{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T15:09:08Z","timestamp":1747321748249},"reference-count":16,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,10,1]]},"abstract":"Abstract<\/jats:title>\n The aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble\u2013Hilbert lemma, we recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two 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>). Then we analyze the asymptotic relation between these two probabilistic laws when the difference \n \n \n \n m<\/m:mi>\n -<\/m:mo>\n k<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n {m-k}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> goes to infinity. New insights which qualify the relative accuracy in the case of high order finite elements are also obtained.<\/jats:p>","DOI":"10.1515\/cmam-2019-0089","type":"journal-article","created":{"date-parts":[[2020,5,26]],"date-time":"2020-05-26T15:53:14Z","timestamp":1590508394000},"page":"799-813","source":"Crossref","is-referenced-by-count":3,"title":["A New Mixed Functional-probabilistic Approach for Finite Element Accuracy"],"prefix":"10.1515","volume":"20","author":[{"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":[[2020,4,15]]},"reference":[{"key":"2023033110450852562_j_cmam-2019-0089_ref_001","doi-asserted-by":"crossref","unstructured":"R. Arcang\u00e9li and J. L. Gout,\nSur l\u2019\u00e9valuation de l\u2019erreur d\u2019interpolation de Lagrange dans un ouvert de \n \n \n \n \ud835\udc11\n n\n \n \n \n {\\mathbf{R}}^{n}\n \n ,\nESAIM Math. Model. Numer. Anal. 10 (1976), 5\u201327.","DOI":"10.1051\/m2an\/197610R100051"},{"key":"2023033110450852562_j_cmam-2019-0089_ref_002","doi-asserted-by":"crossref","unstructured":"F. Assous and J. Chaskalovic,\nData mining techniques for scientific computing: Application to asymptotic paraxial approximations to model ultrarelativistic particles,\nJ. Comput. Phys. 230 (2011), no. 12, 4811\u20134827.","DOI":"10.1016\/j.jcp.2011.03.005"},{"key":"2023033110450852562_j_cmam-2019-0089_ref_003","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":"2023033110450852562_j_cmam-2019-0089_ref_004","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka,\nError-bounds for finite element method,\nNumer. Math. 16 (1970\/71), 322\u2013333.","DOI":"10.1007\/BF02165003"},{"key":"2023033110450852562_j_cmam-2019-0089_ref_005","doi-asserted-by":"crossref","unstructured":"J. H. Bramble and S. R. Hilbert,\nEstimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation,\nSIAM J. Numer. Anal. 7 (1970), 112\u2013124.","DOI":"10.1137\/0707006"},{"key":"2023033110450852562_j_cmam-2019-0089_ref_006","unstructured":"H. Brezis,\nAnalyse Fonctionnelle,\nMasson, Paris, 1983."},{"key":"2023033110450852562_j_cmam-2019-0089_ref_007","doi-asserted-by":"crossref","unstructured":"J. Chaskalovic,\nMathematical and Numerical Methods for Partial Differential Equations,\nMath. Eng.,\nSpringer, Cham, 2014.","DOI":"10.1007\/978-3-319-03563-5"},{"key":"2023033110450852562_j_cmam-2019-0089_ref_008","doi-asserted-by":"crossref","unstructured":"J. Chaskalovic and F. Assous,\nData mining and probabilistic models for error estimate analysis of finite element method,\nMath. Comput. Simulation 129 (2016), 50\u201368.","DOI":"10.1016\/j.matcom.2016.03.013"},{"key":"2023033110450852562_j_cmam-2019-0089_ref_009","doi-asserted-by":"crossref","unstructured":"J. Chaskalovic and F. Assous,\nProbabilistic approach to characterize quantitative uncertainty in numerical approximations,\nMath. Model. Anal. 22 (2017), no. 1, 106\u2013120.","DOI":"10.3846\/13926292.2017.1272499"},{"key":"2023033110450852562_j_cmam-2019-0089_ref_010","doi-asserted-by":"crossref","unstructured":"J. Chaskalovic and F. Assous,\nA new probabilistic interpretation of the Bramble\u2013Hilbert lemma,\nComput. Methods Appl. Math. 20 (2020), no. 1, 79\u201387.","DOI":"10.1515\/cmam-2018-0270"},{"key":"2023033110450852562_j_cmam-2019-0089_ref_011","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":"2023033110450852562_j_cmam-2019-0089_ref_012","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet and P.-A. Raviart,\nGeneral Lagrange and Hermite interpolation in \n \n \n \n \ud835\udc11\n n\n \n \n \n {\\mathbf{R}}^{n}\n \n with applications to finite element methods,\nArch. Ration. Mech. Anal. 46 (1972), 177\u2013199.","DOI":"10.1007\/BF00252458"},{"key":"2023033110450852562_j_cmam-2019-0089_ref_013","doi-asserted-by":"crossref","unstructured":"O. Furdui,\nLimits, Series, and Fractional Part Integrals,\nSpringer, New York, 2013.","DOI":"10.1007\/978-1-4614-6762-5"},{"key":"2023033110450852562_j_cmam-2019-0089_ref_014","doi-asserted-by":"crossref","unstructured":"W. F. Mitchell,\nHow high a degree is high enough for high order finite elements?,\nProc. Comp. Sci. 15 (2015), 246\u2013255.","DOI":"10.1016\/j.procs.2015.05.235"},{"key":"2023033110450852562_j_cmam-2019-0089_ref_015","unstructured":"P.-A. Raviart and J.-M. Thomas,\nIntroduction \u00e0 l\u2019analyse num\u00e9rique des \u00e9quations aux d\u00e9riv\u00e9es partielles,\nMasson, Paris, 1983."},{"key":"2023033110450852562_j_cmam-2019-0089_ref_016","unstructured":"G. Strang and G. J. Fix,\nAn Analysis of the Finite Element Method,\nPrentice-Hall, Englewood Cliffs, 1973."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/20\/4\/article-p799.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0089\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0089\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T12:58:19Z","timestamp":1680267499000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0089\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,15]]},"references-count":16,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,8,5]]},"published-print":{"date-parts":[[2020,10,1]]}},"alternative-id":["10.1515\/cmam-2019-0089"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2019-0089","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,15]]}}}