{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T04:17:13Z","timestamp":1771042633119,"version":"3.50.1"},"reference-count":42,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-17-CE40-0029"],"award-info":[{"award-number":["ANR-17-CE40-0029"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This work deals with the a posteriori error estimates for the Darcy\u2013Forchheimer problem.\nWe first introduce the corresponding variational formulation for the nonlinear problem and discretize it by using the finite-element method. We then\npropose a linear iterative scheme to solve the nonlinear variational problem\nfor a fixed mesh step. Finally, a posteriori error estimate with two types of computable error indicators is showed. The first one is linked to the linearization and the second one to the discretization.\nNumerical computations are performed to show the effectiveness of the derived error indicators.<\/jats:p>","DOI":"10.1515\/cmam-2022-0047","type":"journal-article","created":{"date-parts":[[2022,11,11]],"date-time":"2022-11-11T06:09:14Z","timestamp":1668146954000},"page":"517-544","source":"Crossref","is-referenced-by-count":6,"title":["A Posteriori Error Estimates for Darcy\u2013Forchheimer\u2019s Problem"],"prefix":"10.1515","volume":"23","author":[{"given":"Toni","family":"Sayah","sequence":"first","affiliation":[{"name":"Laboratoire de \u201cMath\u00e9matiques et Applications\u201d , Unit\u00e9 de Recherche \u201cMath\u00e9matiques et Mod\u00e9lisation\u201d, CAR, Facult\u00e9 des sciences , Universit\u00e9 Saint-Joseph de Beyrouth , Beirut , Lebanon"}]},{"given":"Georges","family":"Semaan","sequence":"additional","affiliation":[{"name":"Laboratoire de \u201cMath\u00e9matiques et Applications\u201d , Unit\u00e9 de Recherche \u201cMath\u00e9matiques et Mod\u00e9lisation\u201d, CAR, Facult\u00e9 des sciences , Universit\u00e9 Saint-Joseph de Beyrouth , Beirut , Lebanon"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7181-6299","authenticated-orcid":false,"given":"Faouzi","family":"Triki","sequence":"additional","affiliation":[{"name":"Laboratoire Jean Kuntzmann , UMR CNRS 5224 , Universit\u00e9 Grenoble-Alpes , 700 Avenue Centrale, 38401 Saint-Martin-d\u2019H\u00e8res , France"}]}],"member":"374","published-online":{"date-parts":[[2022,11,11]]},"reference":[{"key":"2023033113095699611_j_cmam-2022-0047_ref_001","doi-asserted-by":"crossref","unstructured":"H.  Abboud, F. E.  Chami and T.  Sayah,\nA priori and a posteriori estimates for three-dimensional Stokes equations with nonstandard boundary conditions,\nNumer. Methods Partial Differential Equations 28 (2012), no. 4, 1178\u20131193.","DOI":"10.1002\/num.20676"},{"key":"2023033113095699611_j_cmam-2022-0047_ref_002","doi-asserted-by":"crossref","unstructured":"M.  Ainsworth and J. T.  Oden,\nA posteriori error estimation in finite element analysis,\nComput. Methods Appl. Mech. Engrg. 142 (1997), no. 1\u20132, 1\u201388.","DOI":"10.1016\/S0045-7825(96)01107-3"},{"key":"2023033113095699611_j_cmam-2022-0047_ref_003","doi-asserted-by":"crossref","unstructured":"A.  Alonso,\nError estimators for a mixed method,\nNumer. Math. 74 (1996), no. 4, 385\u2013395.","DOI":"10.1007\/s002110050222"},{"key":"2023033113095699611_j_cmam-2022-0047_ref_004","unstructured":"K.  Aziz and A.  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Anal. 55 (2021), no. 6, 2921\u20132947.","DOI":"10.1051\/m2an\/2021074"},{"key":"2023033113095699611_j_cmam-2022-0047_ref_008","doi-asserted-by":"crossref","unstructured":"Z.  Belhachmi, C.  Bernardi and S.  Deparis,\nWeighted Cl\u00e9ment operator and application to the finite element discretization of the axisymmetric Stokes problem,\nNumer. Math. 105 (2006), no. 2, 217\u2013247.","DOI":"10.1007\/s00211-006-0039-9"},{"key":"2023033113095699611_j_cmam-2022-0047_ref_009","doi-asserted-by":"crossref","unstructured":"C.  Bernardi, J.  Dakroub, G.  Mansour and T.  Sayah,\nA posteriori analysis of iterative algorithms for Navier\u2013Stokes problem,\nESAIM Math. Model. Numer. Anal. 50 (2016), no. 4, 1035\u20131055.","DOI":"10.1051\/m2an\/2015062"},{"key":"2023033113095699611_j_cmam-2022-0047_ref_010","doi-asserted-by":"crossref","unstructured":"C.  Bernardi, F.  Hecht and R.  Verf\u00fcrth,\nA finite element discretization of the three-dimensional Navier\u2013Stokes equations with mixed boundary conditions,\nM2AN Math. Model. Numer. Anal. 43 (2009), no. 6, 1185\u20131201.","DOI":"10.1051\/m2an\/2009035"},{"key":"2023033113095699611_j_cmam-2022-0047_ref_011","doi-asserted-by":"crossref","unstructured":"C.  Bernardi and T.  Sayah,\nA posteriori error analysis of the time dependent Navier\u2013Stokes equations with mixed boundary conditions,\nSeMA J. 69 (2015), 1\u201323.","DOI":"10.1007\/s40324-015-0033-1"},{"key":"2023033113095699611_j_cmam-2022-0047_ref_012","doi-asserted-by":"crossref","unstructured":"C.  Bernardi and T.  Sayah,\nA posteriori error analysis of the time-dependent Stokes equations with mixed boundary conditions,\nIMA J. Numer. Anal. 35 (2015), no. 1, 179\u2013198.","DOI":"10.1093\/imanum\/drt067"},{"key":"2023033113095699611_j_cmam-2022-0047_ref_013","doi-asserted-by":"crossref","unstructured":"D.  Braess and R.  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