{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T21:16:46Z","timestamp":1648761406282},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,8,18]],"date-time":"2020-08-18T00:00:00Z","timestamp":1597708800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,8,18]],"date-time":"2020-08-18T00:00:00Z","timestamp":1597708800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11501529"],"award-info":[{"award-number":["11501529"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2021,6]]},"DOI":"10.1007\/s11075-020-00991-w","type":"journal-article","created":{"date-parts":[[2020,8,18]],"date-time":"2020-08-18T02:02:54Z","timestamp":1597716174000},"page":"873-894","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A plane wave least squares method for the Maxwell equations in anisotropic media"],"prefix":"10.1007","volume":"87","author":[{"given":"Long","family":"Yuan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,8,18]]},"reference":[{"key":"991_CR1","doi-asserted-by":"publisher","first-page":"879","DOI":"10.1007\/s11075-017-0465-z","volume":"79","author":"P Antunes","year":"2018","unstructured":"Antunes, P.: A numerical algorithm to reduce ill-conditioning in meshless methods for the Helmholtz equation. Numer. Algor. 79, 879\u2013897 (2018)","journal-title":"Numer. Algor."},{"key":"991_CR2","unstructured":"Cessenat, O.: Application D\u2019une Nouvelle Formulation Variationnelle Aux \u00e9quations D\u2019ondes Harmoniques Probl\u00e8mes De Helmholtz 2D et de Maxwell 3D, Ph.D. Thesis, Universit\u00e9 Paris IX Dauphine (1996)"},{"key":"991_CR3","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1137\/S0036142995285873","volume":"35","author":"O Cessenat","year":"1998","unstructured":"Cessenat, O., Despres, B.: Application of an ultra weak variational formulation of elliptic pdes to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal. 35, 255\u2013299 (1998)","journal-title":"SIAM J. Numer. Anal."},{"key":"991_CR4","doi-asserted-by":"publisher","unstructured":"Congreve, S., Houston, P., Perugia, I.: Adaptive refinement for hp \u2212version Trefftz discontinuous Galerkin methods for the homogeneous Helmholtz problem. Adv. Comput. Math. https:\/\/doi.org\/10.1007\/s10444-018-9621-9 (2018)","DOI":"10.1007\/s10444-018-9621-9"},{"key":"991_CR5","doi-asserted-by":"publisher","first-page":"1961","DOI":"10.1016\/j.jcp.2007.02.030","volume":"225","author":"G Gabard","year":"2007","unstructured":"Gabard, G.: Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys. 225, 1961\u20131984 (2007)","journal-title":"J. Comput. Phys."},{"key":"991_CR6","doi-asserted-by":"publisher","first-page":"1309","DOI":"10.1002\/nme.2579","volume":"79","author":"C Geuzaine","year":"2009","unstructured":"Geuzaine, C., Remacle, J.-F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Meth. Engng 79, 1309\u20131331 (2009)","journal-title":"Int. J. Numer. Meth. Engng"},{"key":"991_CR7","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1051\/m2an\/2009002","volume":"43","author":"C Gittelson","year":"2009","unstructured":"Gittelson, C., Hiptmair, R., Perugia, I.: Plane wave discontinuous Galerkin methods: analysis of the h-version, ESAIM: m2AN. Math. Model. Numer. Anal. 43, 297\u2013331 (2009)","journal-title":"Math. Model. Numer. Anal."},{"key":"991_CR8","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1137\/090761057","volume":"49","author":"R Hiptmair","year":"2011","unstructured":"Hiptmair, R., Moiola, A., Perugia, I.: Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49, 264\u2013284 (2011)","journal-title":"SIAM J. Numer. Anal."},{"key":"991_CR9","doi-asserted-by":"publisher","first-page":"2263","DOI":"10.1142\/S021820251100574X","volume":"21","author":"R Hiptmair","year":"2011","unstructured":"Hiptmair, R., Moiola, A., Perugia, I.: Stability results for the time-harmonic Maxwell equationis with impedance boundary conditions. Math. Mod. Meth. Appl. Sci. 21, 2263\u20132287 (2011)","journal-title":"Math. Mod. Meth. Appl. Sci."},{"key":"991_CR10","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1090\/S0025-5718-2012-02627-5","volume":"82","author":"R Hiptmair","year":"2013","unstructured":"Hiptmair, R., Moiola, A., Perugia, I.: Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comp. 82, 247\u2013268 (2013)","journal-title":"Math. Comp."},{"key":"991_CR11","doi-asserted-by":"crossref","unstructured":"Hiptmair, R., Moiola, A., Perugia, I.: A survey of Trefftz methods for the Helmholtz equation. In: Building bridges: connections and challenges in modern approaches to numerical partial differential equations, pp 237\u2013279. Springer International Publishing, New York (2015)","DOI":"10.1007\/978-3-319-41640-3_8"},{"issue":"3","key":"991_CR12","doi-asserted-by":"publisher","first-page":"1242","DOI":"10.1137\/16M1084791","volume":"15","author":"Q Hu","year":"2017","unstructured":"Hu, Q., Li, X.: Efficient multilevel preconditioners for three-dimensional plane wave Helmholtz systems with large wave numbers. Multiscale Model Simul. 15(3), 1242\u20131266 (2017)","journal-title":"Multiscale Model Simul."},{"key":"991_CR13","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1051\/m2an\/2018043","volume":"53","author":"Q Hu","year":"2019","unstructured":"Hu, Q., Song, R.: A variant of the plane wave least squares method for the time-harmonic Maxwell\u2019s equations, ESAIM: m2AN. Math. Model. Numer. Anal. 53, 85\u2013103 (2019)","journal-title":"Math. Model. Numer. Anal."},{"key":"991_CR14","first-page":"587","volume":"11","author":"Q Hu","year":"2014","unstructured":"Hu, Q., Yuan, L.: A weighted variational formulation based on plane wave basis for discretization of Helmholtz equations. Int. J. Numer. Anal. Model. 11, 587\u2013607 (2014)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"991_CR15","doi-asserted-by":"publisher","first-page":"A1911","DOI":"10.1137\/130928509","volume":"36","author":"Q Hu","year":"2014","unstructured":"Hu, Q., Yuan, L.: A plane wave Least-Squares method for Time-Harmonic maxwell\u2019s equations in absorbing media. SIAM J. Sci. Comput. 36, A1911\u2013A1936 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"991_CR16","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/s10444-017-9542-z","volume":"44","author":"Q Hu","year":"2018","unstructured":"Hu, Q., Yuan, L.: A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations. Adv. Comput. Math. 44, 245\u2013275 (2018)","journal-title":"Adv. Comput. Math."},{"key":"991_CR17","doi-asserted-by":"publisher","first-page":"A2232","DOI":"10.1137\/151003040","volume":"38","author":"Q Hu","year":"2016","unstructured":"Hu, Q., Zhang, H.: Substructuring preconditioners for the systems arising from plane wave discretization of Helmholtz equations. SIAM J. Sci. Comput. 38, A2232\u2013A2261 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"991_CR18","doi-asserted-by":"publisher","first-page":"731","DOI":"10.1016\/j.jcp.2006.10.016","volume":"223","author":"T Huttunen","year":"2007","unstructured":"Huttunen, T., Malinen, M., Monk, P.: Solving Maxwell\u2019s equations using the ultra weak variational formulation. J. Comput. Phys. 223, 731\u2013758 (2007)","journal-title":"J. Comput. Phys."},{"key":"991_CR19","first-page":"350","volume":"25","author":"T Huttunen","year":"2007","unstructured":"Huttunen, T., Monk, P.: The use of plane waves to approximate wave propagation in anisotropic media. J. Comput. Math. 25, 350\u2013367 (2007)","journal-title":"J. Comput. Math."},{"key":"991_CR20","doi-asserted-by":"publisher","first-page":"A1525","DOI":"10.1137\/140967696","volume":"37","author":"S Kapita","year":"2015","unstructured":"Kapita, S., Monk, P., Warburton, T.: Residual-based adapitivity and PWDG methods for the Helmholtz equation. SIAM J. Sci. Comput. 37, A1525\u20131553 (2015)","journal-title":"SIAM J. Sci. Comput."},{"key":"991_CR21","unstructured":"Luostari, T.: Non-polynomial approximation methods in acoustics and elasticity, Ph.D. Thesis, University of Eastern Finland. Available at https:\/\/core.ac.uk\/download\/pdf\/19163531.pdf (2013)"},{"key":"991_CR22","first-page":"65","volume-title":"On approximation in meshless methods, frontiers of numerical analysis, universitext","author":"J Melenk","year":"2005","unstructured":"Melenk, J.: On approximation in meshless methods, frontiers of numerical analysis, universitext, pp 65\u2013141. Springer, New York (2005)"},{"key":"991_CR23","doi-asserted-by":"publisher","first-page":"809","DOI":"10.1007\/s00033-011-0147-y","volume":"62","author":"A Moiola","year":"2011","unstructured":"Moiola, A., Hiptmair, R., Perugia, I.: Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys. 62, 809\u2013837 (2011)","journal-title":"Z. Angew. Math. Phys."},{"key":"991_CR24","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198508885.001.0001","volume-title":"Finite Element Methods for Maxwell\u2019s Equations","author":"P Monk","year":"2003","unstructured":"Monk, P.: Finite Element Methods for Maxwell\u2019s Equations. Clarendon Press, Oxford (2003)"},{"key":"991_CR25","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1016\/S0045-7825(98)00326-0","volume":"175","author":"P Monk","year":"1999","unstructured":"Monk, P., Wang, D.: A least-squares method for the Helmholtz equation. Comput. Meth. Appl. Mech. Engng. 175, 121\u2013136 (1999)","journal-title":"Comput. Meth. Appl. Mech. Engng."},{"key":"991_CR26","doi-asserted-by":"publisher","first-page":"683","DOI":"10.1002\/nme.5939","volume":"116","author":"J Peng","year":"2018","unstructured":"Peng, J., Wang, J., Shu, S.: Adaptive BDDC algorithms for the system arising from plane wave discretization of Helmholtz equations. Int. J. Numer. Methods Eng. 116, 683\u2013707 (2018)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"991_CR27","unstructured":"Peng, J., Shu, S., Wang, J., Zhong, L.: Adaptive-Multilevel BDDC algorithm for three-dimensional plane wave Helmholtz systems. J. Comput. Appl. Math. arXiv:1909.04426 (2019)"},{"key":"991_CR28","unstructured":"Sloane, N.: Tables of spherical codes (with collaboration of R.H. Hardin, W.D. Smith and others) published electronically at http:\/\/www2.research.att.com\/njas\/packings (2000)"},{"key":"991_CR29","unstructured":"Trefftz, E.: Ein gegenst\u00fcck zum Ritzschen Verfahren. Sec. Inte. Cong. Appl. Mech. 131\u2013137 (1926)"},{"key":"991_CR30","doi-asserted-by":"publisher","first-page":"1496","DOI":"10.4208\/cicp.OA-2018-0104","volume":"25","author":"L Yuan","year":"2019","unstructured":"Yuan, L., Hu, Q.: Error analysis of the plane wave discontinuous Galerkin method for Maxwell\u2019s equations in anisotropic media. Commun. Comput. Phys. 25, 1496\u20131522 (2019)","journal-title":"Commun. Comput. Phys."},{"key":"991_CR31","unstructured":"Yuan, L, Hu, Q.: Plane wave discontinuous Galerkin methods for the Helmholtz equation and Maxwell equations in anisotropic media, available at arXiv:1909.05997v2"},{"key":"991_CR32","first-page":"802","volume":"13","author":"L Yuan","year":"2016","unstructured":"Yuan, L., Hu, Q., Hengbin, A: Parallel preconditioners for plane wave Helmholtz and Maxwell systems with large wave numbers. Int. J. Numer. Anal. Model 13, 802\u2013819 (2016)","journal-title":"Int. J. Numer. Anal. Model"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00991-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-020-00991-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00991-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,17]],"date-time":"2021-08-17T23:23:41Z","timestamp":1629242621000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-020-00991-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,18]]},"references-count":32,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["991"],"URL":"https:\/\/doi.org\/10.1007\/s11075-020-00991-w","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,18]]},"assertion":[{"value":"25 October 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 July 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 August 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}