{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:21:05Z","timestamp":1787329265988,"version":"build-2736575974"},"reference-count":14,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>The classical error analysis for the Raviart\u2013Thomas interpolation on triangular elements requires the so-called regularity of the elements, or equivalently, the minimum angle condition. However, in the lowest order case, optimal order error estimates have been obtained in [G. Acosta and R. G. Dur\u00e1n, SIAM J. Numer. Anal., 37 (2000), pp. 18\u201336] replacing the regularity hypothesis by the maximum angle condition, which was known to be sufficient to prove estimates for the standard Lagrange interpolation. In this paper we prove error estimates on triangular elements for the Raviart\u2013Thomas interpolation of any order under the maximum angle condition. Also, we show how our arguments can be extended to the three-dimensional case to obtain error estimates for tetrahedral elements under the regular vertex property introduced in [G. Acosta and R. G. Dur\u00e1n, SIAM J. Numer. Anal., 37 (2000), pp. 18\u201336].<\/jats:p>","DOI":"10.1137\/060665312","type":"journal-article","created":{"date-parts":[[2008,4,4]],"date-time":"2008-04-04T18:08:07Z","timestamp":1207332487000},"page":"1442-1453","source":"Crossref","is-referenced-by-count":11,"title":["Error Estimates for the Raviart\u2013Thomas Interpolation Under the Maximum Angle Condition"],"prefix":"10.1137","volume":"46","author":[{"given":"Ricardo G.","family":"Dur\u00e1n","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ariel L.","family":"Lombardi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,4,4]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142997331293"},{"key":"R2","unstructured":"T. Apel,\n                      Anisotropic Finite Elements: Local Estimates and Applications\n                      , Adv. Numer. Math., Teubner, Stuttgart, 1999."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/0713021"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"S. Brenner and L. R. Scott,\n                      The Mathematical Theory of Finite Element Methods\n                      , Springer, New York, 1994.","DOI":"10.1007\/978-1-4757-4338-8"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , North\u2013Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"R. G. Dur\u00e1n,\n                      Error estimates for 3-d narrow finite elements\n                      , Math. Comp. 68 (1999), pp. 187\u2013199.","DOI":"10.1090\/S0025-5718-99-00994-1"},{"key":"R7","unstructured":"R. G. Dur\u00e1n,\n                      Error estimates for anisotropic finite elements and applications\n                      , in Proceedings of the International Congress of Mathematicians, 2006."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-05-01732-1"},{"key":"R9","first-page":"46","volume":"10","author":"Jamet P.","year":"1976","journal-title":"RAIRO Anal. Num\u00e9r.","ISSN":"https:\/\/id.crossref.org\/issn\/0399-0516","issn-type":"print"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0729031"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396415"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"P. A. Raviart and J. M. Thomas,\n                      A mixed finite element method for second order elliptic problems\n                      , in Mathematical Aspects of the Finite Element Method, Lecture Notes in Math. 606, I. Galligani and E. Magenes, eds., Springer, Berlin, 1977, pp. 292\u2013315.","DOI":"10.1007\/BFb0064470"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1994-1226816-5"},{"key":"R14","unstructured":"J. M. Thomas,\n                      Sur l'Analyse Num\u00e9rique des M\u00e9thodes d'\u00c9l\u00e9ments Finis Hybrides et Mixtes\n                      , Th\u00e8se d'Etat, Universit\u00e9 Pierre et Marie Curie, Paris, 1977."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060665312","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:13:41Z","timestamp":1787325221000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060665312"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":14,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060665312"],"URL":"https:\/\/doi.org\/10.1137\/060665312","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}