{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T22:03:26Z","timestamp":1766181806547,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T00:00:00Z","timestamp":1648166400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12061075","12061076"],"award-info":[{"award-number":["12061075","12061076"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, four stabilized methods based on the lowest equal-order finite element pair for the steady micropolar Navier\u2013Stokes equations (MNSE) are presented, which are penalty, regular, multiscale enrichment, and local Gauss integration methods. A priori properties, existence, uniqueness, stability, and error estimation based on Fem approximation of all the methods are proven for the physical variables. Finally, some numerical examples are displayed to show the numerical characteristics of these methods.<\/jats:p>","DOI":"10.3390\/e24040454","type":"journal-article","created":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T15:31:21Z","timestamp":1648222281000},"page":"454","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Numerical Analysis and Comparison of Four Stabilized Finite Element Methods for the Steady Micropolar Equations"],"prefix":"10.3390","volume":"24","author":[{"given":"Jingnan","family":"Liu","sequence":"first","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2284-2861","authenticated-orcid":false,"given":"Demin","family":"Liu","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":[[2022,3,25]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Theory of micropolar fluids","volume":"16","author":"Eringen","year":"1966","journal-title":"J. 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