{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T03:40:17Z","timestamp":1777520417638,"version":"3.51.4"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,11,29]],"date-time":"2018-11-29T00:00:00Z","timestamp":1543449600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"the NSF of China","award":["11271313"],"award-info":[{"award-number":["11271313"]}]},{"name":"the NSF of China","award":["11471329"],"award-info":[{"award-number":["11471329"]}]},{"name":"The National Key Research and Development Program of China","award":["2016YFB0201304"],"award-info":[{"award-number":["2016YFB0201304"]}]},{"name":"National Magnetic Confinement Fusion Science Program of China","award":["2015GB110003"],"award-info":[{"award-number":["2015GB110003"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2019,5]]},"DOI":"10.1007\/s10915-018-0883-7","type":"journal-article","created":{"date-parts":[[2018,11,29]],"date-time":"2018-11-29T09:43:14Z","timestamp":1543484594000},"page":"1078-1110","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Optimal Error Estimates of Penalty Based Iterative Methods for Steady Incompressible Magnetohydrodynamics Equations with Different Viscosities"],"prefix":"10.1007","volume":"79","author":[{"given":"Haiyan","family":"Su","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shipeng","family":"Mao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinlong","family":"Feng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,11,29]]},"reference":[{"key":"883_CR1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511626333","volume-title":"An Introduction to Magnetohydrodynamics","author":"P Davidson","year":"2001","unstructured":"Davidson, P.: An Introduction to Magnetohydrodynamics. Cambridge University Press, Cambridge (2001)"},{"key":"883_CR2","volume-title":"Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Numerical Mathematics and Scientific Computation","author":"J Gerbeau","year":"2006","unstructured":"Gerbeau, J., Bris, C., Leli\u00e8vre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2006)"},{"key":"883_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04405-6","volume-title":"Magnetofluiddynamics in Channels and Containers","author":"U M\u00fcller","year":"2001","unstructured":"M\u00fcller, U., B\u00fchler, L.: Magnetofluiddynamics in Channels and Containers. Springer, Berlin (2001)"},{"key":"883_CR4","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1090\/S0025-5718-1991-1066834-0","volume":"56","author":"M Gunzburger","year":"1991","unstructured":"Gunzburger, M., Meir, A., Peterson, J.: On the existence, uniquess and finite element approximation of solutions of the equations of sationary, incompressible magnetohydrodynamic. Math. Comput. 56, 523\u2013563 (1991)","journal-title":"Math. Comput."},{"key":"883_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-7883-7","volume-title":"Magneto-hydrodynamics","author":"R Moreau","year":"1990","unstructured":"Moreau, R.: Magneto-hydrodynamics. Kluwer Academic Publishers, Dordrecht (1990)"},{"key":"883_CR6","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1002\/cpa.3160360506","volume":"36","author":"M Sermange","year":"1983","unstructured":"Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635\u2013664 (1983)","journal-title":"Commun. Pure Appl. Math."},{"key":"883_CR7","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/s002110000193","volume":"87","author":"J Gerbeau","year":"2000","unstructured":"Gerbeau, J.: A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87, 83\u2013111 (2000)","journal-title":"Numer. Math."},{"key":"883_CR8","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.amc.2017.01.005","volume":"302","author":"J Wu","year":"2017","unstructured":"Wu, J., Liu, D., Feng, X., Huang, P.: An efficient two-step algorithm for the stationary incompressible magnetohydrodynamic equations. Appl. Math. Comput. 302, 21\u201333 (2017)","journal-title":"Appl. Math. Comput."},{"key":"883_CR9","doi-asserted-by":"publisher","first-page":"1877","DOI":"10.1002\/num.21882","volume":"30","author":"G Zhang","year":"2014","unstructured":"Zhang, G., He, Y., Zhang, Y.: Streamline diffusion finite element method for stationary incompressible magnetohydrodynamics. Numer. Methods Partial Differ. Equ. 30, 1877\u20131901 (2014)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"883_CR10","doi-asserted-by":"publisher","first-page":"1479","DOI":"10.1007\/s10483-016-2107-9","volume":"37","author":"J Zhao","year":"2016","unstructured":"Zhao, J., Mao, S., Zheng, W.: Anisotropic adaptive finite element method for magnetohydrodynamic flow at high Hartmann numbers. Appl. Math. Mech. 37, 1479\u20131500 (2016)","journal-title":"Appl. Math. Mech."},{"key":"883_CR11","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1007\/s00211-016-0803-4","volume":"135","author":"K Hu","year":"2017","unstructured":"Hu, K., Ma, Y., Xu, J.: Stable finite element methods preserving $$\\nabla \\cdot { B}=0$$ \u2207 \u00b7 B = 0 exactly for MHD models. Numer. Math. 135, 371\u2013396 (2017)","journal-title":"Numer. Math."},{"key":"883_CR12","doi-asserted-by":"publisher","first-page":"254","DOI":"10.1016\/j.jcp.2017.09.025","volume":"351","author":"L Li","year":"2017","unstructured":"Li, L., Zheng, W.: A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D. J. Comput. Phys. 351, 254\u2013270 (2017)","journal-title":"J. Comput. Phys."},{"key":"883_CR13","doi-asserted-by":"publisher","first-page":"524","DOI":"10.1016\/j.cma.2008.09.002","volume":"198","author":"S Ravindran","year":"2008","unstructured":"Ravindran, S.: Linear feedback control and approximation for a system governed by unsteady MHD equations. Comput. Methods Appl. Mech. Eng. 198, 524\u2013541 (2008)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"883_CR14","doi-asserted-by":"publisher","first-page":"5867","DOI":"10.1016\/S0045-7825(01)00196-7","volume":"190","author":"N Salah","year":"2001","unstructured":"Salah, N., Soulaimani, A., Habashi, W.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867\u20135892 (2001)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"883_CR15","doi-asserted-by":"publisher","first-page":"1304","DOI":"10.1137\/S003614299732615X","volume":"36","author":"A Meir","year":"1999","unstructured":"Meir, A., Schmidt, P.: Analysis and numerical approximation of a stationary MHD flow problem with nonideal boundary. SIAM J. Numer. Anal. 36, 1304\u20131332 (1999)","journal-title":"SIAM J. Numer. Anal."},{"key":"883_CR16","doi-asserted-by":"publisher","first-page":"1351","DOI":"10.1016\/j.cma.2008.12.001","volume":"198","author":"Y He","year":"2009","unstructured":"He, Y., Li, J.: Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 198, 1351\u20131359 (2009)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"883_CR17","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1016\/j.cma.2014.03.022","volume":"276","author":"X Dong","year":"2014","unstructured":"Dong, X., He, Y., Zhang, Y.: Convergence analysis of three finite element iterative methods for the 2D\/3D stationary incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 276, 287\u2013311 (2014)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"883_CR18","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1007\/s10915-014-9900-7","volume":"63","author":"X Dong","year":"2015","unstructured":"Dong, X., He, Y.: Two-level Newton iterative method for the 2D\/3D stationary incompressible magnetohydrodynamics. J. Sci. Comput. 63, 426\u2013451 (2015)","journal-title":"J. Sci. Comput."},{"key":"883_CR19","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1016\/j.cma.2016.02.039","volume":"304","author":"H Su","year":"2016","unstructured":"Su, H., Feng, X., Huang, P.: Iterative methods in penalty finite element discretization for the steady MHD equations. Comput. Methods Appl. Mech. Eng. 304, 521\u2013545 (2016)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"3","key":"883_CR20","doi-asserted-by":"publisher","first-page":"1144","DOI":"10.1007\/s10915-016-0276-8","volume":"70","author":"H Su","year":"2017","unstructured":"Su, H., Feng, X., Zhao, J.: Two-level penalty Newton iterative method for the 2D\/3D stationary incompressible magnetohydrodynamics equations. J. Sci. Comput. 70(3), 1144\u20131179 (2017)","journal-title":"J. Sci. Comput."},{"key":"883_CR21","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1007\/BF01396220","volume":"62","author":"J Shen","year":"1992","unstructured":"Shen, J.: On error estimates of some higher order projection and penalty-projection methods for Navier\u2013Stokes equations. Numer. Math. 62, 49\u201373 (1992)","journal-title":"Numer. Math."},{"key":"883_CR22","doi-asserted-by":"publisher","first-page":"386","DOI":"10.1137\/0732016","volume":"32","author":"J Shen","year":"1995","unstructured":"Shen, J.: On error estimates of the penalty method for unsteady Navier\u2013Stokes equations. SIAM J. Numer. Anal. 32, 386\u2013403 (1995)","journal-title":"SIAM J. Numer. Anal."},{"key":"883_CR23","doi-asserted-by":"publisher","first-page":"1201","DOI":"10.1090\/S0025-5718-05-01751-5","volume":"74","author":"Y He","year":"2005","unstructured":"He, Y.: Optimal error estimate of the penalty finite element method for the time-dependent Navier\u2013Stokes equations. Math. Comput. 74, 1201\u20131216 (2005)","journal-title":"Math. Comput."},{"key":"883_CR24","first-page":"131","volume":"2","author":"Y He","year":"2006","unstructured":"He, Y., Li, J., Yang, X.: Two-level penalized finite element methods for the stationary Navier\u2013Stoke equations. Int. J. Inf. Syst. Sci. 2, 131\u2013143 (2006)","journal-title":"Int. J. Inf. Syst. Sci."},{"key":"883_CR25","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1016\/j.apnum.2016.10.010","volume":"112","author":"R An","year":"2017","unstructured":"An, R., Li, Y.: Error analysis of first-order projection method for time-dependent magnetohydrodynamics equations. Appl. Numer. Math. 112, 167\u2013181 (2017)","journal-title":"Appl. Numer. Math."},{"key":"883_CR26","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.amc.2017.01.003","volume":"302","author":"T Zhu","year":"2017","unstructured":"Zhu, T., Su, H., Feng, X.: Some Uzawa-type finite element iterative methods for the steady incompressible magnetohydrodynamic equations. Appl. Math. Comput. 302, 34\u201347 (2017)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"883_CR27","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/s11075-017-0376-z","volume":"78","author":"Q Zhang","year":"2018","unstructured":"Zhang, Q., Su, H., Feng, X.: A partitioned finite element scheme based on Gauge\u2013Uzawa method for time-dependent MHD equations. Numer. Algorithms 78(1), 277\u2013295 (2018)","journal-title":"Numer. Algorithms"},{"key":"883_CR28","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1137\/0915016","volume":"15","author":"J Xu","year":"1994","unstructured":"Xu, J.: A novel two two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231\u2013237 (1994)","journal-title":"SIAM J. Sci. Comput."},{"key":"883_CR29","doi-asserted-by":"publisher","first-page":"1759","DOI":"10.1137\/S0036142992232949","volume":"33","author":"J Xu","year":"1996","unstructured":"Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759\u20131777 (1996)","journal-title":"SIAM J. Numer. Anal."},{"key":"883_CR30","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/0898-1221(94)00137-5","volume":"28","author":"W Layton","year":"1994","unstructured":"Layton, W., Lenferink, H., Peterson, J.: A two-level Newton finite element algorithm for approximating electrically conducting incompressible fluid flows. Comput. Math. Appl. 28, 21\u201331 (1994)","journal-title":"Comput. Math. Appl."},{"key":"883_CR31","first-page":"198","volume":"6","author":"W Layton","year":"1997","unstructured":"Layton, W., Meir, A., Schmidtz, P.: A two-level discretization method for the stationary MHD equations. Electron. Trans. Numer. Anal. 6, 198\u2013210 (1997)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"883_CR32","doi-asserted-by":"publisher","first-page":"920","DOI":"10.1007\/s10915-015-9994-6","volume":"65","author":"G Zhang","year":"2015","unstructured":"Zhang, G., Zhang, Y., He, Y.: Two-level coupled and decoupled parallel correction methods for stationary incompressible magnetohydrodynamics. J. Sci. Comput. 65, 920\u2013939 (2015)","journal-title":"J. Sci. Comput."},{"key":"883_CR33","doi-asserted-by":"publisher","first-page":"767","DOI":"10.1093\/imanum\/dru015","volume":"35","author":"Y He","year":"2014","unstructured":"He, Y.: Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations. IMA J. Numer. Anal. 35, 767\u2013801 (2014)","journal-title":"IMA J. Numer. Anal."},{"key":"883_CR34","doi-asserted-by":"publisher","first-page":"1263","DOI":"10.1137\/S0036142901385659","volume":"41","author":"Y He","year":"2003","unstructured":"He, Y.: Two-level method based on fnite element and Crank\u2013Nicolson extrapolation for the time-dependent Navier\u2013Stokes equations. SIAM J. Numer. Anal. 41, 1263\u20131285 (2003)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0883-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-018-0883-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0883-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,7]],"date-time":"2022-09-07T04:04:20Z","timestamp":1662523460000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-018-0883-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,29]]},"references-count":34,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,5]]}},"alternative-id":["883"],"URL":"https:\/\/doi.org\/10.1007\/s10915-018-0883-7","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,29]]},"assertion":[{"value":"25 February 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 November 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 November 2018","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}