{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T11:29:12Z","timestamp":1747135752142,"version":"3.37.3"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T00:00:00Z","timestamp":1681171200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T00:00:00Z","timestamp":1681171200000},"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":["12131005","11871092"],"award-info":[{"award-number":["12131005","11871092"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010906","name":"NSAF Joint Fund","doi-asserted-by":"publisher","award":["U1930402","12101036"],"award-info":[{"award-number":["U1930402","12101036"]}],"id":[{"id":"10.13039\/501100010906","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":[[2023,5]]},"DOI":"10.1007\/s10915-023-02182-w","type":"journal-article","created":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T00:02:31Z","timestamp":1681171351000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Superconvergence Analysis of Curlcurl-Conforming Elements on Rectangular Meshes"],"prefix":"10.1007","volume":"95","author":[{"given":"Lixiu","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1503-193X","authenticated-orcid":false,"given":"Qian","family":"Zhang","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":[[2023,4,11]]},"reference":[{"key":"2182_CR1","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1002\/num.1690120303","volume":"12","author":"I Babu\u0161ka","year":"1996","unstructured":"Babu\u0161ka, I., Strouboulis, T., Upadhyay, C.S., Gangaraj, S.K.: Computer-based proof of the existence of superconvergence points in the finite element method: superconvergence of the derivatives in finite element solutions of Laplace\u2019s, Poisson\u2019s, and the elasticity equations. Numer. Methods Part. Differ. Equ. 12, 347\u2013392 (1996)","journal-title":"Numer. Methods Part. Differ. Equ."},{"issue":"2","key":"2182_CR2","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1007\/s41478-018-0082-9","volume":"27","author":"SK Bhal","year":"2019","unstructured":"Bhal, S.K., Danumjaya, P.: A fourth-order orthogonal spline collocation solution to 1D-Helmholtz equation with discontinuity. J. Anal. 27(2), 377\u2013390 (2019)","journal-title":"J. Anal."},{"key":"2182_CR3","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1090\/S0025-5718-1977-0431744-9","volume":"31","author":"J Bramble","year":"1997","unstructured":"Bramble, J., Schatz, A.: High order local accuracy by averaging in the finite element method. Math. Comput. 31, 94\u2013111 (1997)","journal-title":"Math. Comput."},{"issue":"2","key":"2182_CR4","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1515\/cmam-2019-0011","volume":"19","author":"SC Brenner","year":"2019","unstructured":"Brenner, S.C., Cui, J., Sung, L.: Multigrid methods based on Hodge decomposition for a quad-curl problem. Comput. Methods Appl. Math. 19(2), 215\u2013232 (2019)","journal-title":"Comput. Methods Appl. Math."},{"key":"2182_CR5","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1007\/s10915-017-0449-0","volume":"73","author":"SC Brenner","year":"2017","unstructured":"Brenner, S.C., Sun, J., Sung, L.: Hodge decomposition methods for a quad-curl problem on planar domains. J. Sci. Comput. 73, 495\u2013513 (2017)","journal-title":"J. Sci. Comput."},{"issue":"3","key":"2182_CR6","doi-asserted-by":"crossref","first-page":"443","DOI":"10.3934\/ipi.2007.1.443","volume":"1","author":"F Cakoni","year":"2017","unstructured":"Cakoni, F., Haddar, H.: A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Probl. Imaging 1(3), 443\u2013456 (2017)","journal-title":"Inverse Probl. Imaging"},{"issue":"1","key":"2182_CR7","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1137\/21M1391006","volume":"60","author":"W Cao","year":"2022","unstructured":"Cao, W., Jia, L., Zhang, Z.: A $${C}^1$$-conforming Petrov-Galerkin method for convection-diffusion equations and superconvergence analysis over rectangular meshes. SIAM J. Numer. Anal. 60(1), 274\u2013311 (2022)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"2182_CR8","doi-asserted-by":"crossref","first-page":"1651","DOI":"10.1137\/140996203","volume":"53","author":"W Cao","year":"2015","unstructured":"Cao, W., Shu, C.W., Yang, Y., Zhang, Z.: Superconvergence of discontinuous Galerkin methods for two-dimensional hyperbolic equations. SIAM J. Numer. Anal. 53(4), 1651\u20131671 (2015)","journal-title":"SIAM J. Numer. Anal."},{"issue":"5","key":"2182_CR9","doi-asserted-by":"crossref","first-page":"2555","DOI":"10.1137\/130946873","volume":"52","author":"W Cao","year":"2013","unstructured":"Cao, W., Zhang, Z., Zou, Q.: Superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 52(5), 2555\u20132573 (2013)","journal-title":"SIAM J. Numer. Anal."},{"issue":"23","key":"2182_CR10","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.99.235001","volume":"99","author":"L Chac\u00f3n","year":"2007","unstructured":"Chac\u00f3n, L., Simakov, A.N., Zocco, A.: Steady-state properties of driven magnetic reconnection in 2D electron magnetohydrodynamics. Phys. Rev. Lett. 99(23), 235001 (2007)","journal-title":"Phys. Rev. Lett."},{"key":"2182_CR11","volume-title":"Structure Theory of Superconvergence of Finite Elements","author":"C Chen","year":"2001","unstructured":"Chen, C.: Structure Theory of Superconvergence of Finite Elements. Hunan science and Technology Press (2001). (in Chinese)"},{"key":"2182_CR12","doi-asserted-by":"crossref","first-page":"1337","DOI":"10.1090\/S0025-5718-2012-02653-6","volume":"82","author":"C Chen","year":"2013","unstructured":"Chen, C., Hu, S.: The highest order superconvergence for bi-$$k$$ degree rectangular elements at nodes: a proof of 2$$k$$-conjecture. Math. Comput. 82, 1337\u20131355 (2013)","journal-title":"Math. Comput."},{"key":"2182_CR13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10915-020-01404-9","volume":"87","author":"G Chen","year":"2021","unstructured":"Chen, G., Cui, J., Xu, L.: A hybridizable discontinuous Galerkin method for the quad-curl problem. J. Sci. Comput. 87, 1\u201323 (2021)","journal-title":"J. Sci. Comput."},{"issue":"4","key":"2182_CR14","doi-asserted-by":"crossref","first-page":"2990","DOI":"10.1093\/imanum\/draa034","volume":"41","author":"G Chen","year":"2021","unstructured":"Chen, G., Qiu, W., Xu, L.: Analysis of an interior penalty DG method for the quad-curl problem. IMA J. Numer. Anal. 41(4), 2990\u20133023 (2021)","journal-title":"IMA J. Numer. Anal."},{"key":"2182_CR15","volume":"367","author":"J Han","year":"2020","unstructured":"Han, J.: Shifted inverse iteration based multigrid methods for the quad-curl eigenvalue problem. Appl. Math. Comput. 367, 124770 (2020)","journal-title":"Appl. Math. Comput."},{"key":"2182_CR16","doi-asserted-by":"crossref","first-page":"565","DOI":"10.4208\/jcm.1206-m3572","volume":"30","author":"Q Hong","year":"2012","unstructured":"Hong, Q., Hu, J., Shu, S., Xu, J.: A discontinuous Galerkin method for the fourth-order curl problem. J. Comput. Math. 30, 565\u2013578 (2012)","journal-title":"J. Comput. Math."},{"issue":"6","key":"2182_CR17","doi-asserted-by":"crossref","first-page":"A3859","DOI":"10.1137\/20M1333390","volume":"42","author":"K Hu","year":"2020","unstructured":"Hu, K., Zhang, Q., Zhang, Z.: Simple curl-curl-conforming finite elements in two dimensions. SIAM J. Sci. Comput. 42(6), A3859\u2013A3877 (2020). arXiv:2004.12507v2","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"2182_CR18","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1137\/20M1358700","volume":"60","author":"K Hu","year":"2022","unstructured":"Hu, K., Zhang, Q., Zhang, Z.: A family of finite element stokes complexes in three dimensions. SIAM J. Numer. Anal. 60(1), 222\u2013243 (2022)","journal-title":"SIAM J. Numer. Anal."},{"unstructured":"Huang, X.: Nonconforming finite element stokes complexes in three dimensions. arXiv preprint arXiv:2007.14068 (2020)","key":"2182_CR19"},{"issue":"6","key":"2182_CR20","doi-asserted-by":"crossref","first-page":"1794","DOI":"10.1002\/num.20703","volume":"28","author":"Y Huang","year":"2012","unstructured":"Huang, Y., Li, J., Lin, Q.: Superconvergence analysis for time-dependent Maxwell\u2019s equations in metamaterials. Numer. Methods Part. Differ. Equ. 28(6), 1794\u20131816 (2012)","journal-title":"Numer. Methods Part. Differ. Equ."},{"key":"2182_CR21","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.cma.2012.11.008","volume":"255","author":"Y Huang","year":"2013","unstructured":"Huang, Y., Li, J., Wu, C.: Averaging for superconvergence: verification and application of 2D edge elements to Maxwell\u2019s equations in metamaterials. Comput. Methods Appl. Mech. Eng. 255, 121\u2013132 (2013)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"2182_CR22","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1007\/s10915-014-9848-7","volume":"62","author":"Y Huang","year":"2015","unstructured":"Huang, Y., Li, J., Wu, C., Yang, W.: Superconvergence analysis for linear tetrahedral edge elements. J. Sci. Comput. 62(1), 122\u2013145 (2015)","journal-title":"J. Sci. Comput."},{"issue":"22","key":"2182_CR23","doi-asserted-by":"crossref","first-page":"8275","DOI":"10.1016\/j.jcp.2011.07.025","volume":"230","author":"Y Huang","year":"2011","unstructured":"Huang, Y., Li, J., Yang, W., Sun, S.: Superconvergence of mixed finite element approximations to 3-D Maxwell\u2019s equations in metamaterials. J. Comput. Phys. 230(22), 8275\u20138289 (2011)","journal-title":"J. Comput. Phys."},{"key":"2182_CR24","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-33789-5","volume-title":"Time-Domain Finite Element Methods for Maxwell\u2019s Equations in Metamaterials","author":"J Li","year":"2013","unstructured":"Li, J., Huang, Y.: Time-Domain Finite Element Methods for Maxwell\u2019s Equations in Metamaterials. Springer, Berlin (2013)"},{"key":"2182_CR25","volume-title":"Construction and Analysis of High Efficient Finite Elements","author":"Q Lin","year":"1996","unstructured":"Lin, Q., Yan, N.: Construction and Analysis of High Efficient Finite Elements. Hebei University Press (1996). (in Chinese)"},{"issue":"1","key":"2182_CR26","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/BF00253946","volume":"11","author":"RD Mindlin","year":"1962","unstructured":"Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415\u2013448 (1962)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"6","key":"2182_CR27","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1002\/num.1690100611","volume":"10","author":"P Monk","year":"1994","unstructured":"Monk, P.: Superconvergence of finite element approximations to Maxwell\u2019s equations. Numer. Methods Part. Differ. Equ. 10(6), 793\u2013812 (1994)","journal-title":"Numer. Methods Part. Differ. Equ."},{"issue":"3","key":"2182_CR28","doi-asserted-by":"crossref","first-page":"B247","DOI":"10.1137\/110839990","volume":"34","author":"P Monk","year":"2012","unstructured":"Monk, P., Sun, J.: Finite element methods for Maxwell\u2019s transmission eigenvalues. SIAM J. Sci. Comput. 34(3), B247\u2013B264 (2012)","journal-title":"SIAM J. Sci. Comput."},{"key":"2182_CR29","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/BF01396415","volume":"35","author":"JC N\u00e9d\u00e9lec","year":"1980","unstructured":"N\u00e9d\u00e9lec, J.C.: Mixed finite elements in $$\\mathbb{R} ^{3}$$. Numer. Math. 35, 315\u2013341 (1980)","journal-title":"Numer. Math."},{"key":"2182_CR30","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/BF01389668","volume":"50","author":"JC N\u00e9d\u00e9lec","year":"1986","unstructured":"N\u00e9d\u00e9lec, J.C.: A new family of mixed finite elements in $$\\mathbb{R} ^{3}$$. Numer. Math. 50, 57\u201381 (1986)","journal-title":"Numer. Math."},{"issue":"5","key":"2182_CR31","doi-asserted-by":"crossref","first-page":"904","DOI":"10.1007\/s00033-006-6073-8","volume":"59","author":"SK Park","year":"2008","unstructured":"Park, S.K., Gao, X.: Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z. Angew. Math. Phys. 59(5), 904\u2013917 (2008)","journal-title":"Z. Angew. Math. Phys."},{"key":"2182_CR32","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/s00211-015-0708-7","volume":"132","author":"J Sun","year":"2016","unstructured":"Sun, J.: A mixed FEM for the quad-curl eigenvalue problem. Numer. Math. 132, 185\u2013200 (2016)","journal-title":"Numer. Math."},{"issue":"4","key":"2182_CR33","doi-asserted-by":"crossref","first-page":"1093","DOI":"10.1007\/s10543-019-00764-5","volume":"59","author":"J Sun","year":"2019","unstructured":"Sun, J., Zhang, Q., Zhang, Z.: A curl-conforming weak Galerkin method for the quad-curl problem. BIT Numer. Math. 59(4), 1093\u20131114 (2019)","journal-title":"BIT Numer. Math."},{"issue":"1","key":"2182_CR34","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1007\/s10915-017-0544-2","volume":"75","author":"M Sun","year":"2018","unstructured":"Sun, M., Li, J., Wang, P., Zhang, Z.: Superconvergence analysis of high-order rectangular edge elements for time-harmonic Maxwell\u2019s equations. J. Sci. Comput. 75(1), 510\u2013535 (2018)","journal-title":"J. Sci. Comput."},{"doi-asserted-by":"crossref","unstructured":"Wahlbin, L.: Superconvergence In Galerkin Finite Element Methods, vol. 1605. Springer-Verlag (1995)","key":"2182_CR35","DOI":"10.1007\/BFb0096835"},{"key":"2182_CR36","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.amc.2018.12.027","volume":"349","author":"C Wang","year":"2019","unstructured":"Wang, C., Sun, Z., Cui, J.: A new error analysis of a mixed finite element method for the quad-curl problem. Appl. Math. Comput. 349, 23\u201338 (2019)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"2182_CR37","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1515\/cmam-2020-0152","volume":"21","author":"L Wang","year":"2021","unstructured":"Wang, L., Li, H., Zhang, Z.: H($$\\text{ curl}^2$$)-conforming spectral element method for quad-curl problems. Comput. Methods Appl. Math. 21(3), 661\u2013681 (2021)","journal-title":"Comput. Methods Appl. Math."},{"issue":"10","key":"2182_CR38","doi-asserted-by":"crossref","first-page":"1951","DOI":"10.1142\/S0218202521500433","volume":"31","author":"L Wang","year":"2021","unstructured":"Wang, L., Shan, W., Li, H., Zhang, Z.: H($$\\text{ curl}^2$$)-conforming quadrilateral spectral element method for quad-curl problems. Math. Models Methods Appl. Sci. 31(10), 1951\u20131986 (2021)","journal-title":"Math. Models Methods Appl. Sci."},{"unstructured":"Wang, L., Zhang, Q., Sun, J., Zhang, Z.: A priori and a posterior error estimations of quad-curl eigenvalue problems in 2D. arXiv:2007.01330 (2020)","key":"2182_CR39"},{"key":"2182_CR40","doi-asserted-by":"crossref","first-page":"1207","DOI":"10.1007\/s10915-018-0805-8","volume":"78","author":"L Wang","year":"2019","unstructured":"Wang, L., Zhang, Q., Zhang, Z.: Superconvergence analysis and PPR recovery of arbitrary order edge elements for Maxwell\u2019s equations. J. Sci. Comput. 78, 1207\u20131230 (2019)","journal-title":"J. Sci. Comput."},{"issue":"3","key":"2182_CR41","doi-asserted-by":"crossref","first-page":"A1527","DOI":"10.1137\/18M1199988","volume":"41","author":"Q Zhang","year":"2019","unstructured":"Zhang, Q., Wang, L., Zhang, Z.: H($$\\text{ curl}^2$$)-conforming finite elements in 2 dimensions and applications to the quad-curl problem. SIAM J. Sci. Comput. 41(3), A1527\u2013A1547 (2019)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"2182_CR42","doi-asserted-by":"crossref","first-page":"639","DOI":"10.4208\/csiam-am.2020-0023","volume":"1","author":"Q Zhang","year":"2020","unstructured":"Zhang, Q., Zhang, Z.: A family of curl-curl conforming finite elements on tetrahedral meshes. CSIAM Trans. Appl. Math. 1(4), 639\u2013663 (2020)","journal-title":"CSIAM Trans. Appl. Math."},{"key":"2182_CR43","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1051\/m2an\/2018005","volume":"52","author":"S Zhang","year":"2018","unstructured":"Zhang, S.: Mixed schemes for quad-curl equations. ESAIM Math. Modell. Numer. Anal. 52, 147\u2013161 (2018)","journal-title":"ESAIM Math. Modell. Numer. Anal."},{"doi-asserted-by":"crossref","unstructured":"Zhang, S.: Regular decomposition and a framework of order reduced methods for fourth order problems. Numer. Math. 138, 241\u2013271 (2018)","key":"2182_CR44","DOI":"10.1007\/s00211-017-0902-x"},{"issue":"3","key":"2182_CR45","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1002\/num.10001","volume":"18","author":"Z Zhang","year":"2002","unstructured":"Zhang, Z.: Finite element superconvergence approximation for one-dimensional singularity perturbed problems. Numer. Methods Part. Differ. Equ. 18(3), 374\u2013395 (2002)","journal-title":"Numer. Methods Part. Differ. Equ."},{"issue":"276","key":"2182_CR46","doi-asserted-by":"crossref","first-page":"1871","DOI":"10.1090\/S0025-5718-2011-02480-4","volume":"80","author":"B Zheng","year":"2011","unstructured":"Zheng, B., Hu, Q., Xu, J.: A nonconforming finite element method for fourth order curl equations in $$\\mathbb{R} ^3$$. Math. Comput. 80(276), 1871\u20131886 (2011)","journal-title":"Math. Comput."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02182-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-023-02182-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02182-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T20:37:58Z","timestamp":1682714278000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-023-02182-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,11]]},"references-count":46,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,5]]}},"alternative-id":["2182"],"URL":"https:\/\/doi.org\/10.1007\/s10915-023-02182-w","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2023,4,11]]},"assertion":[{"value":"14 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 March 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 March 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 April 2023","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":"62"}}