{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T11:16:52Z","timestamp":1760267812574,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,30]],"date-time":"2022-09-30T00:00:00Z","timestamp":1664496000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSF of China","award":["12061076","12126361","11701493","XJEDU2020I 001","2020D04002"],"award-info":[{"award-number":["12061076","12126361","11701493","XJEDU2020I 001","2020D04002"]}]},{"name":"Scientific Research Plan of Universities in the Autonomous Region","award":["12061076","12126361","11701493","XJEDU2020I 001","2020D04002"],"award-info":[{"award-number":["12061076","12126361","11701493","XJEDU2020I 001","2020D04002"]}]},{"name":"Key Laboratory Open Project of Xinjiang Province","award":["12061076","12126361","11701493","XJEDU2020I 001","2020D04002"],"award-info":[{"award-number":["12061076","12126361","11701493","XJEDU2020I 001","2020D04002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this article, we mainly consider a first order penalty finite element method (PFEM) for the 2D\/3D unsteady incompressible magnetohydrodynamic (MHD) equations. The penalty method applies a penalty term to relax the constraint \u201c\u2207\u00b7u=0\u201d, which allows us to transform the saddle point problem into two smaller problems to solve. The Euler semi-implicit scheme is based on a first order backward difference formula for time discretization and semi-implicit treatments for nonlinear terms. It is worth mentioning that the error estimates of the fully discrete PFEM are rigorously derived, which depend on the penalty parameter \u03f5, the time-step size \u03c4, and the mesh size h. Finally, two numerical tests show that our scheme is effective.<\/jats:p>","DOI":"10.3390\/e24101395","type":"journal-article","created":{"date-parts":[[2022,10,8]],"date-time":"2022-10-08T04:04:56Z","timestamp":1665201896000},"page":"1395","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Error Analysis of a PFEM Based on the Euler Semi-Implicit Scheme for the Unsteady MHD Equations"],"prefix":"10.3390","volume":"24","author":[{"given":"Kaiwen","family":"Shi","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":[[2022,9,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"956","DOI":"10.1002\/num.22132","article-title":"Decoupled schemes for unsteady MHD equations. 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