{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T18:14:14Z","timestamp":1648577654589},"reference-count":16,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,6,25]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The aim of this paper is to provide new perspectives on the relative finite elements accuracy. Starting from a geometrical interpretation of the error estimate which can be deduced from Bramble\u2013Hilbert lemma, we derive a probability law that evaluates the relative accuracy, considered as a random variable, between two finite elements <jats:italic>P<jats:sub>k<\/jats:sub><\/jats:italic> and <jats:italic>P<jats:sub>m<\/jats:sub><\/jats:italic>, <jats:italic>k<\/jats:italic> &lt; <jats:italic>m<\/jats:italic>. We extend this probability law to get a cumulated probabilistic law for two main applications. The first one concerns a family of meshes, the second one is dedicated to a sequence of simplexes constituting a given mesh. Both of these applications could be considered as a first step toward application for adaptive mesh refinement with probabilistic methods.<\/jats:p>","DOI":"10.1515\/jnma-2019-0001","type":"journal-article","created":{"date-parts":[[2020,6,15]],"date-time":"2020-06-15T17:47:19Z","timestamp":1592243239000},"page":"63-74","source":"Crossref","is-referenced-by-count":0,"title":["On generalized binomial laws to evaluate finite element accuracy: preliminary probabilistic results for adaptive mesh refinement"],"prefix":"10.1515","volume":"28","author":[{"given":"Jo\u00ebl","family":"Chaskalovic","sequence":"first","affiliation":[{"name":"D\u2019Alembert, Sorbonne University, Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Franck","family":"Assous","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ariel University, 40700, Ariel, Isra\u00ebl"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","reference":[{"key":"ref71","volume-title":"Multivariable Calculus","year":"2008"},{"key":"ref141","volume-title":"Introduction \u00e0 l\u2019analyse num\u00e9rique des \u00e9quations aux d\u00e9riv\u00e9es partielles","year":"1982"},{"key":"ref31","article-title":"A new probabilistic interpretation of Bramble\u2013Hilbert lemma","year":"2018","journal-title":"arXiv: 1803.09547"},{"key":"ref61","volume-title":"Introduction \u00e0 l\u2019analyse num\u00e9rique des \u00e9quations aux d\u00e9riv\u00e9es partielles","year":"1982"},{"key":"ref91","volume-title":"Mathematical and numerical methods for partial differential equations","year":"2013"},{"key":"ref21","first-page":"79","article-title":"A new probabilistic interpretation of Bramble\u2013Hilbert lemma","volume":"20","year":"2018","journal-title":"Computational Methods in Applied Mathematics"},{"key":"ref51","volume-title":"Handbook of Numerical Analysis","volume":"II","year":"1991"},{"key":"ref81","doi-asserted-by":"crossref","first-page":"4811","DOI":"10.1016\/j.jcp.2011.03.005","article-title":"Data mining techniques for scientific computing: Application to asymptotic paraxial approximations to model ultra-relativistic particles","volume":"230","year":"2011","journal-title":"J. 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