{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T04:00:38Z","timestamp":1744689638696,"version":"3.40.4"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,3,13]],"date-time":"2025-03-13T00:00:00Z","timestamp":1741824000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,3,13]],"date-time":"2025-03-13T00:00:00Z","timestamp":1741824000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,6]]},"DOI":"10.1007\/s40314-025-03155-5","type":"journal-article","created":{"date-parts":[[2025,3,13]],"date-time":"2025-03-13T02:49:56Z","timestamp":1741834196000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Local Fourier analysis of multigrid methods with ILU smoother for triangular grids using aggressive coarsening"],"prefix":"10.1007","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0688-6987","authenticated-orcid":false,"given":"Grazielli","family":"Vassoler-Rutz","sequence":"first","affiliation":[]},{"given":"Michely L.","family":"Oliveira","sequence":"additional","affiliation":[]},{"given":"Marcio A. V.","family":"Pinto","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,3,13]]},"reference":[{"key":"3155_CR1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511624100","volume-title":"Iterative solution methods","author":"O Axelsson","year":"1994","unstructured":"Axelsson O (1994) Iterative solution methods. Cambridge University Press, New York"},{"issue":"1","key":"3155_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4208\/nmtma.2015.w01si","volume":"8","author":"J Brannick","year":"2015","unstructured":"Brannick J, Hu X, Rodrigo C, Zikatanov L (2015) Local Fourier analysis of multigrid methods with polynomial smoothers and aggressive coarsening. Numer Math Theory Methods Appl 8(1):1\u201321","journal-title":"Numer Math Theory Methods Appl"},{"key":"3155_CR3","first-page":"227","volume":"51","author":"NI Buleev","year":"1960","unstructured":"Buleev NI (1960) A numerical method for the solution of two-dimensional and three-dimensional equations of diffusion. Math Sb 51:227\u2013238","journal-title":"Math Sb"},{"issue":"3","key":"3155_CR4","doi-asserted-by":"publisher","first-page":"668","DOI":"10.1137\/0912035","volume":"12","author":"TF Chan","year":"1991","unstructured":"Chan TF (1991) Fourier analysis of relaxed incomplete factorization preconditioners. SIAM J Sci Stat Comput 12(3):668\u2013680","journal-title":"SIAM J Sci Stat Comput"},{"issue":"3","key":"3155_CR5","doi-asserted-by":"publisher","first-page":"2081","DOI":"10.1137\/080713483","volume":"31","author":"FJ Gaspar","year":"2009","unstructured":"Gaspar FJ, Gracia JL, Lisbona FJ (2009) Fourier analysis for multigrid methods on triangular grids. SIAM J Sci Comput 31(3):2081\u20132102","journal-title":"SIAM J Sci Comput"},{"issue":"7","key":"3155_CR6","doi-asserted-by":"publisher","first-page":"1693","DOI":"10.1016\/j.apnum.2009.01.003","volume":"59","author":"FJ Gaspar","year":"2009","unstructured":"Gaspar FJ, Gracia JL, Lisbona FJ, Rodrigo C (2009) On geometric multigrid methods for triangular grids using three-coarsening strategy. Appl Numer Math 59(7):1693\u20131708","journal-title":"Appl Numer Math"},{"key":"3155_CR7","doi-asserted-by":"crossref","unstructured":"Hackbusch W (1994) Iterative solution of large sparse linear systems of equations. Applied mathematical sciences series, no. 95. Springer","DOI":"10.1007\/978-1-4612-4288-8"},{"key":"3155_CR8","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02427-0","volume-title":"Multi-grid methods and applications","author":"W Hackbusch","year":"1985","unstructured":"Hackbusch W (1985) Multi-grid methods and applications. Springer, Berlin"},{"key":"3155_CR9","unstructured":"Johannsen K (2005) A robust 9-point ILU smoother for anisotropic problems. IWR preprint"},{"key":"3155_CR10","doi-asserted-by":"crossref","unstructured":"Khalil M (1989) Local mode smoothing analysis of various incomplete factorization iterative methods. In: Robust multi-grid methods. Springer, pp 155\u2013164","DOI":"10.1007\/978-3-322-86200-6_14"},{"key":"3155_CR11","doi-asserted-by":"crossref","unstructured":"Oertel K-D, St\u00fcben K (1989) Multigrid with ILU-smoothing: systematic tests and improvements. In: Robust multi-grid methods. Springer, pp 188\u2013199","DOI":"10.1007\/978-3-322-86200-6_17"},{"key":"3155_CR12","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1090\/qam\/134855","volume":"19","author":"TA Oliphant","year":"1961","unstructured":"Oliphant TA (1961) An implicit numerical method for solving two-dimensional time-dependent diffusion problems. Q Appl Math 19:221\u2013229","journal-title":"Q Appl Math"},{"key":"3155_CR13","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1090\/qam\/143350","volume":"20","author":"TA Oliphant","year":"1962","unstructured":"Oliphant TA (1962) An extrapolation process for solving linear systems. Q Appl Math 20:257\u2013267","journal-title":"Q Appl Math"},{"issue":"2","key":"3155_CR14","first-page":"251","volume":"117","author":"ML Oliveira","year":"2018","unstructured":"Oliveira ML, Pinto MAV, Gon\u00e7alves SFT, Rutz GV (2018) On the robustness of the XY-Zebra-Gauss-Seidel smoother on an anisotropic diffusion problem. Comput Model Eng Sci 117(2):251\u2013270","journal-title":"Comput Model Eng Sci"},{"key":"3155_CR15","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/j.apnum.2016.02.007","volume":"106","author":"M Pinto","year":"2016","unstructured":"Pinto M, Rodrigo C, Gaspar F, Oosterlee C (2016) On the robustness of ILU smoothers on triangular grids. Appl Numer Math 106:37\u201352","journal-title":"Appl Numer Math"},{"key":"3155_CR16","volume-title":"Geometric multigrid methods on triangular grids: application to semi-structured meshes","author":"C Rodrigo","year":"2012","unstructured":"Rodrigo C, Gaspar FJ, Lisbona FJ (2012) Geometric multigrid methods on triangular grids: application to semi-structured meshes. Lambert Academic Publishing, Saarbr\u00fccken"},{"key":"3155_CR17","doi-asserted-by":"crossref","unstructured":"Rutz GV, Pinto MAV, Gon\u00e7alves SFT (2019) On the robustness of the multigrid method combining ILU and partial weight applied in an orthotropic diffusion problem. Revista Internacional de M\u00e9todos Num\u00e9ricos para C\u00e1lculo y Dise\u00f1o en Ingenier\u00eda 35(1)","DOI":"10.23967\/j.rimni.2019.03.001"},{"issue":"2","key":"3155_CR18","first-page":"699","volume":"128","author":"LP Silva","year":"2021","unstructured":"Silva LP, Rutyna BB, Santos Righi AR, Villela Pinto MA (2021) High order of accuracy for Poisson equation obtained by grouping of repeated Richardson extrapolation with fourth order schemes. Comput Model Eng Sci 128(2):699\u2013715","journal-title":"Comput Model Eng Sci"},{"key":"3155_CR19","doi-asserted-by":"crossref","unstructured":"Stevenson R (1994) Robust multi-grid with 7-point ILU smoothing. In: Multigrid methods IV. Springer, pp 295\u2013307","DOI":"10.1007\/978-3-0348-8524-9_22"},{"issue":"2","key":"3155_CR20","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/s002110050063","volume":"68","author":"R Stevenson","year":"1994","unstructured":"Stevenson R (1994) Modified ILU as a smoother. Numer Math 68(2):295\u2013309","journal-title":"Numer Math"},{"key":"3155_CR21","doi-asserted-by":"crossref","unstructured":"St\u00fcben K, Trottenberg U (1982) Multigrid methods: fundamental algorithms, model problem analysis and applications. In: Multigrid methods. Springer, pp 1\u2013176","DOI":"10.1007\/BFb0069928"},{"key":"3155_CR22","volume-title":"Multigrid","author":"U Trottenberg","year":"2001","unstructured":"Trottenberg U, Oosterlee CW, Sch\u00fcller A (2001) Multigrid. Academic Press, New York"},{"key":"3155_CR23","first-page":"121","volume-title":"Boundary problems in differential equations","author":"RS Varga","year":"1960","unstructured":"Varga RS (1960) Factorization and normalized iterative methods. In: Langer RE (ed) Boundary problems in differential equations. University of Wisconsin Press, Madison, pp 121\u2013142"},{"key":"3155_CR24","volume-title":"Matrix iterative analysis","author":"RS Varga","year":"1962","unstructured":"Varga RS (1962) Matrix iterative analysis. Prentice-Hall Inc., Englewood Cliffs"},{"issue":"4","key":"3155_CR25","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1137\/0903025","volume":"3","author":"P Wesseling","year":"1982","unstructured":"Wesseling P (1982) Theoretical and practical aspects of a multigrid method. SIAM J Sci Stat Comput 3(4):387\u2013407","journal-title":"SIAM J Sci Stat Comput"},{"key":"3155_CR26","volume-title":"An introduction to multigrid methods","author":"P Wesseling","year":"1992","unstructured":"Wesseling P (1992) An introduction to multigrid methods. Wiley, Chichester"},{"key":"3155_CR27","doi-asserted-by":"crossref","unstructured":"Wesseling P, Sonneveld P (1980) Numerical experiments with a multiple grid and a preconditioned Lanczos type method. In: Approximation methods for Navier\u2013Stokes problems. Springer, pp 543\u2013562","DOI":"10.1007\/BFb0086930"},{"key":"3155_CR28","volume-title":"Practical Fourier analysis for multigrid methods","author":"R Wienands","year":"2005","unstructured":"Wienands R, Joppich W (2005) Practical Fourier analysis for multigrid methods. Chapman and Hall\/CRC Press"},{"issue":"2","key":"3155_CR29","doi-asserted-by":"publisher","first-page":"180","DOI":"10.1016\/0899-8248(89)90029-3","volume":"1","author":"G Wittum","year":"1989","unstructured":"Wittum G (1989) Linear iterations as smoothers in multigrid methods: theory with applications to incomplete decompositions. IMPACT Comput Sci Eng 1(2):180\u2013215","journal-title":"IMPACT Comput Sci Eng"},{"issue":"4","key":"3155_CR30","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1137\/0910043","volume":"10","author":"G Wittum","year":"1989","unstructured":"Wittum G (1989) On the robustness of ILU smoothing. SIAM J Sci Stat Comput 10(4):699\u2013717","journal-title":"SIAM J Sci Stat Comput"},{"issue":"4","key":"3155_CR31","doi-asserted-by":"publisher","first-page":"1613","DOI":"10.1137\/060665695","volume":"29","author":"HB Zubair","year":"2007","unstructured":"Zubair HB, Oosterlee CW, Wienands R (2007) Multigrid for high-dimensional elliptic partial differential equations on non-equidistant grids. SIAM J Sci Comput 29(4):1613\u20131636","journal-title":"SIAM J Sci Comput"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03155-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03155-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03155-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,14]],"date-time":"2025-04-14T04:26:42Z","timestamp":1744604802000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03155-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,13]]},"references-count":31,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["3155"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03155-5","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,3,13]]},"assertion":[{"value":"16 May 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 February 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 February 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 March 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there are no Conflict of interest related to this study.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"184"}}