{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:23Z","timestamp":1787336903437,"version":"build-2736575974"},"reference-count":25,"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":[[2000,1]]},"abstract":"<jats:p>Standard multigrid methods are not well suited for problems with anisotropic discrete operators, which can occur, for example, on grids that are stretched in order to resolve a boundary layer. One of the most efficient approaches to yield robust methods is the combination of standard coarsening with alternating-direction plane relaxation in the three dimensions. However, this approach may be difficult to implement in codes with multiblock structured grids because there may be no natural definition of global lines or planes. This inherent obstacle limits the range of an implicit smoother to only the portion of the computational domain in the current block. This report studies in detail, both numerically and analytically, the behavior of blockwise plane smoothers in order to provide guidance to engineers who use block-structured grids. The results obtained so far show alternating-direction plane smoothers to be very robust, even on multiblock grids. In common computational fluid dynamics multiblock simulations, where the number of subdomains crossed by the line of a strong anisotropy is low (up to four), textbook multigrid convergence rates can be obtained with a small overlap of cells between neighboring blocks.<\/jats:p>","DOI":"10.1137\/s106482759935736x","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"218-242","source":"Crossref","is-referenced-by-count":2,"title":["Alternating Plane Smoothers For Multiblock Grids"],"prefix":"10.1137","volume":"22","author":[{"given":"Ignacio M.","family":"Llorente","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Boris","family":"Diskin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"N. Duane","family":"Melson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"A. Brandt,\n                      Multi\u2010grid solvers on parallel computers\n                      , in Elliptic Problem Solvers, M. Schultz, ed., Academic Press, New York, 1981, pp. 39\u201384.","DOI":"10.1016\/B978-0-12-632620-8.50008-5"},{"key":"R2","unstructured":"A. Brandt,\n                      Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics\n                      , Tech. Report, GMD\u2010Studien 85, 1984."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827598332205"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"Achi Brandt, Boris Diskin, Multigrid solvers on decomposed domains, Contemp. Math., Vol. 157, Amer. Math. Soc., Providence, RI, 1994, 135\u201315594m:65185","DOI":"10.1090\/conm\/157\/01415"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7930(98)00043-7"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.6036"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"J. E. Dendy, S. F. McCormick, J. Ruge, T. Russell, and S. Schaffer,\n                      Multigrid methods for three\u2010dimensional petroleum reservoir simulation\n                      , in Tenth SPE Symposium on Reservoir Simulation, SPE paper 18049, Houston, TX, 1989.","DOI":"10.2118\/18409-MS"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1080\/10637199608915575"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0908059"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0913082"},{"key":"R11","unstructured":"J. E. Jones and N. D. Melson,\n                      A Note on Multi\u2010Block Relaxation Schemes for Multigrid Solvers\n                      , Tech. Report, ICASE 97\u201015, 1997."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/0167-739X(94)90007-8"},{"key":"R13","first-page":"92","volume":"10","author":"Llorente I. M.","year":"2000","journal-title":"Electron. Trans. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1097-4067","issn-type":"print"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1109\/71.595573"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(89)90121-6"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/0730010"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1996.5442"},{"key":"R18","unstructured":"A. Overman and J. V. Rosendale,\n                      Mapping robust parallel multigrid algorithms to scalable memory architectures\n                      , in Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, 1993."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595281587"},{"key":"R20","unstructured":"B. Smith, P. Bjorstad, and W. Gropp,\n                      Domain Decomposition, Parallel Multilevel Methods for Elliptic Partial Differential Equations\n                      , Cambridge University Press, Cambridge, UK, 1996."},{"key":"R21","unstructured":"K. St\u00fcben,\n                      Parallel and robust multigrid with applications\n                      , in Proceedings of the Euroconference on Supercomputation in Nonlinear and Disordered Systems: Algorithms, Applications and Architectures, L. V\u00e1zquez, F. Tirado, and I. M. Llorente, eds., Madrid, Spain, World Scientific, River Edge, NJ, 1997."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(86)90112-8"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596302825"},{"key":"R24","volume-title":"An introduction to multigrid methods","author":"Wesseling Pieter","year":"1992"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/0732051"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S106482759935736X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:53Z","timestamp":1787333753000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S106482759935736X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,1]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2000,1]]}},"alternative-id":["10.1137\/S106482759935736X"],"URL":"https:\/\/doi.org\/10.1137\/s106482759935736x","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,1]]}}}