{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T08:47:19Z","timestamp":1773823639774,"version":"3.50.1"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2017,12,27]],"date-time":"2017-12-27T00:00:00Z","timestamp":1514332800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2018,8]]},"DOI":"10.1007\/s10444-017-9582-4","type":"journal-article","created":{"date-parts":[[2017,12,27]],"date-time":"2017-12-27T00:31:07Z","timestamp":1514334667000},"page":"1295-1319","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Local and parallel finite element algorithm based on the partition of unity method for the incompressible MHD flow"],"prefix":"10.1007","volume":"44","author":[{"given":"Xiaojing","family":"Dong","sequence":"first","affiliation":[]},{"given":"Yinnian","family":"He","sequence":"additional","affiliation":[]},{"given":"Hongbo","family":"Wei","sequence":"additional","affiliation":[]},{"given":"Yuhong","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,12,27]]},"reference":[{"key":"9582_CR1","first-page":"881","volume":"69","author":"JC Xu","year":"2000","unstructured":"Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69, 881\u2013909 (2000)","journal-title":"Math. Comput."},{"key":"9582_CR2","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1023\/A:1012284322811","volume":"14","author":"JC Xu","year":"2001","unstructured":"Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems. Adv. Comput. Math. 14, 293\u2013327 (2001)","journal-title":"Adv. Comput. Math."},{"key":"9582_CR3","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/s102550200018","volume":"18","author":"JC Xu","year":"2002","unstructured":"Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms for eigenvalue problems. Acta Math. Appl. Sinica. 18, 185\u2013200 (2002)","journal-title":"Acta Math. Appl. Sinica."},{"key":"9582_CR4","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/s00211-008-0141-2","volume":"109","author":"YN He","year":"2008","unstructured":"He, Y.N., Xu, J.C., Zhou, A.H., et al.: Local and parallel finite element algorithms for the Stokes problem. Numer. Math. 109, 415\u2013434 (2008)","journal-title":"Numer. Math."},{"key":"9582_CR5","first-page":"227","volume":"24","author":"YN He","year":"2006","unstructured":"He, Y.N., Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms for the Navier-Stokes problem. J. Comput. Math. 24, 227\u2013238 (2006)","journal-title":"J. Comput. Math."},{"key":"9582_CR6","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.cma.2011.11.003","volume":"209","author":"YQ Shang","year":"2012","unstructured":"Shang, Y.Q., He, Y.N.: A parallel Oseen-linearized algorithm for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 209, 172\u2013183 (2012)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"9582_CR7","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1016\/j.apnum.2010.03.013","volume":"60","author":"YQ Shang","year":"2010","unstructured":"Shang, Y.Q., He, Y.N.: Parallel iterative finite element algorithms based on full domain partition for the stationary Navier-Stokes equations. Appl. Numer. Math. 60, 719\u2013737 (2010)","journal-title":"Appl. Numer. Math."},{"key":"9582_CR8","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1007\/s10915-010-9371-4","volume":"44","author":"YN He","year":"2010","unstructured":"He, Y.N., Mei, L.Q., Shang, Y.Q., et al.: Newton iterative parallel finite element algorithm for the steady Navier-Stokes equations. J. Sci. Comput. 44, 92\u2013106 (2010)","journal-title":"J. Sci. Comput."},{"key":"9582_CR9","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/s10915-016-0246-1","volume":"70","author":"QL Tang","year":"2017","unstructured":"Tang, Q.L., Huang, Y.Q.: Local and parallel finite element algorithm based on Oseen-type iteration for the stationary incompressible MHD flow. J. Sci. Comput. 70, 149\u2013174 (2017)","journal-title":"J. Sci. Comput."},{"key":"9582_CR10","doi-asserted-by":"crossref","first-page":"1513","DOI":"10.1002\/num.22151","volume":"33","author":"YH Zhang","year":"2017","unstructured":"Zhang, Y.H., Hou, Y.R., Shan, L., Dong, X.J.: Local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics. Numer. Meth. PDE. 33, 1513\u20131539 (2017)","journal-title":"Numer. Meth. PDE."},{"key":"9582_CR11","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/s10483-008-0411-x","volume":"29","author":"C Wang","year":"2008","unstructured":"Wang, C., Huang, Z., Li, L.: Two-grid partition of unity method for second order elliptic problems. Appl. Math. Mech. -Engl. Ed 29, 527\u2013533 (2008)","journal-title":"Appl. Math. Mech. -Engl. Ed"},{"key":"9582_CR12","doi-asserted-by":"crossref","first-page":"C547","DOI":"10.1137\/130925748","volume":"36","author":"JP Yu","year":"2014","unstructured":"Yu, J.P., Shi, F., Zheng, H.B.: Local and parallel finite element algorithms based on the partition of unity for the Stokes problem. SIAM J. Sci. Comput. 36, C547\u2013C567 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"9582_CR13","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1007\/s10915-014-9979-x","volume":"65","author":"HB Zheng","year":"2015","unstructured":"Zheng, H.B., Yu, J.P., Shi, F.: Local and parallel finite element algorithm based on the partition of unity for incompressible flows. J. Sci. Comput. 65, 512\u2013532 (2015)","journal-title":"J. Sci. Comput."},{"key":"9582_CR14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmaa.2015.09.077","volume":"435","author":"HB Zheng","year":"2015","unstructured":"Zheng, H.B., Shi, F., Hou, Y.R., et al.: New local and parallel finite element algorithm based on the partition of unity. J. Math. Anal. Appl. 435, 1\u201319 (2015)","journal-title":"J. Math. Anal. Appl."},{"key":"9582_CR15","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/S0045-7825(96)01087-0","volume":"139","author":"J Melenk","year":"1996","unstructured":"Melenk, J., Babu\u0161ka, I.: The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139, 289\u2013314 (1996)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"9582_CR16","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1002\/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N","volume":"40","author":"I Babu\u0161ka","year":"1997","unstructured":"Babu\u0161ka, I., Melenk, J.: The partition of unity method. Int. J. Numer. Methods Enrg. 40, 727\u2013758 (1997)","journal-title":"Int. J. Numer. Methods Enrg."},{"key":"9582_CR17","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1023\/A:1014246117321","volume":"15","author":"M Holst","year":"2001","unstructured":"Holst, M.: Adaptive numerical treatment of elliptic systems on manifolds. Adv. Comput. Math. 15, 139\u2013191 (2001)","journal-title":"Adv. Comput. Math."},{"key":"9582_CR18","unstructured":"Holst, M.: Applications of domain decomposition and partition of unity methods in physics and geometry (plenary paper). In: Herrera, I., Keyes, D.E., Widlund, O.B., Yates, R. (eds.) Proceedings of the Fourteenth International Conference on Domain Decomposition Methods, pp 63\u201378. Cocoyoc, Mexico City (2002)"},{"key":"9582_CR19","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1090\/S0025-5718-02-01459-X","volume":"72","author":"YQ Huang","year":"2003","unstructured":"Huang, Y.Q., Xu, J.C.: A conforming finite element method for overlapping and nonmatching grids. Math. Comput. 72, 1057\u20131066 (2003)","journal-title":"Math. Comput."},{"key":"9582_CR20","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1002\/num.20554","volume":"27","author":"C Bacuta","year":"2011","unstructured":"Bacuta, C., Sun, J., Zheng, C.: Partition of unity refinement for local approximation. Numer. Meth. PDE. 27, 803\u2013817 (2011)","journal-title":"Numer. Meth. PDE."},{"key":"9582_CR21","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.1080\/00207160.2015.1026338","volume":"93","author":"GZ Du","year":"2016","unstructured":"Du, G.Z., Hou, Y.R., Zuo, L.Y.: Local and parallel finite element methods for the mixed Navier-Stokes\/Darcy Model. Int. J. Comp. Math. 93, 1155\u20131172 (2016)","journal-title":"Int. J. Comp. Math."},{"key":"9582_CR22","doi-asserted-by":"crossref","first-page":"2541","DOI":"10.1016\/j.camwa.2016.04.001","volume":"71","author":"YR Hou","year":"2016","unstructured":"Hou, Y.R., Du, G.Z.: An expandable local and parallel two-grid finite element scheme. Comput. Math. Appl. 71, 2541\u20132556 (2016)","journal-title":"Comput. Math. Appl."},{"key":"9582_CR23","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1090\/S0025-5718-1991-1066834-0","volume":"56","author":"MD Gunzburger","year":"1991","unstructured":"Gunzburger, M.D., Meir, A.J., Peterson, J.S.: On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics. Math. Comput. 56, 523\u2013563 (1991)","journal-title":"Math. Comput."},{"key":"9582_CR24","doi-asserted-by":"crossref","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, USA (1990)"},{"key":"9582_CR25","volume-title":"Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation","author":"J-F Gerbeau","year":"2006","unstructured":"Gerbeau, J.-F., Le 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":"9582_CR26","doi-asserted-by":"crossref","first-page":"2840","DOI":"10.1016\/j.cma.2010.05.007","volume":"199","author":"C Greif","year":"2010","unstructured":"Greif, C., Li, D., Sch\u00f6tzau, D., Wei. X.X.: A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Engrg. 199, 2840\u20132855 (2010)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"9582_CR27","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1007\/s00211-003-0487-4","volume":"96","author":"D Sch\u00f6tzau","year":"2004","unstructured":"Sch\u00f6tzau, D.: Mixed finite element methods for stationary incompressible magneto-hydrodynamics. Numer. Math. 96, 771\u2013800 (2004)","journal-title":"Numer. Math."},{"key":"9582_CR28","doi-asserted-by":"crossref","first-page":"5867","DOI":"10.1016\/S0045-7825(01)00196-7","volume":"190","author":"NB Salah","year":"2001","unstructured":"Salah, N.B., Soulaimani, A., Habashi, W.G.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Engrg. 190, 5867\u20135892 (2001)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"9582_CR29","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1016\/j.cma.2016.02.039","volume":"304","author":"HY Su","year":"2016","unstructured":"Su, H.Y., Feng, X.L., Huang, P.Z.: Iterative methods in penalty finite element discretization for the steady MHD equations. Comput. Methods Appl. Mech. Engrg. 304, 521\u2013545 (2016)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"9582_CR30","doi-asserted-by":"crossref","first-page":"2977","DOI":"10.1016\/j.jcp.2010.12.046","volume":"230","author":"R Planas","year":"2011","unstructured":"Planas, R., Badia, S., Codina, R.: Approximation of the inductionless MHD problem using a stabilized finite element method. J. Comput. Phys. 230, 2977\u20132996 (2011)","journal-title":"J. Comput. Phys."},{"key":"9582_CR31","first-page":"198","volume":"6","author":"WJ Layton","year":"1997","unstructured":"Layton, W.J., Meir, A.J., Schmidt, P.G.: A two-level discretization method for the stationary MHD equations. Elec. Tran. Numer. Anal. 6, 198\u2013210 (1997)","journal-title":"Elec. Tran. Numer. Anal."},{"key":"9582_CR32","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1007\/s10915-014-9900-7","volume":"63","author":"XJ Dong","year":"2015","unstructured":"Dong, X.J., He, Y.N.: Two-level Newton iterative method for the 2D\/3D stationary incompressible magnetohydrodynamics. J. Sc. Comput. 63, 426\u2013451 (2015)","journal-title":"J. Sc. Comput."},{"key":"9582_CR33","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1007\/s11425-015-5087-0","volume":"59","author":"XJ Dong","year":"2016","unstructured":"Dong, X.J., He, Y.N.: Convergence of some finite element iterative methods related to different Reynolds numbers for the 2D\/3D stationary incompressible magnetohydrodynamics. Sci. China Math. 59, 589\u2013608 (2016)","journal-title":"Sci. China Math."},{"key":"9582_CR34","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/s00211-016-0803-4","volume":"135","author":"KB Hu","year":"2017","unstructured":"Hu, K.B., Ma, Y.C., Xu, J.C.: Stable finite element methods preserving \u2207\u22c5B =\u20090 exactly for MHD models. Numer. Math. 135, 371\u2013396 (2017)","journal-title":"Numer. Math."},{"key":"9582_CR35","doi-asserted-by":"publisher","unstructured":"Cai, W., Hu, J., Zhang, S.Y.: High order hierarchical divergence-free constrained transport h(d i v) finite element method for magnetic induction equation. Numerical Mathematics Theory Methods & Applications. https:\/\/doi.org\/10.4208\/nmtma.2017.s03 (2017)","DOI":"10.4208\/nmtma.2017.s03"},{"key":"9582_CR36","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1016\/j.jcp.2016.04.019","volume":"316","author":"YC Ma","year":"2016","unstructured":"Ma, Y.C., Hu, K.B., Hu, X.Z., Xu, J.C.: Robust preconditioners for incompressible MHD models. J. Comput. Phy. 316, 721\u2013746 (2016)","journal-title":"J. Comput. Phy."},{"key":"9582_CR37","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/j.jcp.2017.09.025","volume":"351","author":"LX Li","year":"2017","unstructured":"Li, L.X., Zheng, W.Y.: A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D. J. Comput. Phy. 351, 254\u2013270 (2017)","journal-title":"J. Comput. Phy."},{"key":"9582_CR38","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1137\/0725032","volume":"25","author":"JG Heywood","year":"1988","unstructured":"Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem III: Smoothing property and high order error estimates for spatial discretization. SIAM J. Numer. Anal. 25, 489\u2013512 (1988)","journal-title":"SIAM J. Numer. Anal."},{"key":"9582_CR39","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/j.cma.2014.03.022","volume":"276","author":"XJ Dong","year":"2014","unstructured":"Dong, X.J., He, Y.N., Zhang, Y.: Convergence analysis of three finite element iterative methods for the 2D\/3D stationary incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Engrg. 276, 287\u2013311 (2014)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"9582_CR40","first-page":"635","volume":"XXXIV","author":"M Sermane","year":"1984","unstructured":"Sermane, M., Temam, R.: Some mathematics questions related to the MHD equations. Comm. Pure Appl. Math. XXXIV, 635\u2013664 (1984)","journal-title":"Comm. Pure Appl. Math."},{"key":"9582_CR41","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1007\/BF02576171","volume":"21","author":"DN Arnold","year":"1984","unstructured":"Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo 21, 337\u2013344 (1984)","journal-title":"Calcolo"},{"key":"9582_CR42","unstructured":"Fortin, M.: Calcul Num\u00e9rique des Ecoulements Fluides de Bingham et des Fluides Newtoniens Incompressible par des M\u00e9thodes D\u2019el\u00e9ments Finis. Universit\u00e9 de Paris VI, Doctoral Thesis (1972)"},{"key":"9582_CR43","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0045-7930(73)90027-3","volume":"1","author":"P Hood","year":"1973","unstructured":"Hood, P., Taylor, C.: A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1, 73\u2013100 (1973)","journal-title":"Comput. Fluids"},{"key":"9582_CR44","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1051\/m2an\/1982160100491","volume":"16","author":"L Mansfield","year":"1982","unstructured":"Mansfield, L.: Finite element subspaces with optimal rates of convergence for stationary Stokes problem. RAIRO Anal. Num\u00e9r. 16, 49\u201366 (1982)","journal-title":"RAIRO Anal. Num\u00e9r."},{"key":"9582_CR45","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1137\/070693230","volume":"7","author":"J Nolen","year":"2008","unstructured":"Nolen, J., Papanicolaou, G., Pironneau, O.: A framework for adaptive multiscale methods for elliptic problems. Multiscale Model Simul. 7, 171\u2013196 (2008)","journal-title":"Multiscale Model Simul."},{"key":"9582_CR46","unstructured":"Hecht, F., Pironneau, O., Ohtsuka, K.: FreeFEM++. http:\/\/www.freefem.org\/ (2011)"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-017-9582-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-017-9582-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-017-9582-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,8]],"date-time":"2019-10-08T18:36:44Z","timestamp":1570559804000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-017-9582-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,27]]},"references-count":46,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2018,8]]}},"alternative-id":["9582"],"URL":"https:\/\/doi.org\/10.1007\/s10444-017-9582-4","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,12,27]]}}}