{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T12:20:50Z","timestamp":1772367650005,"version":"3.50.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,1,1]],"date-time":"2020-01-01T00:00:00Z","timestamp":1577836800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,1,1]],"date-time":"2020-01-01T00:00:00Z","timestamp":1577836800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971366"],"award-info":[{"award-number":["11971366"]}],"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":["11571266"],"award-info":[{"award-number":["11571266"]}],"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":["11661161017"],"award-info":[{"award-number":["11661161017"]}],"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":[[2020,1]]},"DOI":"10.1007\/s10915-019-01111-0","type":"journal-article","created":{"date-parts":[[2020,1,8]],"date-time":"2020-01-08T09:02:42Z","timestamp":1578474162000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Mixed Method for Maxwell Eigenproblem"],"prefix":"10.1007","volume":"82","author":[{"given":"Zhijie","family":"Du","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huoyuan","family":"Duan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,1,8]]},"reference":[{"key":"1111_CR1","volume-title":"Sobolev Spaces","author":"RA Adams","year":"1975","unstructured":"Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)"},{"key":"1111_CR2","doi-asserted-by":"publisher","first-page":"823","DOI":"10.1002\/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B","volume":"21","author":"C Amrouche","year":"1998","unstructured":"Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21, 823\u2013864 (1998)","journal-title":"Math. Methods Appl. Sci."},{"key":"1111_CR3","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1007\/BF02165003","volume":"16","author":"I Babus\u0306ka","year":"1971","unstructured":"Babus\u0306ka, I.: Error-bounds for finite element method. Numer. Math. 16, 322\u2013333 (1971)","journal-title":"Numer. Math."},{"key":"1111_CR4","first-page":"641","volume-title":"Handbook of Numerical Analysis II","author":"I Babus\u0306ka","year":"1991","unstructured":"Babus\u0306ka, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis II, pp. 641\u2013787. North-Holland, Amsterdam (1991)"},{"key":"1111_CR5","doi-asserted-by":"publisher","first-page":"398","DOI":"10.1137\/110835360","volume":"50","author":"S Badia","year":"2012","unstructured":"Badia, S., Codina, R.: A nodal-based finite element approximation of the Maxwell problem suitable for singular solutions. SIAM J. Numer. Anal. 50, 398\u2013417 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"1111_CR6","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1070\/RM1987v042n06ABEH001505","volume":"42","author":"M Birman","year":"1987","unstructured":"Birman, M., Solomyak, M.: $$L^2$$-theory of the Maxwell operator in arbitrary domains. Russ. Math. Surv. 42, 75\u201396 (1987)","journal-title":"Russ. Math. Surv."},{"key":"1111_CR7","doi-asserted-by":"publisher","first-page":"3672","DOI":"10.1016\/j.cma.2006.10.024","volume":"196","author":"D Boffi","year":"2007","unstructured":"Boffi, D.: Approximation of eigenvalues in mixed form, discrete compactness property, and application to HP mixed finite elements. Comput. Methods Appl. Mech. Eng. 196, 3672\u20133681 (2007)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1111_CR8","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, 1264\u20131290 (1999)","journal-title":"SIAM J. Numer. Anal."},{"key":"1111_CR9","doi-asserted-by":"publisher","first-page":"1887","DOI":"10.1090\/S0025-5718-2011-02464-6","volume":"80","author":"A Bonito","year":"2011","unstructured":"Bonito, A., Guermond, J.-L.: Approximation of the eigenvalue problem for the time-harmonic Maxwell system by continuous Lagrange finite elements. Math. Comp. 80, 1887\u20131910 (2011)","journal-title":"Math. Comp."},{"key":"1111_CR10","doi-asserted-by":"publisher","first-page":"1457","DOI":"10.1051\/m2an\/2015086","volume":"50","author":"A Bonito","year":"2016","unstructured":"Bonito, A., Guermond, J.-L., Luddens, F.: An interior penalty method with C0 finite elements for the approximation of the Maxwell equations in heterogeneous media: convergence analysis with minimal regularity, ESAIM: M2AN Math. Model. Numer. Anal. 50, 1457\u20131489 (2016)","journal-title":"Model. Numer. Anal."},{"key":"1111_CR11","doi-asserted-by":"publisher","first-page":"1575","DOI":"10.1090\/S0025-5718-05-01759-X","volume":"74","author":"J Bramble","year":"2005","unstructured":"Bramble, J., Kolev, T., Pasciak, J.E.: The approximation of the Maxwell eigenvalue problem using a least squares method. Math. Comput. 74, 1575\u20131598 (2005)","journal-title":"Math. Comput."},{"key":"1111_CR12","doi-asserted-by":"publisher","first-page":"1739","DOI":"10.1090\/S0025-5718-03-01616-8","volume":"73","author":"J Bramble","year":"2004","unstructured":"Bramble, J., Pasciak, J.: A new approximation technique for div-curl systems. Math. Comp. 73, 1739\u20131762 (2004)","journal-title":"Math. Comp."},{"key":"1111_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and Hybrid Finite Element Methods","author":"F Brezzi","year":"1991","unstructured":"Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New-York (1991)"},{"key":"1111_CR14","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1007\/s00211-009-0246-2","volume":"113","author":"A Buffa","year":"2009","unstructured":"Buffa, A., Ciarlet Jr., P., Jamelot, E.: Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements. Numer. Math. 113, 497\u2013518 (2009)","journal-title":"Numer. Math."},{"key":"1111_CR15","doi-asserted-by":"publisher","first-page":"2198","DOI":"10.1137\/050636887","volume":"44","author":"A Buffa","year":"2006","unstructured":"Buffa, A., Perugia, I.: Discontinuous Galerkin approximation of the Maxwell eigenproblem. SIAM J. Numer. Anal. 44, 2198\u20132226 (2006)","journal-title":"SIAM J. Numer. Anal."},{"key":"1111_CR16","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s00211-018-0970-6","volume":"140","author":"S Christiansen","year":"2018","unstructured":"Christiansen, S., Hu, K.: Generalized finite element systems for smooth differential forms and Stokes\u2019 problem. Numer. Math. 140, 327\u2013371 (2018)","journal-title":"Numer. Math."},{"key":"1111_CR17","volume-title":"Handbook of Numerical Analysis, Vol. II, Finite Element Methods (part 1)","author":"PG Ciarlet","year":"1991","unstructured":"Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Ciarlet, P.G., Lions, J.-L. (eds.) Handbook of Numerical Analysis, Vol. II, Finite Element Methods (part 1). North-Holland, Amsterdam (1991)"},{"key":"1111_CR18","doi-asserted-by":"publisher","first-page":"358","DOI":"10.1016\/j.cma.2008.08.005","volume":"198","author":"P Ciarlet Jr","year":"2008","unstructured":"Ciarlet Jr., P., Hechme, G.: Computing electromagnetic eigenmodes with continuous Galerkin approximations. Comput. Methods Appl. Mech. Eng. 198, 358\u2013365 (2008)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1111_CR19","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1002\/(SICI)1099-1476(199902)22:3<243::AID-MMA37>3.0.CO;2-0","volume":"22","author":"M Costabel","year":"1999","unstructured":"Costabel, M., Dauge, M.: Maxwell and Lam\u00e9 eigenvalues on polyhedra. Math. Methods Appl. Sci. 22, 243\u2013258 (1999)","journal-title":"Math. Methods Appl. Sci."},{"key":"1111_CR20","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/s002110100388","volume":"93","author":"M Costabel","year":"2002","unstructured":"Costabel, M., Dauge, M.: Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math. 93, 239\u2013277 (2002)","journal-title":"Numer. Math."},{"key":"1111_CR21","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1051\/m2an\/1979130302271","volume":"13","author":"J Douglas","year":"1979","unstructured":"Douglas, J., Dupont, T., Percell, P., Scott, R.: A family of C1 finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. RAIRO-Numer. Anal. 13, 227\u2013255 (1979)","journal-title":"RAIRO-Numer. Anal."},{"key":"1111_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0025-5718-03-01520-5","volume":"73","author":"HY Duan","year":"2004","unstructured":"Duan, H.Y., Liang, G.P.: Nonconforming elements in least-squares mixed finite element methods. Math. Comp. 73, 1\u201318 (2004)","journal-title":"Math. Comp."},{"key":"1111_CR23","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1007\/s00211-012-0500-x","volume":"123","author":"HY Duan","year":"2013","unstructured":"Duan, H.Y., Lin, P., Tan, R.C.E.: Analysis of a continuous finite element method for $$H({ curl}, {\\rm div})$$-elliptic interface problem. Numer. Math. 123, 671\u2013707 (2013)","journal-title":"Numer. Math."},{"key":"1111_CR24","doi-asserted-by":"publisher","first-page":"1274","DOI":"10.1137\/070707749","volume":"47","author":"HY Duan","year":"2009","unstructured":"Duan, H.Y., Jia, F., Lin, P., Tan, Roger C.E.: The local $$L^2$$ projected $$C^0$$ finite element method for Maxwell problem. SIAM J. Numer. Anal. 47, 1274\u20131303 (2009)","journal-title":"SIAM J. Numer. Anal."},{"key":"1111_CR25","doi-asserted-by":"publisher","first-page":"1678","DOI":"10.1137\/100812136","volume":"51","author":"HY Duan","year":"2013","unstructured":"Duan, H.Y., Lin, P., Tan, Roger C.E.: Error estimates for a vectorial second-order elliptic eigenproblem by the local $$L^2$$ projected $$C^0$$ finite element method. SIAM J. Numer. Anal. 51, 1678\u20131714 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"1111_CR26","doi-asserted-by":"publisher","first-page":"2537","DOI":"10.1137\/050640102","volume":"45","author":"HY Duan","year":"2007","unstructured":"Duan, H.Y., Lin, P., Saikrishnan, P., Tan, R.C.E.: A least squares finite element method for the magnetostatic problem in a multiply-connected Lipschitz domain. SIAM J. Numer. Anal. 45, 2537\u20132563 (2007)","journal-title":"SIAM J. Numer. Anal."},{"key":"1111_CR27","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/s00211-012-0456-x","volume":"122","author":"HY Duan","year":"2012","unstructured":"Duan, H.Y., Lin, P., Tan, Roger C.E.: $$C^0$$ elements for generalized indefinite Maxwell\u2019s equations. Numer. Math. 122, 61\u201399 (2012)","journal-title":"Numer. Math."},{"key":"1111_CR28","doi-asserted-by":"publisher","first-page":"3208","DOI":"10.1137\/110850578","volume":"50","author":"HY Duan","year":"2012","unstructured":"Duan, H.Y., Li, S., Tan, Roger C.E., Zheng, W.Y.: A delta-regularization finite element method for a double curl problem with divergence-free constraint. SIAM J. Numer. Anal. 50, 3208\u20133230 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"1111_CR29","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1016\/j.jcp.2014.02.044","volume":"268","author":"HY Duan","year":"2014","unstructured":"Duan, H.Y., Tan, Roger C.E., Yang, S.-Y., You, C.-S.: Computation of Maxwell singular solution by nodal-continuous elements. J. Comput. Phys. 268, 63\u201383 (2014)","journal-title":"J. Comput. Phys."},{"key":"1111_CR30","doi-asserted-by":"publisher","first-page":"A224","DOI":"10.1137\/16M1078082","volume":"40","author":"HY Duan","year":"2018","unstructured":"Duan, H.Y., Tan, Roger C.E., Yang, S.-Y., You, C.-S.: A mixed $$H^1$$-conforming finite element method for solving Maxwell\u2019s equations with non-$$H^1$$ solution. SIAM J. Sci. Comput. 40, A224\u2013A250 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"1111_CR31","doi-asserted-by":"publisher","first-page":"1193","DOI":"10.1137\/140980004","volume":"54","author":"HY Duan","year":"2016","unstructured":"Duan, H.Y., Lin, P., Tan, Roger C.E.: A finite element method for a curlcurl-graddiv eigenvalue interface problem. SIAM J. Numer. Anal. 54, 1193\u20131228 (2016)","journal-title":"SIAM J. Numer. Anal."},{"key":"1111_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61623-5","volume-title":"Finite Element Methods for Navier\u2013Stokes Equations, Theory and Algorithms","author":"V Girault","year":"1986","unstructured":"Girault, V., Raviart, P.A.: Finite Element Methods for Navier\u2013Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)"},{"key":"1111_CR33","unstructured":"Grisvard, P.: Boundary value problems in non-smooth domains, Univ. of Maryland, Dept. of Math., Lecture Notes no. 19 (1980)"},{"key":"1111_CR34","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1017\/S0962492902000041","volume":"21","author":"R Hiptmair","year":"2002","unstructured":"Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 21, 237\u2013339 (2002)","journal-title":"Acta Numer."},{"key":"1111_CR35","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1016\/0045-7825(87)90053-3","volume":"64","author":"F Kikuchi","year":"1987","unstructured":"Kikuchi, F.: Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism. Comput. Methods Appl. Mech. Eng. 64, 509\u2013521 (1987)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1111_CR36","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1090\/S0025-5718-1981-0606505-9","volume":"36","author":"B Mercier","year":"1981","unstructured":"Mercier, B., Osborn, J., Rappaz, J., Raviart, P.-A.: Eigenvalue approximation by mixed and hybrid methods. Math. Comp. 36, 427\u2013453 (1981)","journal-title":"Math. Comp."},{"key":"1111_CR37","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198508885.001.0001","volume-title":"Finite Element Methods for Maxwell Equations","author":"P Monk","year":"2003","unstructured":"Monk, P.: Finite Element Methods for Maxwell Equations. Clarendon Press, Oxford (2003)"},{"key":"1111_CR38","doi-asserted-by":"publisher","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 $$R^3$$. Numer. Math. 35, 315\u2013341 (1980)","journal-title":"Numer. Math."},{"key":"1111_CR39","doi-asserted-by":"publisher","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 $$R^3$$. Numer. Math. 50, 57\u201381 (1986)","journal-title":"Numer. Math."},{"key":"1111_CR40","unstructured":"Qin, J.: On the convergence of some low order mixed finite elements for incompressible fluids, Thesis (Ph.D.) The Pennsylvania State University, 158 pp (1994)"},{"key":"1111_CR41","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1051\/m2an\/1985190101111","volume":"19","author":"LR Scott","year":"1985","unstructured":"Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. Math. Modell. Numer. Anal. 19, 111\u2013143 (1985)","journal-title":"Math. Modell. Numer. Anal."},{"key":"1111_CR42","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1002\/mma.1670020103","volume":"2","author":"C Weber","year":"1980","unstructured":"Weber, C.: A local compactness theorem for Maxwell\u2019s equations. Math. Method Appl. Sci. 2, 12\u201325 (1980)","journal-title":"Math. Method Appl. Sci."},{"key":"1111_CR43","doi-asserted-by":"publisher","first-page":"2685","DOI":"10.1109\/20.92213","volume":"6","author":"SH Wong","year":"1988","unstructured":"Wong, S.H., Cendes, Z.J.: Combined finite element-modal solution of three-dimensional eddy current problems. IEEE Trans. Magn. MAG-24 6, 2685\u20132687 (1988)","journal-title":"IEEE Trans. Magn. MAG-24"},{"key":"1111_CR44","doi-asserted-by":"publisher","first-page":"855","DOI":"10.1137\/090751049","volume":"32","author":"X Xu","year":"2010","unstructured":"Xu, X., Zhang, S.: A new divergence-free interpolation operator with applications to the Darcy\u2013Stokes\u2013Brinkman equations. SIAM J. Sci. Comput. 32, 855\u2013874 (2010)","journal-title":"SIAM J. Sci. Comput."},{"key":"1111_CR45","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1090\/S0025-5718-04-01711-9","volume":"74","author":"S Zhang","year":"2004","unstructured":"Zhang, S.: A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comp. 74, 543\u2013554 (2004)","journal-title":"Math. Comp."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-01111-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-019-01111-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-01111-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,7]],"date-time":"2021-01-07T01:06:10Z","timestamp":1609981570000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-019-01111-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":45,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["1111"],"URL":"https:\/\/doi.org\/10.1007\/s10915-019-01111-0","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]},"assertion":[{"value":"7 January 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 October 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 November 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 January 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"8"}}