{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T07:41:08Z","timestamp":1787384468237,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2001,1]]},"abstract":"<jats:p>In this paper, we present three-grid Fourier analysis for multigrid methods. Due to the recursive structure of a multigrid iteration, this analysis can be deduced from the well-known two-grid Fourier analysis. The coarse grid correction part of multigrid algorithms can be more accurately evaluated with the three-grid analysis. We apply the analysis to several scalar equations and discretizations with an emphasis on problems with a multigrid coarse grid correction difficulty like upwind discretizations of the convection diffusion equation. The main focus lies on possible improvements by carefully chosen Galerkin operators and\/or by an additional acceleration with restarted GMRES, GMRES(m). Numerical test calculations validate the theoretical predictions.<\/jats:p>","DOI":"10.1137\/s106482750037367x","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"651-671","source":"Crossref","is-referenced-by-count":44,"title":["On Three-Grid Fourier Analysis for Multigrid"],"prefix":"10.1137","volume":"23","author":[{"given":"Roman","family":"Wienands","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cornelis W.","family":"Oosterlee","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,8,4]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02238488"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1977-0431719-X"},{"key":"R3","unstructured":"A. Brandt,\n                      Rigorous local mode analysis of multigrid\n                      , in Preliminary Proceedings of the 4th Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, 1989, pp. 55\u2013133."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0731087"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(92)90049-5"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/0914039"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(83)90016-4"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(99)00037-7"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1002\/fld.1650040704"},{"key":"R10","unstructured":"W. Hackbusch,\n                      Multi\u2010Grid Methods and Applications\n                      , Springer\u2010Verlag, Berlin, 1985."},{"key":"R11","unstructured":"P. W. Hemker,\n                      Fourier Analysis of Grid Functions, Prolongations and Restrictions\n                      , Report NW 98, CWI, Amsterdam, 1980."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(90)90047-4"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1997.5854"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827598338093"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"R16","unstructured":"J. Varah, A survey of iterative methods for sparse linear systems, Proceedings of the fifteenth Manitoba conference on numerical mathematics and computing (Winnipeg, Man., 1985), Vol. 51, 1986, 83\u20139288h:65080"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-1506(200003)7:2<67::AID-NLA186>3.0.CO;2-8"},{"key":"R18","unstructured":"R. Stevenson,\n                      On the Validity of Local Mode Analysis of Multi\u2010Grid Methods\n                      , Ph.D. thesis, Rijks University, Utrecht, the Netherlands, 1990."},{"key":"R19","unstructured":"K. St\u00fcben,\n                      Algebraic Multigrid (AMG): An Introduction with Applications\n                      , Tech. report 53, GMD, St. Augustin, Germany, 1999."},{"key":"R19","unstructured":"This report appeared as an appendix in [21]."},{"key":"R20","unstructured":"Klaus St\u00fcben, Ulrich Trottenberg, Multigrid methods: fundamental algorithms, model problem analysis and applications, Lecture Notes in Math., Vol. 960, Springer, Berlin, 1982, 1\u201317684m:65129"},{"key":"R21","unstructured":"U. Trottenberg, C. W. Oosterlee, and A. Sch\u00fcller,\n                      Multigrid\n                      , Academic Press, New York, 2001."},{"key":"R22","unstructured":"P. Vanek, J. Mandel, and M. Brezina,\n                      Algebraic Multigrid on Unstructured Meshes\n                      , UCD\/CCM Report 34, University of Colorado, Denver, CO, 1994."},{"key":"R23","volume-title":"An introduction to multigrid methods","author":"Wesseling Pieter","year":"1992"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827599353014"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1997.5659"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596310998"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(90)90252-U"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S106482750037367X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:42:37Z","timestamp":1787334157000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S106482750037367X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,1]]},"references-count":28,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2001,1]]}},"alternative-id":["10.1137\/S106482750037367X"],"URL":"https:\/\/doi.org\/10.1137\/s106482750037367x","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,1]]}}}