CR12","doi-asserted-by":"publisher","first-page":"510","DOI":"10.1007\/s10915-019-00936-z","volume":"80","author":"Z Fang","year":"2019","unstructured":"Fang, Z., Li, J., Tang, T., Zhou, T.: Efficient stochastic Galerkin methods for Maxwell\u2019s equations with random inputs. J. Sci. Comput. 80(1), 510\u2013535 (2019)","journal-title":"J. Sci. Comput."},{"key":"10272_CR13","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1017\/S0962492902000041","volume":"11","author":"R Hiptmair","year":"2002","unstructured":"Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer 11, 237\u2013339 (2002)","journal-title":"Acta Numer"},{"key":"10272_CR14","doi-asserted-by":"publisher","first-page":"5890","DOI":"10.1016\/j.jcp.2012.05.013","volume":"231","author":"K Hiremath","year":"2012","unstructured":"Hiremath, K., Zschiedrich, L., Schmidt, F.: Numerical solution of nonlocal hydrodynamic Drude model for arbitrary shaped nano-plasmonic structures using Nedelec finite elements. J. Comput. Phys. 231, 5890\u20135896 (2012)","journal-title":"J. Comput. Phys."},{"key":"10272_CR15","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1007\/s00211-005-0604-7","volume":"100","author":"P Houston","year":"2005","unstructured":"Houston, P., Perugia, I., Schneebeli, A., Sch\u00f6tzau, D.: Interior penalty method for the indefinite time harmonic Maxwell Equations. Numer. Math. 100, 485\u2013518 (2005)","journal-title":"Numer. Math."},{"key":"10272_CR16","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1007\/s10915-024-02585-3","volume":"100","author":"Y Huang","year":"2024","unstructured":"Huang, Y., Li, J., He, B.: Developing and analyzing some novel finite element schemes for the electromagnetic rotation cloak model. J. Sci. Comput. 100, 26 (2024)","journal-title":"J. Sci. Comput."},{"issue":"1","key":"10272_CR17","doi-asserted-by":"publisher","first-page":"B248","DOI":"10.1137\/120869869","volume":"35","author":"Y Huang","year":"2013","unstructured":"Huang, Y., Li, J., Yang, W.: Modeling backward wave propagation in metamaterials by the finite element time domain method. SIAM J. Sci. Comput. 35(1), B248\u2013B274 (2013)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"10272_CR18","doi-asserted-by":"publisher","first-page":"921","DOI":"10.1016\/j.camwa.2016.06.003","volume":"72","author":"Y Huang","year":"2016","unstructured":"Huang, Y., Li, J., Yang, W.: Theoretical and numerical analysis of a non-local dispersion model for light interaction with metallic nanostructures. Comput. Math. Appl. 72(4), 921\u2013932 (2016)","journal-title":"Comput. Math. Appl."},{"issue":"3","key":"10272_CR19","doi-asserted-by":"publisher","first-page":"A831","DOI":"10.1137\/15M105207X","volume":"39","author":"S Lanteri","year":"2017","unstructured":"Lanteri, S., Scheid, C., Viquerat, J.: Analysis of a generalized dispersive model coupled to a DGTD method with application to nanophotonics. SIAM J. Sci. Comput. 39(3), A831\u2013A859 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"10272_CR20","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1007\/s10915-020-01275-0","volume":"84","author":"JJ Lee","year":"2020","unstructured":"Lee, J.J.: A mixed method for time-transient acoustic wave propagation in metamaterials. J. Sci. Comput. 84, 20 (2020)","journal-title":"J. Sci. Comput."},{"key":"10272_CR21","doi-asserted-by":"publisher","first-page":"915","DOI":"10.1016\/j.jcp.2013.11.018","volume":"258","author":"J Li","year":"2014","unstructured":"Li, J., Hesthaven, J.S.: Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials. J. Comput. Phys. 258, 915\u2013930 (2014)","journal-title":"J. Comput. Phys."},{"key":"10272_CR22","doi-asserted-by":"crossref","unstructured":"Li, J., Huang, Y.: Time-domain finite element methods for Maxwell\u2019s equations in metamaterials. Springer Ser. Comput Math, vol. 43. Springer, New York (2013)","DOI":"10.1007\/978-3-642-33789-5"},{"key":"10272_CR23","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1090\/S0025-5718-2014-02911-6","volume":"84","author":"J Li","year":"2015","unstructured":"Li, J., Huang, Y., Yang, W.: Well-posedness study and finite element simulation of time-domain cylindrical and elliptical cloaks. Math. Comp. 84, 543\u2013562 (2015)","journal-title":"Math. Comp."},{"key":"10272_CR24","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1007\/s10444-022-09948-0","volume":"48","author":"J Li","year":"2022","unstructured":"Li, J., Shu, C.-W., Yang, W.: Development and analysis of two new finite element schemes for a time-domain carpet cloak model. Adv. Comput. Math. 48, 24 (2022)","journal-title":"Adv. Comput. Math."},{"issue":"2","key":"10272_CR25","doi-asserted-by":"publisher","first-page":"932","DOI":"10.1002\/num.22911","volume":"39","author":"K Li","year":"2023","unstructured":"Li, K., Huang, T.-Z., Li, L., Lanteri, S.: Simulation of the interaction of light with 3-D metallic nanostructures using a proper orthogonal decomposition-Galerkin reduced-order discontinuous Galerkin time-domain method. Numer. Methods Partial Differential Eqs. 39(2), 932\u2013954 (2023)","journal-title":"Numer. Methods Partial Differential Eqs."},{"key":"10272_CR26","doi-asserted-by":"crossref","unstructured":"Monk, P.: Finite element methods for Maxwell\u2019s equations. Oxford University Press (2003)","DOI":"10.1093\/acprof:oso\/9780198508885.001.0001"},{"key":"10272_CR27","doi-asserted-by":"publisher","first-page":"6795","DOI":"10.1103\/PhysRevB.87.045401","volume":"87","author":"A Moreau","year":"2013","unstructured":"Moreau, A., Ciraci, C., Smith, D.: Impact of nonlocal response on metallodielectric multilayers and optical patch antennas. Phys. Rev. B 87, 6795\u20136820 (2013)","journal-title":"Phys. Rev. B"},{"issue":"3","key":"10272_CR28","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1007\/BF01396415","volume":"35","author":"J-C N\u00e9d\u00e9lec","year":"1980","unstructured":"N\u00e9d\u00e9lec, J.-C.: Mixed finite elements in $$\\cal{R} ^3$$. Numer. Math. 35(3), 315\u2013341 (1980)","journal-title":"Numer. Math."},{"key":"10272_CR29","doi-asserted-by":"publisher","first-page":"3462","DOI":"10.1016\/j.camwa.2020.02.006","volume":"79","author":"S Nicaise","year":"2020","unstructured":"Nicaise, S., Scheid, C.: Stability and asymptotic properties of a linearized hydrodynamic medium model for dispersive media in nanophotonics. Comput. Math. Appl. 79, 3462\u20133494 (2020)","journal-title":"Comput. Math. Appl."},{"key":"10272_CR30","doi-asserted-by":"publisher","first-page":"396","DOI":"10.1016\/j.jcp.2016.04.020","volume":"316","author":"N Schmitt","year":"2016","unstructured":"Schmitt, N., Scheid, C., Lanteri, S., Viquerat, J., Moreau, A.: A DGTD method for the numerical modeling of the interaction of light with nanometer scale metallic structures taking into account non-local dispersion effects. J. Comput. Phys. 316, 396\u2013415 (2016)","journal-title":"J. Comput. Phys."},{"key":"10272_CR31","doi-asserted-by":"publisher","first-page":"548","DOI":"10.1016\/j.jcp.2017.11.025","volume":"355","author":"F Vidal-Codina","year":"2018","unstructured":"Vidal-Codina, F., Nguyen, N.-C., Oh, S.-H., Peraire, J.: A hybridizable discontinuous Galerkin method for computing nonlocal electromagnetic effects in three-dimensional metallic nanostructures. J. Comput. Phys. 355, 548\u2013565 (2018)","journal-title":"J. Comput. Phys."},{"key":"10272_CR32","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1007\/s10915-020-01354-2","volume":"86","author":"B Wang","year":"2021","unstructured":"Wang, B., Yang, Z., Wang, L.-L., Jiang, S.: On time-domain NRBC for Maxwell\u2019s equations and its application in accurate simulation of electromagnetic invisibility cloaks. J. Sci. Comput. 86, 20 (2021)","journal-title":"J. Sci. Comput."},{"issue":"4","key":"10272_CR33","doi-asserted-by":"publisher","first-page":"B907","DOI":"10.1137\/20M1324120","volume":"43","author":"Y Zhang","year":"2021","unstructured":"Zhang, Y., Ma, C., Cao, L.-Q., Shi, D.: Efficient multiscale algorithms for simulating nonlocal optical response of metallic nanostructure arrays. SIAM J. Sci. Comput. 43(4), B907\u2013B936 (2021)","journal-title":"SIAM J. Sci. Comput."},{"key":"10272_CR34","doi-asserted-by":"publisher","first-page":"623","DOI":"10.1090\/S0025-5718-2011-02544-5","volume":"81","author":"L Zhong","year":"2012","unstructured":"Zhong, L., Chen, L., Shu, S., Wittum, G., Xu, J.: Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations. Math. Comp. 81, 623\u2013642 (2012)","journal-title":"Math. Comp."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-025-10272-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-025-10272-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-025-10272-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T12:03:48Z","timestamp":1767960228000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-025-10272-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12]]},"references-count":34,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["10272"],"URL":"https:\/\/doi.org\/10.1007\/s10444-025-10272-6","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12]]},"assertion":[{"value":"16 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 November 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 December 2025","order":3,"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 no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"54"}}