{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T23:03:32Z","timestamp":1777590212520,"version":"3.51.4"},"reference-count":51,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2023,3,21]],"date-time":"2023-03-21T00:00:00Z","timestamp":1679356800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2023,3,31]]},"abstract":"<jats:p>\n            We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier\u2013Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for generalized minimal residual iterations. Its performance properties are demonstrated for 2D and 3D benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the\n            <jats:italic>deal.II<\/jats:italic>\n            finite element library. The concepts are flexible and can be transferred to similar software platforms.\n          <\/jats:p>","DOI":"10.1145\/3582492","type":"journal-article","created":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T12:41:24Z","timestamp":1675255284000},"page":"1-25","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":12,"title":["A Geometric Multigrid Method for Space-Time Finite Element Discretizations of the Navier\u2013Stokes Equations and its Application to 3D Flow Simulation"],"prefix":"10.1145","volume":"49","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5035-4235","authenticated-orcid":false,"given":"Mathias","family":"Anselmann","sequence":"first","affiliation":[{"name":"Helmut Schmidt University, Faculty of Mechanical and Civil Engineering, Hamburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1180-4250","authenticated-orcid":false,"given":"Markus","family":"Bause","sequence":"additional","affiliation":[{"name":"Helmut Schmidt University, Faculty of Mechanical and Civil Engineering, Hamburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2023,3,21]]},"reference":[{"key":"e_1_3_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.03.015"},{"key":"e_1_3_1_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-67202-1_2"},{"key":"e_1_3_1_4_2","doi-asserted-by":"publisher","DOI":"10.1145\/1465482.1465560"},{"key":"e_1_3_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2020.10.027"},{"key":"e_1_3_1_6_2","doi-asserted-by":"publisher","DOI":"10.1002\/fld.5074"},{"key":"e_1_3_1_7_2","doi-asserted-by":"publisher","DOI":"10.1515\/jnma-2020-0043"},{"key":"e_1_3_1_8_2","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2016055"},{"key":"e_1_3_1_9_2","doi-asserted-by":"publisher","DOI":"10.1002\/cnm.529"},{"key":"e_1_3_1_10_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000212"},{"key":"e_1_3_1_11_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000212"},{"key":"e_1_3_1_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(96)00059-1"},{"key":"e_1_3_1_13_2","doi-asserted-by":"crossref","unstructured":"Erik Burman Stefan Frei and Andre Massing. 2020. Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains. Numerische Mathematik 150 (Dec.2020) 423\u2013478. arxiv:1910.03054","DOI":"10.1007\/s00211-021-01264-x"},{"key":"e_1_3_1_14_2","doi-asserted-by":"publisher","DOI":"10.1145\/3425193"},{"key":"e_1_3_1_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2011.09.001"},{"key":"e_1_3_1_16_2","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199678792.001.0001"},{"key":"e_1_3_1_17_2","doi-asserted-by":"publisher","DOI":"10.1137\/0719018"},{"key":"e_1_3_1_18_2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y"},{"key":"e_1_3_1_19_2","doi-asserted-by":"publisher","DOI":"10.2174\/1876389801204010035"},{"key":"e_1_3_1_20_2","doi-asserted-by":"publisher","DOI":"10.1002\/fld.3831"},{"key":"e_1_3_1_21_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-33134-3_54"},{"key":"e_1_3_1_22_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2014.04.011"},{"key":"e_1_3_1_23_2","doi-asserted-by":"publisher","DOI":"10.1002\/fld.377"},{"key":"e_1_3_1_24_2","doi-asserted-by":"publisher","DOI":"10.1002\/fld.1080"},{"key":"e_1_3_1_25_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-45750-5"},{"key":"e_1_3_1_26_2","doi-asserted-by":"publisher","DOI":"10.1002\/fld.195"},{"key":"e_1_3_1_27_2","doi-asserted-by":"publisher","DOI":"10.1002\/1097-0363(20000630)33:4<453::AID-FLD15>3.0.CO;2-0"},{"key":"e_1_3_1_28_2","doi-asserted-by":"publisher","DOI":"10.1515\/jnma-2015-0005"},{"key":"e_1_3_1_29_2","doi-asserted-by":"publisher","DOI":"10.1137\/S106482759935808X"},{"key":"e_1_3_1_30_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.softx.2019.100239"},{"key":"e_1_3_1_31_2","doi-asserted-by":"publisher","DOI":"10.1177\/1094342016671790"},{"key":"e_1_3_1_32_2","doi-asserted-by":"publisher","DOI":"10.1145\/779359.779361"},{"key":"e_1_3_1_33_2","doi-asserted-by":"publisher","DOI":"10.1137\/060655407"},{"key":"e_1_3_1_34_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-002-1451-3"},{"key":"e_1_3_1_35_2","doi-asserted-by":"publisher","DOI":"10.1504\/IJCSM.2007.016537"},{"key":"e_1_3_1_36_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-5712-3_23"},{"key":"e_1_3_1_37_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-015-0710-0"},{"key":"e_1_3_1_38_2","doi-asserted-by":"publisher","DOI":"10.1137\/070679776"},{"key":"e_1_3_1_39_2","doi-asserted-by":"publisher","DOI":"10.1137\/050632166"},{"key":"e_1_3_1_40_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-63970-3"},{"key":"e_1_3_1_41_2","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"e_1_3_1_42_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-322-89849-4_39"},{"key":"e_1_3_1_43_2","volume-title":"The Trilinos Project Website","author":"Team The Trilinos Project","year":"2020","unstructured":"The Trilinos Project Team. 2020. The Trilinos Project Website. Retrieved 6-10-2020 from https:\/\/trilinos.github.io\/index.html."},{"key":"e_1_3_1_44_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58393-3"},{"issue":"1","key":"e_1_3_1_45_2","first-page":"29","article-title":"Numerical studies of Vanka-type smoothers in computational solid mechanics","volume":"1","author":"Turek Stefan","year":"2009","unstructured":"Stefan Turek and Hilmar Wobker. 2009. Numerical studies of Vanka-type smoothers in computational solid mechanics. Advances in Applied Mathematics and Mechanics 1, 1 (2009), 29\u201355.","journal-title":"Advances in Applied Mathematics and Mechanics"},{"key":"e_1_3_1_46_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2012.05.038"},{"key":"e_1_3_1_47_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2012.05.037"},{"key":"e_1_3_1_48_2","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(86)90022-8"},{"key":"e_1_3_1_49_2","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(86)90008-2"},{"key":"e_1_3_1_50_2","unstructured":"Henry von Wahl Thomas Richter and Christoph Lehrenfeld. 2020. An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains. arxiv:2002.02352.https:\/\/arxiv.org\/pdf\/2002.02352.pdf."},{"key":"e_1_3_1_51_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492915000021"},{"key":"e_1_3_1_52_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2017.10.023"}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3582492","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3582492","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T18:08:50Z","timestamp":1750183730000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3582492"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,21]]},"references-count":51,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,3,31]]}},"alternative-id":["10.1145\/3582492"],"URL":"https:\/\/doi.org\/10.1145\/3582492","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"value":"0098-3500","type":"print"},{"value":"1557-7295","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,21]]},"assertion":[{"value":"2021-07-23","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2023-01-18","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2023-03-21","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}