{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T01:36:43Z","timestamp":1772761003607,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T00:00:00Z","timestamp":1659484800000},"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":["11971085"],"award-info":[{"award-number":["11971085"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["cstc2021jcyj-jqX0011"],"award-info":[{"award-number":["cstc2021jcyj-jqX0011"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["cstc2020jcyj-msxm0777"],"award-info":[{"award-number":["cstc2020jcyj-msxm0777"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["cstc2021ycjh-bgzxm0065"],"award-info":[{"award-number":["cstc2021ycjh-bgzxm0065"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["CSSXKFKTM202006"],"award-info":[{"award-number":["CSSXKFKTM202006"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Chongqing","award":["11971085"],"award-info":[{"award-number":["11971085"]}]},{"name":"Natural Science Foundation of Chongqing","award":["cstc2021jcyj-jqX0011"],"award-info":[{"award-number":["cstc2021jcyj-jqX0011"]}]},{"name":"Natural Science Foundation of Chongqing","award":["cstc2020jcyj-msxm0777"],"award-info":[{"award-number":["cstc2020jcyj-msxm0777"]}]},{"name":"Natural Science Foundation of Chongqing","award":["cstc2021ycjh-bgzxm0065"],"award-info":[{"award-number":["cstc2021ycjh-bgzxm0065"]}]},{"name":"Natural Science Foundation of Chongqing","award":["CSSXKFKTM202006"],"award-info":[{"award-number":["CSSXKFKTM202006"]}]},{"name":"Key Laboratory for Optimization and Control Ministry of Education, Chongqing Normal University","award":["11971085"],"award-info":[{"award-number":["11971085"]}]},{"name":"Key Laboratory for Optimization and Control Ministry of Education, Chongqing Normal University","award":["cstc2021jcyj-jqX0011"],"award-info":[{"award-number":["cstc2021jcyj-jqX0011"]}]},{"name":"Key Laboratory for Optimization and Control Ministry of Education, Chongqing Normal University","award":["cstc2020jcyj-msxm0777"],"award-info":[{"award-number":["cstc2020jcyj-msxm0777"]}]},{"name":"Key Laboratory for Optimization and Control Ministry of Education, Chongqing Normal University","award":["cstc2021ycjh-bgzxm0065"],"award-info":[{"award-number":["cstc2021ycjh-bgzxm0065"]}]},{"name":"Key Laboratory for Optimization and Control Ministry of Education, Chongqing Normal University","award":["CSSXKFKTM202006"],"award-info":[{"award-number":["CSSXKFKTM202006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The element-free Galerkin (EFG) method with penalty for Stokes problems is proposed and analyzed in this work. A priori error estimates of the penalty method, which is used to deal with Dirichlet boundary conditions, are derived to illustrate its validity in a continuous sense. Based on a feasible assumption, it is proved that there is a unique weak solution in the modified weak form of penalized Stokes problems. Then, the error bounds with the penalty factor for the EFG discretization are derived, which provide a rationale for choosing an efficient penalty factor. Numerical examples are given to confirm the theoretical results.<\/jats:p>","DOI":"10.3390\/e24081072","type":"journal-article","created":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T20:52:01Z","timestamp":1659559921000},"page":"1072","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Analysis of the Element-Free Galerkin Method with Penalty for Stokes Problems"],"prefix":"10.3390","volume":"24","author":[{"given":"Tao","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics, Physics and Data Science, Chongqing University of Science & Technology, Chongqing 401331, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaolin","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Chongqing Normal University, Chongqing 400047, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0962492902000090","article-title":"Survey of meshless and generalized finite element methods: A unified approach","volume":"12","author":"Banerjee","year":"2003","journal-title":"Acta Numer."},{"key":"ref_2","unstructured":"Li, S.H., and Liu, W.K. 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