{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T03:02:36Z","timestamp":1760151756266,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,22]],"date-time":"2022-04-22T00:00:00Z","timestamp":1650585600000},"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":["12061076","11701493","12126361"],"award-info":[{"award-number":["12061076","11701493","12126361"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Tianshan Youth Project of Xinjiang Province","award":["2017Q079"],"award-info":[{"award-number":["2017Q079"]}]},{"name":"Scientific Research Plan of Universities in the Autonomous Region","award":["XJEDU2020I 001"],"award-info":[{"award-number":["XJEDU2020I 001"]}]},{"name":"Key Laboratory Open Project of Xinjiang Province","award":["2020D04002"],"award-info":[{"award-number":["2020D04002"]}]},{"name":"National Science Foundation of Xinjiang Province","award":["2016D01C073"],"award-info":[{"award-number":["2016D01C073"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Several two-level iterative methods based on nonconforming finite element methods are applied for solving numerically the 2D\/3D stationary incompressible MHD equations under different uniqueness conditions. These two-level algorithms are motivated by applying the m iterations on a coarse grid and correction once on a fine grid. A one-level Oseen iterative method on a fine mesh is further studied under a weak uniqueness condition. Moreover, the stability and error estimate are rigorously carried out, which prove that the proposed methods are stable and effective. Finally, some numerical examples corroborate the effectiveness of our theoretical analysis and the proposed methods.<\/jats:p>","DOI":"10.3390\/e24050587","type":"journal-article","created":{"date-parts":[[2022,4,23]],"date-time":"2022-04-23T08:14:06Z","timestamp":1650701646000},"page":"587","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Optimal Convergence Analysis of Two-Level Nonconforming Finite Element Iterative Methods for 2D\/3D MHD Equations"],"prefix":"10.3390","volume":"24","author":[{"given":"Haiyan","family":"Su","sequence":"first","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiali","family":"Xu","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":[[2022,4,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1007\/s00021-004-0107-9","article-title":"On the global unique solvabiity and initial boundary value problems for coupled modified Navier\u2013Stokes and Maxwell equations","volume":"6","author":"Gunzburger","year":"2004","journal-title":"J. Math. Fluid. Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1090\/S0025-5718-1991-1066834-0","article-title":"On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics","volume":"56","author":"Gunzburger","year":"1991","journal-title":"Math. Comp."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jde.2007.03.023","article-title":"On the regularity criteria for weak solutions to the magnetohydrodynamic equations","volume":"238","author":"He","year":"2007","journal-title":"J. Differ. Equ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1007\/BF01446316","article-title":"Large-time behaviour of solutions to the magneto-hydrodynamics equations","volume":"304","author":"Schonbek","year":"1996","journal-title":"Math. Ann."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2840","DOI":"10.1016\/j.cma.2010.05.007","article-title":"A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics","volume":"199","author":"Greif","year":"2010","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1007\/s00211-003-0487-4","article-title":"Mixed finite element methods for stationary incompressible magneto-hydrodynamics","volume":"96","year":"2004","journal-title":"Numer. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.apnum.2004.02.005","article-title":"Mixed finite element approximation of incompressible MHD problems based on weighted regularization","volume":"51","author":"Hasler","year":"2004","journal-title":"Appl. Numer. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/s10915-008-9265-x","article-title":"A mixed DG method for linearized incompressible magnetohydrodynamics","volume":"40","author":"Houston","year":"2009","journal-title":"J. Sci. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1051\/m2an:2008034","article-title":"Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system","volume":"42","author":"Prohl","year":"2008","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/j.cma.2014.03.022","article-title":"Convergence analysis of three finite element iterative methods for the 2D\/3D stationary incompressible magnetohydrodynamics","volume":"276","author":"Dong","year":"2014","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1016\/j.camwa.2014.07.025","article-title":"Analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain","volume":"68","author":"Zhang","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s100920050031","article-title":"A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier\u2013Stokes equations","volume":"36","author":"Cai","year":"1999","journal-title":"Calcolo"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/S0252-9602(11)60238-5","article-title":"A new nonconforming mixed finite element scheme for the stationary Navier\u2013Stokes equations","volume":"31","author":"Shi","year":"2011","journal-title":"Acta Math. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/s00607-008-0001-z","article-title":"A new local stabilized nonconforming finite element method for the Stokes equations","volume":"82","author":"Li","year":"2008","journal-title":"Computing"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2821","DOI":"10.1016\/j.cam.2010.12.001","article-title":"A new local stabilized nonconforming finite element method for solving stationary Navier\u2013Stokes equations","volume":"235","author":"Zhu","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.matcom.2011.02.015","article-title":"A two-level stabilized nonconforming finite element method for the stationary Navier\u2013Stokes equations","volume":"114","author":"Zhu","year":"2015","journal-title":"Math. Comput. Simulat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/j.apm.2013.06.033","article-title":"An Oseen scheme for the conduction-convection equations based on a stabilized nonconforming method","volume":"38","author":"Huang","year":"2014","journal-title":"Appl. Math. Model."},{"key":"ref_18","first-page":"904","article-title":"Nonconforming mixed finite element methods for stationary incompressible magnetohydrodynamics","volume":"10","author":"Shi","year":"2013","journal-title":"Int. J. Numer. Anal. Model."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1155\/2012\/825609","article-title":"Low-order nonconforming mixed finite element methods for stationary incompressible magnetohydrodynamics equations","volume":"2012","author":"Shi","year":"2012","journal-title":"J. Appl. Math."},{"key":"ref_20","first-page":"263","article-title":"Two-level Picard and modified Picard methods for the Navier\u2013Stokes equations","volume":"69","author":"Layton","year":"1995","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2035","DOI":"10.1137\/S003614299630230X","article-title":"A two-level method with backtracking for the Navier\u2013Stokes equations","volume":"35","author":"Layton","year":"1998","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_22","first-page":"25","article-title":"Two-grid finite element schemes for the steady Navier\u2013Stokes problem in polyhedra","volume":"58","author":"Girault","year":"2001","journal-title":"Portugal. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1051\/m2an:2001145","article-title":"Two-grid finite element schemes for the transient Navier\u2013Stokes problem","volume":"35","author":"Girault","year":"2001","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1002\/fld.2014","article-title":"A variational multiscale Newton-Schur approach for the incompressible Navier\u2013Stokes equations","volume":"62","author":"Turner","year":"2009","journal-title":"Int. J. Numer. Meth. Fluids"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1002\/fld.2019","article-title":"Two-level finite element method with a stabilizing subgrid for the incompressible MHD equations","volume":"62","author":"Aydin","year":"2010","journal-title":"Int. J. Numer. Meth. Fluids"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"920","DOI":"10.1007\/s10915-015-9994-6","article-title":"Two-level coupled and decoupled parallel correction methods for stationary incompressible magnetohydrodynamics","volume":"65","author":"Zhang","year":"2015","journal-title":"J. Sci. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1007\/s10915-014-9900-7","article-title":"Two-level Newton iterative method for the 2D\/3D stationary incompressible magnetohydrodynamics","volume":"63","author":"Dong","year":"2015","journal-title":"J. Sci. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/0898-1221(94)00137-5","article-title":"A two-level Newton finite element algorithm for approximating electrically conducting incompressible fluid flows","volume":"28","author":"Layton","year":"1994","journal-title":"Comput. Math. Appl."},{"key":"ref_29","first-page":"198","article-title":"A two-level discretization method for the stationary MHD equations","volume":"6","author":"Layton","year":"1997","journal-title":"Electron. Trans. Numer. Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1007\/s11425-015-5087-0","article-title":"Convergence of some finite element iterative methods related to different Reynolds numbers for the 2D\/3D stationary incompressible magnetohydrodynamics","volume":"59","author":"Dong","year":"2016","journal-title":"Sci. China Math."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Xu, J., Su, H., and Li, Z. (2021). Optimal convergence of three iterative methods based on nonconforming finite element discretization for 2D\/3D MHD equations. Numer. Algorithms, 1\u201335.","DOI":"10.1007\/s11075-021-01224-4"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Gerbeau, J., Bris, C., and Leli\u00e8vre, T. (2006). Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198566656.001.0001"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1137\/0719018","article-title":"Finite element approximation of the nonstationary Navier\u2013Stokes problem. I. Regularityof solutions and second-order error estimates for spatial discretization","volume":"19","author":"Heywood","year":"1982","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1002\/fld.3848","article-title":"A nonconforming finite element method for the Stokes equations using the Crouzeix-Raviart element for the velocity and the standard linear element for the pressure","volume":"74","author":"Lamichhane","year":"2013","journal-title":"Int. J. Numer. Meth. Fluids"},{"key":"ref_35","first-page":"293","article-title":"Nonconforming mixed finite element method for the stationary conduction-convection problem","volume":"6","author":"Shi","year":"2009","journal-title":"Int. J. Numer. Anal. Model."},{"key":"ref_36","first-page":"561","article-title":"A nonconforming anisotropic finite element approximation with moving grids for Stokes problem","volume":"24","author":"Shi","year":"2006","journal-title":"J. Comput. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.cam.2007.02.015","article-title":"A stabilized finite element method based on two local Gauss integrations for the Stokes equations","volume":"214","author":"Li","year":"2008","journal-title":"J. Comput. Appl. Math."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Ciarlet, P. (1978). The Finite Element Method for Elliptic Problems, North-Holland Publishing.","DOI":"10.1115\/1.3424474"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/5\/587\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:59:10Z","timestamp":1760137150000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/5\/587"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,22]]},"references-count":38,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["e24050587"],"URL":"https:\/\/doi.org\/10.3390\/e24050587","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,4,22]]}}}