{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,29]],"date-time":"2026-03-29T21:44:40Z","timestamp":1774820680821,"version":"3.50.1"},"reference-count":12,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T00:00:00Z","timestamp":1611014400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T00:00:00Z","timestamp":1611014400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2021,10]]},"DOI":"10.1007\/s10208-020-09478-5","type":"journal-article","created":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T18:03:01Z","timestamp":1611079381000},"page":"1153-1180","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Duality in Finite Element Exterior Calculus and Hodge Duality on the Sphere"],"prefix":"10.1007","volume":"21","author":[{"given":"Yakov","family":"Berchenko-Kogan","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,1,19]]},"reference":[{"key":"9478_CR1","doi-asserted-by":"crossref","unstructured":"D. N. Arnold, Spaces of finite element differential forms, in Analysis and numerics of partial differential equations, vol. 4 of Springer INdAM Ser., Springer, Milan, 2013, pp.\u00a0117\u2013140.","DOI":"10.1007\/978-88-470-2592-9_9"},{"key":"9478_CR2","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/s10208-011-9087-3","volume":"11","author":"DN Arnold","year":"2011","unstructured":"D. N. Arnold and G. Awanou, The serendipity family of finite elements, Found. Comput. Math., 11 (2011), pp. 337\u2013344.","journal-title":"Found. Comput. Math."},{"key":"9478_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00211-014-0631-3","volume":"129","author":"DN Arnold","year":"2015","unstructured":"D. N. Arnold, D. Boffi, and F. Bonizzoni, Finite element differential forms on curvilinear cubic meshes and their approximation properties, Numer. Math., 129 (2015), pp. 1\u201320.","journal-title":"Numer. Math."},{"key":"9478_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492906210018","volume":"15","author":"DN Arnold","year":"2006","unstructured":"D. N. Arnold, R. S. Falk, and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15 (2006), pp. 1\u2013155.","journal-title":"Acta Numer."},{"key":"9478_CR5","doi-asserted-by":"publisher","first-page":"1660","DOI":"10.1016\/j.cma.2008.12.017","volume":"198","author":"DN Arnold","year":"2009","unstructured":"D. N. Arnold, R. S. Falk, and R. Winther, Geometric decompositions and local bases for spaces of finite element differential forms, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 1660\u20131672.","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"9478_CR6","unstructured":"Y. Berchenko-Kogan, Duality in finite element exterior calculus, 2018. arXiv:1807.01161."},{"key":"9478_CR7","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/BF01389710","volume":"47","author":"F Brezzi","year":"1985","unstructured":"F. Brezzi, J. Douglas, Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), pp. 217\u2013235.","journal-title":"Numer. Math."},{"key":"9478_CR8","doi-asserted-by":"publisher","first-page":"2631","DOI":"10.1090\/S0025-5718-2014-02827-5","volume":"83","author":"RS Falk","year":"2014","unstructured":"R. S. Falk and R. Winther, Local bounded cochain projections, Math. Comp., 83 (2014), pp. 2631\u20132656.","journal-title":"Math. Comp."},{"key":"9478_CR9","unstructured":"M. W. Licht, On basis constructions in finite element exterior calculus, 2018. arXiv:1810.01896."},{"key":"9478_CR10","doi-asserted-by":"crossref","unstructured":"J.-C. N\u00e9d\u00e9lec, Mixed finite elements in$${\\mathbb{R}}^{3}$$, Numer. Math., 35 (1980), pp. 315\u2013341.","DOI":"10.1007\/BF01396415"},{"key":"9478_CR11","doi-asserted-by":"crossref","unstructured":"J.-C. N\u00e9d\u00e9lec, A new family of mixed finite elements in$${\\mathbb{R}}^{3}$$, Numer. Math., 50 (1986), pp. 57\u201381.","DOI":"10.1007\/BF01389668"},{"key":"9478_CR12","doi-asserted-by":"crossref","unstructured":"P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, (1977), pp. 292\u2013315. Lecture Notes in Math., Vol. 606.","DOI":"10.1007\/BFb0064470"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-020-09478-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10208-020-09478-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-020-09478-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,13]],"date-time":"2021-10-13T03:22:48Z","timestamp":1634095368000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10208-020-09478-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,19]]},"references-count":12,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2021,10]]}},"alternative-id":["9478"],"URL":"https:\/\/doi.org\/10.1007\/s10208-020-09478-5","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,19]]},"assertion":[{"value":"17 July 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 June 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 September 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 January 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}