{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:35:07Z","timestamp":1760243707784,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,7]],"date-time":"2022-10-07T00:00:00Z","timestamp":1665100800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"],"award-info":[{"award-number":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"]}]},{"name":"Young Elite Scientist Sponsorship Program by Cast of CAST","award":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"],"award-info":[{"award-number":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"]}]},{"name":"China Postdoctoral Science Foundation","award":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"],"award-info":[{"award-number":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"]}]},{"name":"Natural Science Foundation of Hunan Province","award":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"],"award-info":[{"award-number":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"]}]},{"name":"Excellent Youth Program of Scientific Research Project of Hunan Provincial Department of Education","award":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"],"award-info":[{"award-number":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"]}]},{"name":"International Scientific and Technological Innovation Cooperation Base of Hunan Province for Computational Science","award":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"],"award-info":[{"award-number":["12071404","11701151","2020QNRC001","2018T110073","2019JJ40279","20B564","2018WK4006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, based on the stabilization technique, the Oseen iterative method and the two-level finite element algorithm are combined to numerically solve the stationary incompressible magnetohydrodynamic (MHD) equations. For the low regularity of the magnetic field, when dealing with the magnetic field sub-problem, the Lagrange multiplier technique is used. The stabilized method is applied to approximate the flow field sub-problem to circumvent the inf-sup condition restrictions. One- and two-level stabilized finite element algorithms are presented, and their stability and convergence analysis is given. The two-level method uses the Oseen iteration to solve the nonlinear MHD equations on a coarse grid of size H, and then employs the linearized correction on a fine grid with grid size h. The error analysis shows that when the grid sizes satisfy h=O(H2), the two-level stabilization method has the same convergence order as the one-level one. However, the former saves more computational cost than the latter one. Finally, through some numerical experiments, it has been verified that our proposed method is effective. The two-level stabilized method takes less than half the time of the one-level one when using the second class N\u00e9d\u00e9lec element to approximate magnetic field, and even takes almost a third of the computing time of the one-level one when adopting the first class N\u00e9d\u00e9lec element.<\/jats:p>","DOI":"10.3390\/e24101426","type":"journal-article","created":{"date-parts":[[2022,10,8]],"date-time":"2022-10-08T04:04:56Z","timestamp":1665201896000},"page":"1426","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Two-Level Finite Element Iterative Algorithm Based on Stabilized Method for the Stationary Incompressible Magnetohydrodynamics"],"prefix":"10.3390","volume":"24","author":[{"given":"Qili","family":"Tang","sequence":"first","affiliation":[{"name":"Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Min","family":"Hou","sequence":"additional","affiliation":[{"name":"Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yajie","family":"Xiao","sequence":"additional","affiliation":[{"name":"Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lina","family":"Yin","sequence":"additional","affiliation":[{"name":"Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gerbeau, J., Le Bris, C., and Leli\u00e8vre, T. 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