{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T13:38:06Z","timestamp":1778852286409,"version":"3.51.4"},"reference-count":27,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,16]],"date-time":"2020-08-16T00:00:00Z","timestamp":1597536000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>We review a number of preconditioners for the advection-diffusion operator and for the Schur complement matrix, which, in turn, constitute the building blocks for Constraint and Triangular Preconditioners to accelerate the iterative solution of the discretized and linearized Navier-Stokes equations. An intensive numerical testing is performed onto the driven cavity problem with low values of the viscosity coefficient. We devise an efficient multigrid preconditioner for the advection-diffusion matrix, which, combined with the commuted BFBt Schur complement approximation, and inserted in a 2\u00d72 block preconditioner, provides convergence of the Generalized Minimal Residual (GMRES) method in a number of iteration independent of the meshsize for the lowest values of the viscosity parameter. The low-rank acceleration of such preconditioner is also investigated, showing its great potential.<\/jats:p>","DOI":"10.3390\/a13080199","type":"journal-article","created":{"date-parts":[[2020,8,17]],"date-time":"2020-08-17T04:35:51Z","timestamp":1597638951000},"page":"199","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Scalable Block Preconditioners for Linearized Navier-Stokes Equations at High Reynolds Number"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6157-0360","authenticated-orcid":false,"given":"Filippo","family":"Zanetti","sequence":"first","affiliation":[{"name":"School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8273-9674","authenticated-orcid":false,"given":"Luca","family":"Bergamaschi","sequence":"additional","affiliation":[{"name":"Department of Civil Environmental and Architectural Engineering, University of Padua, 35122 Padova, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2095","DOI":"10.1137\/050646421","article-title":"An augmented Lagrangian-based approach to the Oseen problem","volume":"28","author":"Benzi","year":"2006","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Elman, H., Sylvester, D., and Wathen, A. (2014). Finite Elements and Fast Iterative Solvers, Oxford University Press. [2nd ed.].","DOI":"10.1093\/acprof:oso\/9780199678792.001.0001"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"A3073","DOI":"10.1137\/18M1219370","article-title":"An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier-Stokes equations at high Reynolds number","volume":"41","author":"Farrell","year":"2019","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1137\/S106482759935808X","article-title":"A Preconditioner for the Steady-State Navier-Stokes Equations","volume":"24","author":"Kay","year":"2002","journal-title":"SIAM J. Sci. 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