{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:43:33Z","timestamp":1787330613541,"version":"build-2736575974"},"reference-count":33,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>This paper develops a nested iteration algorithm to solve time-dependent nonlinear systems of partial differential equations. For each time step, Newton's method is used to form approximate solutions from a sequence of nested spaces, where the resolution of the approximations increases as the algorithm progresses. Nested iteration results in most of the iterations being performed on coarser grids, where minimal work is needed to reduce error to the level of discretization error. The approximate solution on a given coarse grid is interpolated to a refined grid and is used as an initial guess for the problem posed there. The approximation is then already close enough to the solution on the current grid that a minimal amount of work is needed to solve the refined problem due to the rapid convergence of Newton's method near a solution. The paper develops an algorithm that attempts to optimize accuracy-per-computational-cost on each grid, so that essentially no unnecessary work is done on any grid. The nested iteration algorithm is then applied to a reduced two-dimensional model of the incompressible, resistive magnetohydrodynamic (MHD) equations. Using this algorithm on the MHD equations in the context of a first-order system least squares finite element discretization and algebraic multigrid to solve the linearized systems, instabilities in a model tokamak fusion reactor are simulated. Numerical results show that this highly complex nonlinear problem is solved in an equivalent of 30\u201380 fine-grid relaxation sweeps per time step.<\/jats:p>","DOI":"10.1137\/090766905","type":"journal-article","created":{"date-parts":[[2010,5,26]],"date-time":"2010-05-26T18:52:18Z","timestamp":1274899938000},"page":"1506-1526","source":"Crossref","is-referenced-by-count":15,"title":["Nested Iteration and 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"}]},{"given":"G. D.","family":"Sanders","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,5,26]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"J. H. Adler, T. A. Manteuffel, S. F. McCormick, J. Nolting, J. W. Ruge, and L. Tang,\n                      An efficiency-based adaptive refinement scheme applied to incompressible, resistive magnetohydrodynamics\n                      , in Large-Scale Scientific Computing, Lecture Notes in Comput. 5910, Sci. Springer, Berlin, 2009, pp. 1\u201313.","DOI":"10.1007\/978-3-642-12535-5_1"},{"key":"R2","unstructured":"J. H. Adler,\n                      Nested Iteration and First Order Systems Least Squares on Incompressible Resistive Magnetohydrodynamics\n                      , Ph.D. thesis, University of Colorado at Boulder, Boulder, CO, 2009."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/080727282"},{"key":"R4","unstructured":"G. Bateman,\n                      MHD Instabilities\n                      , The MIT Press, Cambridge, MA, 1978."},{"key":"R5","unstructured":"A. Brandt, S. F. McCormick, and J. W. Ruge,\n                      Algebraic Multigrid (AMG) for Sparse Matrix Equations\n                      , Cambridge University Press, London, 1984."},{"key":"R6","unstructured":"A. Brandt, S. McCormick, and J. Ruge,\n                      Algebraic Multigrid (AMG) for Automatic Multigrid Solutions with Application to Geodetic Computations\n                      , report, Institute for Computational Studies, Fort Collins, CO, 1982."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(86)90095-0"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/050626272"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/040614402"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"W. L. Briggs, V. E. Henson, and S. F. McCormick,\n                      A Multigrid Tutorial\n                      , 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2000.","DOI":"10.1137\/1.9780898719505"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/0731091"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994266066"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.7015"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"L. Chacon, D. A. Knoll, and J. M. Finn,\n                      Nonlinear study of the curvature-driven parallel velocity shear-tearing instability\n                      , Phys. Plasmas, 9 (2002), pp. 1164\u20131176.","DOI":"10.1063\/1.1460885"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902404406"},{"key":"R16","unstructured":"A. Codd,\n                      Elasticity-Fluid Coupled Systems and Elliptic Grid Generation (EGG) Basedon First-Order System Least Squares (FOSLS)\n                      , Ph.D. thesis, University of Colorado atBoulder, Boulder, CO, 2001."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1002\/nla.575"},{"key":"R18","unstructured":"P. Deuflhard,\n                      Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms,\n                      Springer, Berlin, 2004."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(86)90004-5"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1996.5580"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1063\/1.2173515"},{"key":"R22","unstructured":"S. MacLachlan,\n                      Improving Robustness in Multiscale Methods\n                      , Ph.D. thesis, University of Colorado at Boulder, Boulder, CO, 2004."},{"key":"R23","unstructured":"D. R. Nicholson,\n                      Introduction to Plasma Theory\n                      , John Wiley and Sons, New York, 1983."},{"key":"R24","unstructured":"J. Nolting,\n                      Efficiency-based Local Adaptive Refinement for FOSLS Finite Elements\n                      , Ph.D. thesis, University of Colorado at Boulder, Boulder, CO, 2008."},{"key":"R25","unstructured":"U. Trottenberg, C. W. Oosterlee, and A. Schuller,\n                      Multigrid\n                      , Academic Press, New York, 2000."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2008.06.029"},{"key":"R27","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":"R28","doi-asserted-by":"crossref","unstructured":"J. Ruge and K. St$\\ddot{u}$ben,\n                      Algebraic Multigrid (AMG)\n                      , in Multigrid Methods, Frontiers Appl. Math. 3, S.F. McCormick, ed., SIAM, Philadelphia, PA, 1987, pp. 73\u2013130.","DOI":"10.1137\/1.9781611971057.ch4"},{"key":"R29","unstructured":"J. Ruge, Fospack users manual, version 1.0. unpublished, 2000."},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1007\/s006070070028"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1063\/1.861310"},{"key":"R32","unstructured":"P. Ullrich,\n                      Dynamics and Properties of the Magnetohydrodynamics Equations\n                      , unpublished, 2005."},{"key":"R33","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\/090766905","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:15:50Z","timestamp":1787328950000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090766905"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":33,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090766905"],"URL":"https:\/\/doi.org\/10.1137\/090766905","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}