{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T00:50:48Z","timestamp":1773708648883,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,9]],"date-time":"2022-02-09T00:00:00Z","timestamp":1644364800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we propose an adaptive defect-correction method for natural convection (NC) equations. A defect-correction method (DCM) is proposed for solving NC equations to overcome the convection dominance problem caused by a high Rayleigh number. To solve the large amount of computation and the discontinuity of the gradient of the numerical solution, we combine a new recovery-type posteriori estimator in view of the gradient recovery and superconvergent theory. The presented reliability and efficiency analysis shows that the true error can be effectively bounded by the recovery-based error estimator. Finally, the stability, accuracy and efficiency of the proposed method are confirmed by several numerical investigations.<\/jats:p>","DOI":"10.3390\/e24020255","type":"journal-article","created":{"date-parts":[[2022,2,9]],"date-time":"2022-02-09T21:19:06Z","timestamp":1644441546000},"page":"255","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Recovery-Based Error Estimator for Natural Convection Equations Based on Defect-Correction Methods"],"prefix":"10.3390","volume":"24","author":[{"given":"Lulu","family":"Li","sequence":"first","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haiyan","family":"Su","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinlong","family":"Feng","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1764","DOI":"10.1002\/nme.1620121113","article-title":"Upwinding of high order Galerkin methods in conduction-convection problems","volume":"12","author":"Christie","year":"2010","journal-title":"Int. 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