{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T05:54:04Z","timestamp":1789710844308,"version":"4.0.1"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>We consider preconditioning techniques for a mixed finite element discretization of an incompressible magnetohydrodynamics (MHD) problem. Upon discretization and linearization, a 4-by-4 nonsymmetric block-structured linear system needs to be (repeatedly) solved. One of the principal challenges is the presence of a skew-symmetric term that couples the fluid velocity with the magnetic field. We propose a preconditioner that exploits the block structure of the underlying linear system, utilizing and combining effective solvers for the mixed Maxwell and the Navier--Stokes subproblems. We perform a spectral analysis for an ideal version of the preconditioner, and develop and test a practical version of it. Large-scale numerical results for linear systems of dimensions up to $10^7$ in two and three dimensions validate the effectiveness of our approach.<\/jats:p>","DOI":"10.1137\/16m1098991","type":"journal-article","created":{"date-parts":[[2017,12,19]],"date-time":"2017-12-19T12:43:41Z","timestamp":1513687421000},"page":"A2993-A3013","source":"Crossref","is-referenced-by-count":18,"title":["Preconditioners for Mixed Finite Element Discretizations of Incompressible MHD Equations"],"prefix":"10.1137","volume":"39","author":[{"given":"Michael","family":"Wathen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chen","family":"Greif","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dominik","family":"Sch\u00f6tzau","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,12,19]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(95)00931-0"},{"key":"atypb2","unstructured":"S. Balay, M. F. Adams, J. Brown, P. Brune, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, and H. Zhang,\n                      PETSc User's Manual\n                      , Technical Report ANL-95\/11 - Revision 3.4, Argonne National Laboratory, 2013,http:\/\/www.mcs.anl.gov\/petsc."},{"key":"atypb3","unstructured":"S. Balay, M. F. Adams, J. Brown, P. Brune, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, and H. Zhang,\n                      PETSc web page\n                      ,http:\/\/www.mcs.anl.gov\/petsc(2014)."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199678792.001.0001"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-47789-6_66"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198566656.001.0001"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2010.05.007"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1002\/nla.515"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1991-1066834-0"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/060660588"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.4208\/jcm.2009.27.5.013"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"D. Li,\n                      Numerical Solution of the Time-Harmonic Maxwell Equations and Incompressible Magnetohydrodynamics Problems\n                      , Ph.D. thesis, University of British Columbia, 2010,https:\/\/doi.org\/10.14288\/1.0051989.","DOI":"10.1007\/978-3-642-11842-5_30"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1002\/nla.782"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-23099-8"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198508885.001.0001"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827599355153"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/140955082"},{"key":"atypb18","volume-title":"Longmans","author":"Roberts P. H.","year":"1967"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-003-0487-4"},{"key":"atypb20","volume-title":"Master's thesis","author":"Wathen M.","year":"2014"},{"key":"atypb21","unstructured":"X. Wei,\n                      Mixed Discontinuous Galerkin Finite Element Methods for Incompressible Magnetohydrodynamics\n                      , Ph.D. thesis, University of British Columbia, 2011,https:\/\/doi.org\/10.14288\/1.0080512."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/BF02238513"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1098991","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:39:17Z","timestamp":1787333957000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1098991"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":22,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1098991"],"URL":"https:\/\/doi.org\/10.1137\/16m1098991","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}