{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T10:40:41Z","timestamp":1769856041085,"version":"3.49.0"},"reference-count":34,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2020M670117"],"award-info":[{"award-number":["2020M670117"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871145"],"award-info":[{"award-number":["11871145"]}],"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":["11971016"],"award-info":[{"award-number":["11971016"]}],"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":["U1930402"],"award-info":[{"award-number":["U1930402"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,7,1]]},"abstract":"Abstract<\/jats:title>\n In this paper, we propose an \n \n \n \n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n curl<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n \\boldsymbol{H}(\\mathbf{curl}^{2})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-conforming spectral elements to solve the quad-curl problem on cubic meshes in three dimensions.\nStarting with generalized vectorial Jacobi polynomials, we first construct the basis functions of the \n \n \n \n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n curl<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n \\boldsymbol{H}(\\mathbf{curl}^{2})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-conforming spectral elements using the contravariant transform together with the affine mapping from the reference cube onto each physical element.\nFalling into four categories, interior modes, face modes, edge modes, and vertex modes, these \n \n \n \n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n curl<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n \\boldsymbol{H}(\\mathbf{curl}^{2})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-conforming basis functions are constructed in an arbitrarily high degree with a hierarchical structure.\nNext, \n \n \n \n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n curl<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n \\boldsymbol{H}(\\mathbf{curl}^{2})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-conforming spectral element approximation schemes are established to solve the boundary value problem as well as the eigenvalue problem of quad-curl equations.\nNumerical experiments demonstrate the effectiveness and efficiency of the \u210e-version and the \ud835\udc5d-version of our \n \n \n \n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n curl<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n \\boldsymbol{H}(\\mathbf{curl}^{2})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-conforming spectral element method.<\/jats:p>","DOI":"10.1515\/cmam-2020-0152","type":"journal-article","created":{"date-parts":[[2021,5,5]],"date-time":"2021-05-05T21:27:41Z","timestamp":1620250061000},"page":"661-681","source":"Crossref","is-referenced-by-count":11,"title":["\ud835\udc6f(curl<\/b>\n 2<\/sup>)-Conforming Spectral Element Method for Quad-Curl Problems"],"prefix":"10.1515","volume":"21","author":[{"given":"Lixiu","family":"Wang","sequence":"first","affiliation":[{"name":"Beijing Computational Science Research Center , Beijing 100193 , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huiyuan","family":"Li","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Computer Science\/Laboratory of Parallel Computing , Institute of Software, Chinese Academy of Sciences , Beijing 100190 , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhimin","family":"Zhang","sequence":"additional","affiliation":[{"name":"Beijing Computational Science Research Center , Beijing 100193 , P. R. China ; and Department of Mathematics, Wayne State University, MI 48202, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2021,5,6]]},"reference":[{"key":"2023033111373212183_j_cmam-2020-0152_ref_001","doi-asserted-by":"crossref","unstructured":"F. Ben Belgacem and C. Bernardi,\nSpectral element discretization of the Maxwell equations,\nMath. Comp. 68 (1999), no. 228, 1497\u20131520.","DOI":"10.1090\/S0025-5718-99-01086-8"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_002","doi-asserted-by":"crossref","unstructured":"M. Benzi, G. H. Golub and J. Liesen,\nNumerical solution of saddle point problems,\nActa Numer. 14 (2005), 1\u2013137.","DOI":"10.1017\/S0962492904000212"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_003","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, J. Cui and L.-y. Sung,\nMultigrid methods based on Hodge decomposition for a quad-curl problem,\nComput. Methods Appl. Math. 19 (2019), no. 2, 215\u2013232.","DOI":"10.1515\/cmam-2019-0011"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_004","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, J. Sun and L.-y. Sung,\nHodge decomposition methods for a quad-curl problem on planar domains,\nJ. Sci. Comput. 73 (2017), no. 2\u20133, 495\u2013513.","DOI":"10.1007\/s10915-017-0449-0"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_005","doi-asserted-by":"crossref","unstructured":"W. Cai,\nComputational Methods for Electromagnetic Phenomena,\nCambridge University, Cambridge, 2013.","DOI":"10.1017\/CBO9781139108157"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_006","doi-asserted-by":"crossref","unstructured":"F. Cakoni and H. Haddar,\nA variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media,\nInverse Probl. Imaging 1 (2007), no. 3, 443\u2013456.","DOI":"10.3934\/ipi.2007.1.443"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_007","doi-asserted-by":"crossref","unstructured":"G. C. Cohen,\nHigher-Order Numerical Methods for Transient Wave Equations,\nSpringer, Berlin, 2002.","DOI":"10.1007\/978-3-662-04823-8"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_008","doi-asserted-by":"crossref","unstructured":"C. Greif and D. Sch\u00f6tzau,\nPreconditioners for the discretized time-harmonic Maxwell equations in mixed form,\nNumer. Linear Algebra Appl. 14 (2007), no. 4, 281\u2013297.","DOI":"10.1002\/nla.515"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_009","doi-asserted-by":"crossref","unstructured":"B.-Y. Guo, J. Shen and L.-L. Wang,\nGeneralized Jacobi polynomials\/functions and their applications,\nAppl. Numer. Math. 59 (2009), no. 5, 1011\u20131028.","DOI":"10.1016\/j.apnum.2008.04.003"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_010","doi-asserted-by":"crossref","unstructured":"B.-Y. Guo and T.-J. Wang,\nComposite Laguerre-Legendre spectral method for fourth-order exterior problems,\nJ. Sci. Comput. 44 (2010), no. 3, 255\u2013285.","DOI":"10.1007\/s10915-010-9367-0"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_011","doi-asserted-by":"crossref","unstructured":"Q. Hong, J. Hu, S. Shu and J. Xu,\nA discontinuous Galerkin method for the fourth-order curl problem,\nJ. Comput. Math. 30 (2012), no. 6, 565\u2013578.","DOI":"10.4208\/jcm.1206-m3572"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_012","doi-asserted-by":"crossref","unstructured":"K. Hu, Q. Zhang and Z. Zhang,\nSimple curl-curl-conforming finite elements in two dimensions,\nSIAM J. Sci. Comput. 42 (2020), no. 6, A3859\u2013A3877.","DOI":"10.1137\/20M1333390"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_013","doi-asserted-by":"crossref","unstructured":"H. Li, W. Shan and Z. Zhang,\n\n \n \n \n C\n 1\n \n \n \n C^{1}\n \n -conforming quadrilateral spectral element method for fourth-order equations,\nCommun. Appl. Math. Comput. 1 (2019), no. 3, 403\u2013434.","DOI":"10.1007\/s42967-019-00041-w"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_014","doi-asserted-by":"crossref","unstructured":"J. Li and Y. Huang,\nTime-Domain Finite Element Methods for Maxwell\u2019s Equations in Metamaterials,\nSpringer Ser. Comput. Math. 43,\nSpringer, Heidelberg, 2013.","DOI":"10.1007\/978-3-642-33789-5"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_015","doi-asserted-by":"crossref","unstructured":"Y. Liu, J. Lee, X. Tian and Q. Liu,\nA spectral-element time-domain solution of Maxwell\u2019s equations,\nMicrowave Optical Technol. Lett. 48 (2006), no. 4, 673\u2013680.","DOI":"10.1002\/mop.21440"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_016","doi-asserted-by":"crossref","unstructured":"P. Monk and J. Sun,\nFinite element methods for Maxwell\u2019s transmission eigenvalues,\nSIAM J. Sci. Comput. 34 (2012), no. 3, B247\u2013B264.","DOI":"10.1137\/110839990"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_017","doi-asserted-by":"crossref","unstructured":"L. Na, L. Tob\u00f3n, Y. Zhao, Y. Tang and Q. Liu,\nMixed spectral-element method for 3-D Maxwell\u2019s eigenvalue problem,\nIEEE Trans. Microwave Theory Tech. 63 (2015), no. 2, 317\u2013325.","DOI":"10.1109\/TMTT.2014.2387839"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_018","doi-asserted-by":"crossref","unstructured":"S. Nicaise,\nSingularities of the quad curl problem,\nJ. Differential Equations 264 (2018), no. 8, 5025\u20135069.","DOI":"10.1016\/j.jde.2017.12.032"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_019","doi-asserted-by":"crossref","unstructured":"A. T. Patera,\nA spectral element method for fluid dynamics: Laminar flow in a channel expansion,\nJ. Comput. Phys. 54 (1984), no. 3, 468\u2013488.","DOI":"10.1016\/0021-9991(84)90128-1"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_020","doi-asserted-by":"crossref","unstructured":"J. Shen, T. Tang and L. Wang,\nSpectral Methods,\nSpringer, Berlin, 2011.","DOI":"10.1007\/978-3-540-71041-7"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_021","doi-asserted-by":"crossref","unstructured":"J. Sun,\nA mixed FEM for the quad-curl eigenvalue problem,\nNumer. Math. 132 (2016), no. 1, 185\u2013200.","DOI":"10.1007\/s00211-015-0708-7"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_022","doi-asserted-by":"crossref","unstructured":"J. Sun and L. Xu,\nComputation of Maxwell\u2019s transmission eigenvalues and its applications in inverse medium problems,\nInverse Problems 29 (2013), no. 10, Article ID 104013.","DOI":"10.1088\/0266-5611\/29\/10\/104013"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_023","doi-asserted-by":"crossref","unstructured":"J. Sun, Q. Zhang and Z. Zhang,\nA curl-conforming weak Galerkin method for the quad-curl problem,\nBIT 59 (2019), no. 4, 1093\u20131114.","DOI":"10.1007\/s10543-019-00764-5"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_024","doi-asserted-by":"crossref","unstructured":"J. Sun and A. Zhou,\nFinite Element Methods for Eigenvalue Problems,\nChapman and Hall\/CRC, Boca Raton, 2016.","DOI":"10.1201\/9781315372419"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_025","doi-asserted-by":"crossref","unstructured":"Z. Sun, J. Cui, F. Gao and C. Wang,\nMultigrid methods for a quad-curl problem based on \n \n \n \n C\n 0\n \n \n \n C^{0}\n \n interior penalty method,\nComput. Math. Appl. 76 (2018), no. 9, 2192\u20132211.","DOI":"10.1016\/j.camwa.2018.07.048"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_026","unstructured":"G. Szeg\u00f6,\nOrthogonal Polynomials,\nAmer. Math. Soc. Colloq. Publ. 23,\nAmerican Mathematical Society, New York, 1939."},{"key":"2023033111373212183_j_cmam-2020-0152_ref_027","doi-asserted-by":"crossref","unstructured":"C. Wang, Z. Sun and J. Cui,\nA new error analysis of a mixed finite element method for the quad-curl problem,\nAppl. Math. Comput. 349 (2019), 23\u201338.","DOI":"10.1016\/j.amc.2018.12.027"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_028","unstructured":"L. Wang, Q. Zhang, J. Sun and Z. Zhang,\nA priori and a posteriori error estimates for the quad-curl eigenvalue problem,\npreprint (2020), https:\/\/arxiv.org\/abs\/2007.01330."},{"key":"2023033111373212183_j_cmam-2020-0152_ref_029","doi-asserted-by":"crossref","unstructured":"L. Wang, Q. Zhang and Z. Zhang,\nSuperconvergence analysis and PPR recovery of arbitrary order edge elements for Maxwell\u2019s equations,\nJ. Sci. Comput. 78 (2019), no. 2, 1207\u20131230.","DOI":"10.1007\/s10915-018-0805-8"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_030","doi-asserted-by":"crossref","unstructured":"Q. Zhang, L. Wang and Z. Zhang,\n\n \n \n \n H\n \u2062\n \n (\n \n curl\n 2\n \n )\n \n \n \n \n H(\\mathrm{curl}^{2})\n \n -conforming finite elements in 2 dimensions and applications to the quad-curl problem,\nSIAM J. Sci. Comput. 41 (2019), no. 3, A1527\u2013A1547.","DOI":"10.1137\/18M1199988"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_031","unstructured":"Q. Zhang and Z. Zhang,\nCurl-curl conforming elements on tetrahedra,\npreprint (2020), https:\/\/arxiv.org\/abs\/2007.10421."},{"key":"2023033111373212183_j_cmam-2020-0152_ref_032","doi-asserted-by":"crossref","unstructured":"S. Zhang,\nMixed schemes for quad-curl equations,\nESAIM Math. Model. Numer. Anal. 52 (2018), no. 1, 147\u2013161.","DOI":"10.1051\/m2an\/2018005"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_033","doi-asserted-by":"crossref","unstructured":"S. Zhang,\nRegular decomposition and a framework of order reduced methods for fourth order problems,\nNumer. Math. 138 (2018), no. 1, 241\u2013271.","DOI":"10.1007\/s00211-017-0902-x"},{"key":"2023033111373212183_j_cmam-2020-0152_ref_034","doi-asserted-by":"crossref","unstructured":"B. Zheng, Q. Hu and J. Xu,\nA nonconforming finite element method for fourth order curl equations in \n \n \n \n R\n 3\n \n \n \n \\mathbb{R}^{3}\n \n ,\nMath. Comp. 80 (2011), no. 276, 1871\u20131886.","DOI":"10.1090\/S0025-5718-2011-02480-4"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0152\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0152\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T15:12:34Z","timestamp":1680275554000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0152\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,6]]},"references-count":34,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,5,6]]},"published-print":{"date-parts":[[2021,7,1]]}},"alternative-id":["10.1515\/cmam-2020-0152"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2020-0152","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,6]]}}}