{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:40:00Z","timestamp":1776786000569,"version":"3.51.2"},"reference-count":44,"publisher":"American Mathematical Society (AMS)","issue":"257","license":[{"start":{"date-parts":[[2007,8,31]],"date-time":"2007-08-31T00:00:00Z","timestamp":1188518400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>More than a decade ago, Bramble, Pasciak and Xu developed a framework in analyzing the multigrid methods with nonnested spaces or noninherited quadratic forms. It was subsequently known as the BPX multigrid framework, which was widely used in the analysis of multigrid and domain decomposition methods. However, the framework has an apparent limit in the analysis of nonnested V-cycle methods, and it produces a variable V-cycle, or nonuniform convergence rate V-cycle methods, or other nonoptimal results in analysis thus far. This paper completes a long-time effort in extending the BPX multigrid framework so that it truly covers the nonnested V-cycle. We will apply the extended BPX framework to the analysis of many V-cycle nonnested multigrid methods. Some of them were proven previously only for two-level and W-cycle iterations. Some numerical results are presented to support the theoretical analysis of this paper.<\/p>","DOI":"10.1090\/s0025-5718-06-01897-7","type":"journal-article","created":{"date-parts":[[2006,11,1]],"date-time":"2006-11-01T16:50:02Z","timestamp":1162399802000},"page":"137-152","source":"Crossref","is-referenced-by-count":12,"title":["A generalized BPX multigrid framework covering nonnested V-cycle methods"],"prefix":"10.1090","volume":"76","author":[{"given":"Huo-Yuan","family":"Duan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shao-Qin","family":"Gao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Roger","family":"Tan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shangyou","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2006,8,31]]},"reference":[{"issue":"153","key":"1","doi-asserted-by":"publisher","first-page":"35","DOI":"10.2307\/2007724","article-title":"An optimal order process for solving finite element equations","volume":"36","author":"Bank, Randolph E.","year":"1981","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s006070050048","article-title":"A multigrid method for nonconforming FE-discretisations with application to non-matching grids","volume":"63","author":"Braess, D.","year":"1999","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"issue":"5","key":"3","doi-asserted-by":"publisher","first-page":"967","DOI":"10.1137\/0720066","article-title":"A new convergence proof for the multigrid method including the \ud835\udc49-cycle","volume":"20","author":"Braess, D.","year":"1983","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"979","DOI":"10.1137\/0727056","article-title":"Multigrid methods for nonconforming finite element methods","volume":"27","author":"Braess, D.","year":"1990","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"5","series-title":"Pitman Research Notes in Mathematics Series","isbn-type":"print","volume-title":"Multigrid methods","volume":"294","author":"Bramble, James H.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0582234352"},{"issue":"6","key":"6","doi-asserted-by":"publisher","first-page":"1746","DOI":"10.1137\/0731089","article-title":"Uniform convergence of multigrid \ud835\udc49-cycle iterations for indefinite and nonsymmetric problems","volume":"31","author":"Bramble, James H.","year":"1994","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"180","key":"7","doi-asserted-by":"publisher","first-page":"311","DOI":"10.2307\/2008314","article-title":"New convergence estimates for multigrid algorithms","volume":"49","author":"Bramble, James H.","year":"1987","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"198","key":"8","doi-asserted-by":"publisher","first-page":"467","DOI":"10.2307\/2153197","article-title":"The analysis of smoothers for multigrid algorithms","volume":"58","author":"Bramble, James H.","year":"1992","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"202","key":"9","doi-asserted-by":"publisher","first-page":"447","DOI":"10.2307\/2153097","article-title":"New estimates for multilevel algorithms including the \ud835\udc49-cycle","volume":"60","author":"Bramble, James H.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"10","isbn-type":"print","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1090\/conm\/157\/01401","article-title":"Uniform convergence estimates for multigrid \ud835\udc49-cycle algorithms with less than full elliptic regularity","author":"Bramble, James H.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0821851586"},{"issue":"193","key":"11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2008527","article-title":"The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms","volume":"56","author":"Bramble, James H.","year":"1991","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"234","key":"12","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1090\/S0025-5718-00-01222-9","article-title":"Uniform convergence of the multigrid \ud835\udc49-cycle for an anisotropic problem","volume":"70","author":"Bramble, James H.","year":"2001","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"185","key":"13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2008649","article-title":"An optimal-order multigrid method for \ud835\udc431 nonconforming finite elements","volume":"52","author":"Brenner, Susanne C.","year":"1989","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"192","key":"14","doi-asserted-by":"publisher","first-page":"411","DOI":"10.2307\/2008426","article-title":"A nonconforming multigrid method for the stationary Stokes equations","volume":"55","author":"Brenner, Susanne C.","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"225","key":"15","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1090\/S0025-5718-99-01035-2","article-title":"Convergence of nonconforming multigrid methods without full elliptic regularity","volume":"68","author":"Brenner, Susanne C.","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"238","key":"16","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1090\/S0025-5718-01-01361-8","article-title":"Convergence of the multigrid \ud835\udc49-cycle algorithm for second-order boundary value problems without full elliptic regularity","volume":"71","author":"Brenner, Susanne C.","year":"2002","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"247","key":"17","doi-asserted-by":"publisher","first-page":"1041","DOI":"10.1090\/S0025-5718-03-01578-3","article-title":"Convergence of nonconforming V-cycle and F-cycle multigrid algorithms for second order elliptic boundary value problems","volume":"73","author":"Brenner, Susanne C.","year":"2004","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"18","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4338-8","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387941932"},{"key":"19","series-title":"Studies in Mathematics and its Applications, Vol. 4","isbn-type":"print","volume-title":"The finite element method for elliptic problems","author":"Ciarlet, Philippe G.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0444850287"},{"issue":"2","key":"20","first-page":"79","article-title":"On the convergence of Galerkin-multigrid methods for nonconforming finite elements","volume":"7","author":"Chen, Z.","year":"1999","journal-title":"East-West J. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0928-0200","issn-type":"print"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1002\/(SICI)1098-2426(200005)16:3<265::AID-NUM1>3.0.CO;2-8","article-title":"On the convergence of nonnested multigrid methods with nested spaces on coarse grids","volume":"16","author":"Chen, Zhangxin","year":"2000","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"222","key":"22","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1090\/S0025-5718-98-00920-X","article-title":"Multigrid and multilevel methods for nonconforming \ud835\udc44\u2081 elements","volume":"67","author":"Chen, Zhangxin","year":"1998","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"23","first-page":"129","article-title":"Multigrid methods for the generalized Stokes equations based on mixed finite element methods","volume":"20","author":"Deng, Qing-ping","year":"2002","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"},{"issue":"194","key":"24","doi-asserted-by":"publisher","first-page":"409","DOI":"10.2307\/2008389","article-title":"Multigrid analysis of finite element methods with numerical integration","volume":"56","author":"Goldstein, Charles I.","year":"1991","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"25","doi-asserted-by":"publisher","first-page":"1029","DOI":"10.1137\/S0036142998341040","article-title":"Multigrid for the mortar finite element method","volume":"37","author":"Gopalakrishnan, Jayadeep","year":"2000","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"26","isbn-type":"print","volume-title":"Computational fluid and solid mechanics 2003. Vol. 1","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0080440460"},{"issue":"3","key":"27","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1007\/s002110200392","article-title":"A multilevel discontinuous Galerkin method","volume":"95","author":"Gopalakrishnan, J.","year":"2003","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"28","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02427-0","volume-title":"Multigrid methods and applications","volume":"4","author":"Hackbusch, Wolfgang","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/3540127615"},{"key":"29","unstructured":"C. Lee, A nonconforming multigrid method using conforming subspaces, in: Proc. Sixth Copper Mountain Conf. on Multigrid Methods, N. Melson et al., eds, NASA Conference Publication 3224, Part 1, Washington, DC, 1993, 317-330."},{"issue":"3","key":"30","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1016\/0377-0427(94)00106-8","article-title":"Multigrid computation of vector potentials","volume":"62","author":"Monk, Peter","year":"1995","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"31","series-title":"Frontiers in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971057","volume-title":"Multigrid methods","volume":"3","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712149"},{"issue":"198","key":"32","doi-asserted-by":"publisher","first-page":"457","DOI":"10.2307\/2153196","article-title":"Higher-dimensional nonnested multigrid methods","volume":"58","author":"Scott, L. Ridgway","year":"1992","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"33","series-title":"Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-8527-9","volume-title":"Multigrid methods for finite elements","volume":"318","author":"Sha\u012ddurov, V. V.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0792332903"},{"issue":"241","key":"34","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1090\/S0025-5718-02-01410-2","article-title":"An analysis of nonconforming multi-grid methods, leading to an improved method for the Morley element","volume":"72","author":"Stevenson, Rob","year":"2003","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"35","series-title":"Prentice-Hall Series in Automatic Computation","volume-title":"An analysis of the finite element method","author":"Strang, Gilbert","year":"1973"},{"issue":"3","key":"36","first-page":"229","article-title":"Multigrid techniques for a divergence-free finite element discretization","volume":"2","author":"Turek, S.","year":"1994","journal-title":"East-West J. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0928-0200","issn-type":"print"},{"issue":"1","key":"37","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1137\/0730013","article-title":"Convergence analysis of multigrid algorithms for nonselfadjoint and indefinite elliptic problems","volume":"30","author":"Wang, Jun Ping","year":"1993","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"38","unstructured":"J. Xu, Theory of Multilevel Methods, PhD thesis, Cornell University, 1989."},{"issue":"2","key":"39","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/PL00005450","article-title":"Multigrid for the mortar element method for \ud835\udc431 nonconforming element","volume":"88","author":"Xu, Xuejun","year":"2001","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"5","key":"40","doi-asserted-by":"publisher","first-page":"1712","DOI":"10.1137\/S0036142999364728","article-title":"A multigrid method for the mortar-type Morley element approximation of a plate bending problem","volume":"39","author":"Xu, Xuejun","year":"2001","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"41","first-page":"346","article-title":"A multigrid method with nonconforming Wilson finite elements","volume":"15","author":"Yu, Xi Jun","year":"1993","journal-title":"Math. Numer. Sinica","ISSN":"https:\/\/id.crossref.org\/issn\/0254-7791","issn-type":"print"},{"issue":"191","key":"42","doi-asserted-by":"publisher","first-page":"23","DOI":"10.2307\/2008790","article-title":"Optimal-order nonnested multigrid methods for solving finite element equations. I. On quasi-uniform meshes","volume":"55","author":"Zhang, Shangyou","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"219","key":"43","doi-asserted-by":"publisher","first-page":"1055","DOI":"10.1090\/S0025-5718-97-00853-3","article-title":"Treatments of discontinuity and bubble functions in the multigrid method","volume":"66","author":"Zhang, Shangyou","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"44","unstructured":"J. Zhao, Convergence of nonconforming V-cycle and F-cycle multigrid methods for the biharmonic problem using the Morley element, IMI Research Report, Math Dept, University of South Caroline, 2003."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2007-76-257\/S0025-5718-06-01897-7\/S0025-5718-06-01897-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2007-76-257\/S0025-5718-06-01897-7\/S0025-5718-06-01897-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:45:40Z","timestamp":1776782740000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2007-76-257\/S0025-5718-06-01897-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,8,31]]},"references-count":44,"journal-issue":{"issue":"257","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["S0025-5718-06-01897-7"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01897-7","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2006,8,31]]}}}