{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:33:42Z","timestamp":1787337222247,"version":"build-2736575974"},"reference-count":23,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>Magnetohydrodynamics (MHD) is a fluid theory that describes plasma physics by treating the plasma as a fluid of charged particles. Hence, the equations that describe the plasma form a nonlinear system that couples Navier\u2013Stokes equations with Maxwell's equations. This paper shows that the first-order system least squares (FOSLS) finite element method is a viable discretization for these large MHD systems. To solve this system, a nested-iteration\u2013Newton\u2013FOSLS\u2013AMG approach is taken. Most of the work is done on the coarse grid, including most of the linearizations. We show that at most one Newton step and a few V-cycles are all that are needed on the finest grid. Here, we describe how the FOSLS method can be applied to incompressible resistive MHD and how it can be used to solve these MHD problems efficiently. A 3D steady state and a reduced 2D time-dependent test problem are studied. The latter equations can simulate a \u201clarge aspect-ratio\u201d tokamak. The goal is to resolve as much physics from the test problems with the least amount of computational work. We show that this is achieved in a few dozen work units or fine grid residual evaluations.<\/jats:p>","DOI":"10.1137\/080727282","type":"journal-article","created":{"date-parts":[[2010,2,5]],"date-time":"2010-02-05T18:13:52Z","timestamp":1265393632000},"page":"229-248","source":"Crossref","is-referenced-by-count":19,"title":["First-Order System Least Squares for Incompressible Resistive Magnetohydrodynamics"],"prefix":"10.1137","volume":"32","author":[{"given":"J. H.","family":"Adler","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"T. A.","family":"Manteuffel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S. F.","family":"McCormick","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. W.","family":"Ruge","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,2,5]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142996313592"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142997324976"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1515\/156939505775248347"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(86)90095-0"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"W. L. Briggs, V. E. Henson, and S. F. McCormick,\n                      A Multigrid Tutorial\n                      , 2nd ed., SIAM, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719505"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827502407822"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0731091"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994266066"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.7015"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1063\/1.1460885"},{"key":"R11","unstructured":"A. Codd,\n                      Elasticity-Fluid Coupled Systems and Elliptic Grid Generation (EGG) Based on First-Order System Least Squares (FOSLS)\n                      , Ph.D. thesis, University of Colorado at Boulder, Boulder, CO, 2001."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902404406"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1002\/nla.567"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2007.05.005"},{"key":"R15","unstructured":"D. R. Nicholson,\n                      Introduction to Plasma Theory\n                      , John Wiley and Sons, New York, 1983."},{"key":"R16","unstructured":"C. W. Oosterlee, A. Schuller, and U. Trottenberg,\n                      Multigrid\n                      , Academic Press, New York, 2000."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2008.06.029"},{"key":"R18","unstructured":"O. Roehrle\n                      Multilevel First-Order System Least Squares for Quasilinear Elliptic Partial Differential Equations\n                      , Ph.D. thesis, University of Colorado at Boulder, Boulder, CO, 2004."},{"key":"R19","unstructured":"J. Ruge,\n                      Fospack Users Manual, Version\n                      1.0, University of Colorado at Boulder, Boulder, CO, 2000."},{"key":"R20","doi-asserted-by":"crossref","unstructured":"J. Ruge and K. St\u00fcben,\n                      Algebraic multigrid\n                      , in Multigrid Methods, S. F. McCormick, ed., SIAM, Philadelphia, 1987, pp. 73\u2013130.","DOI":"10.1137\/1.9781611971057.ch4"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1063\/1.861310"},{"key":"R22","unstructured":"P. Ullrich,\n                      Dynamics and Properties of the Magnetohydrodynamics Equations\n                      , unpublished notes, University of Waterloo, Waterloo, Ontario, 2005."},{"key":"R23","unstructured":"C. Westphal,\n                      First-Order System Least Squares for Geometrically-Nonlinear Elasticity in Nonsmooth Domains\n                      , Ph.D. thesis, University of Colorado at Boulder, Boulder, CO, 2004."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080727282","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:50:30Z","timestamp":1787334630000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080727282"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080727282"],"URL":"https:\/\/doi.org\/10.1137\/080727282","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}