{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:47:22Z","timestamp":1747198042472,"version":"3.40.5"},"reference-count":6,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,1,1]]},"abstract":"Abstract<\/jats:title>\n The aim of this paper is to provide new perspectives on relative finite element accuracy which is usually based on the asymptotic speed of convergence comparison when the mesh size h<\/jats:italic> goes to zero. Starting from a geometrical reading of the error estimate due to the Bramble\u2013Hilbert lemma, we derive two probability distributions that estimate the relative accuracy, considered as a random variable, between two Lagrange finite elements \n \n \n \n P<\/m:mi>\n k<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n {P_{k}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n P<\/m:mi>\n m<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n {P_{m}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> (\n \n \n \n k<\/m:mi>\n <<\/m:mo>\n m<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n {k<m}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>). We establish mathematical properties of these probabilistic distributions and we get new insights which, among others, show that \n \n \n \n P<\/m:mi>\n k<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n {P_{k}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> or \n \n \n \n P<\/m:mi>\n m<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n {P_{m}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is more likely accurate than the other, depending on the value of the mesh size h<\/jats:italic>.<\/jats:p>","DOI":"10.1515\/cmam-2018-0270","type":"journal-article","created":{"date-parts":[[2019,1,20]],"date-time":"2019-01-20T09:02:06Z","timestamp":1547974926000},"page":"79-87","source":"Crossref","is-referenced-by-count":3,"title":["A New Probabilistic Interpretation of the Bramble\u2013Hilbert Lemma"],"prefix":"10.1515","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1263-5313","authenticated-orcid":false,"given":"Jo\u00ebl","family":"Chaskalovic","sequence":"first","affiliation":[{"name":"D\u2019Alembert , Sorbonne University , Paris , France"}]},{"given":"Franck","family":"Assous","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Ariel University , 40700 Ariel , Israel"}]}],"member":"374","published-online":{"date-parts":[[2019,1,20]]},"reference":[{"key":"2023033110163390576_j_cmam-2018-0270_ref_001","doi-asserted-by":"crossref","unstructured":"F. Assous and J. Chaskalovic,\nData mining techniques for scientific computing: Application to asymptotic paraxial approximations to model ultra-relativistic particles,\nJ. Comput. Phys. 230 (2011), 4811\u20134827.","DOI":"10.1016\/j.jcp.2011.03.005"},{"key":"2023033110163390576_j_cmam-2018-0270_ref_002","doi-asserted-by":"crossref","unstructured":"F. Assous and J. Chaskalovic,\nError estimate evaluation in numerical approximations of partial differential equations: A pilot study using data mining methods,\nC. R. Mecanique 341 (2013), 304\u2013313.","DOI":"10.1016\/j.crme.2013.01.002"},{"key":"2023033110163390576_j_cmam-2018-0270_ref_003","doi-asserted-by":"crossref","unstructured":"F. Assous and J. Chaskalovic,\nIndeterminate constants in numerical approximations of PDEs: A pilot study using data mining techniques,\nJ. Comput. Appl. Math 270 (2014), 462\u2013470.","DOI":"10.1016\/j.cam.2013.12.015"},{"key":"2023033110163390576_j_cmam-2018-0270_ref_004","doi-asserted-by":"crossref","unstructured":"J. Chaskalovic,\nMathematical and Numerical Methods for Partial Differential Equations,\nSpringer, Cham, 2013.","DOI":"10.1007\/978-3-319-03563-5"},{"key":"2023033110163390576_j_cmam-2018-0270_ref_005","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\nBasic error estimates for elliptic problems,\nHandbook of Numerical Analysis. Vol. II,\nNorth Holland, Amsterdam (1991), 17\u2013351.","DOI":"10.1016\/S1570-8659(05)80039-0"},{"key":"2023033110163390576_j_cmam-2018-0270_ref_006","unstructured":"P. A. Raviart and J. M. Thomas,\nIntroduction \u00e0 l\u2019analyse num\u00e9rique des \u00e9quations aux d\u00e9riv\u00e9es partielles,\nMasson, Paris, 1982."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/20\/1\/article-p79.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0270\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0270\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T11:36:25Z","timestamp":1680262585000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0270\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,20]]},"references-count":6,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,1,20]]},"published-print":{"date-parts":[[2020,1,1]]}},"alternative-id":["10.1515\/cmam-2018-0270"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2018-0270","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2019,1,20]]}}}