{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,6]],"date-time":"2024-09-06T10:14:08Z","timestamp":1725617648630},"reference-count":31,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T00:00:00Z","timestamp":1648598400000},"content-version":"am","delay-in-days":365,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#am"},{"start":{"date-parts":[[2021,3,30]],"date-time":"2021-03-30T00:00:00Z","timestamp":1617062400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"DOI":"10.13039\/100000160","name":"Division of Earth Sciences","doi-asserted-by":"publisher","award":["0949446","1550901","1925575"],"id":[{"id":"10.13039\/100000160","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["2028346"],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007707","name":"University of California, Davis","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100007707","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2021,10]]},"abstract":"Abstract<\/jats:title>Problems arising in Earth's mantle convection involve finding the solution to Stokes systems with large viscosity contrasts. These systems contain localized features which, even with adaptive mesh refinement, result in linear systems that can be on the order of 109<\/jats:sup>or more unknowns. One common approach for preconditioning to the velocity block of these systems is to apply an Algebraic Multigrid (AMG) V\u2010cycle (as is done in the ASPECT software, for example), however, we find that AMG is lacking robustness with respect to problem size and number of parallel processes. Additionally, we see an increase in iteration counts with refinement when using AMG. In contrast, the Geometric Multigrid (GMG) method, by using information about the geometry of the problem, should offer a more robust option.Here we present a matrix\u2010free GMG V\u2010cycle which works on adaptively refined, distributed meshes, and we will compare it against the current AMG preconditioner (Trilinos ML) used in theASPECT<\/jats:styled-content>1<\/jats:sup>software. We will demonstrate the robustness of GMG with respect to problem size and show scaling up to 114,688 cores and 217 billion unknowns. All computations are run using the open\u2010source, finite element librarydeal.II<\/jats:styled-content>.2<\/jats:sup><\/jats:p>","DOI":"10.1002\/nla.2375","type":"journal-article","created":{"date-parts":[[2021,3,30]],"date-time":"2021-03-30T07:50:20Z","timestamp":1617090620000},"update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Comparison between algebraic and matrix\u2010free geometric multigrid for a Stokes problem on adaptive meshes with variable viscosity"],"prefix":"10.1002","volume":"28","author":[{"ORCID":"http:\/\/orcid.org\/0000-0002-3340-2482","authenticated-orcid":false,"given":"Thomas C.","family":"Clevenger","sequence":"first","affiliation":[{"name":"School of Mathematical and Statistical Sciences Clemson University Clemson South Carolina USA"}]},{"given":"Timo","family":"Heister","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences Clemson University Clemson South Carolina USA"}]}],"member":"311","published-online":{"date-parts":[[2021,3,30]]},"reference":[{"key":"e_1_2_7_2_1","unstructured":"BangerthW DannbergJ GassmoellerR HeisterT.ASPECT v2.1.0 [software]. Davis CA; 2019.https:\/\/doi.org\/10.5281\/zenodo.2653531."},{"key":"e_1_2_7_3_1","doi-asserted-by":"publisher","DOI":"10.1029\/2009GC002526"},{"key":"e_1_2_7_4_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1365-246X.2012.05609.x"},{"key":"e_1_2_7_5_1","doi-asserted-by":"crossref","unstructured":"SundarH BirosG BursteddeC RudiJ GhattasO StadlerG. Parallel geometric\u2010algebraic multigrid on unstructured forests of octrees. Proceedings of the International Conference on High Performance Computing Networking Storage and Analysis. Location is Salt Lake City UT: IEEE Computer Society Press;2012. p. 43.","DOI":"10.1109\/SC.2012.91"},{"key":"e_1_2_7_6_1","doi-asserted-by":"publisher","DOI":"10.1137\/16M110455X"},{"key":"e_1_2_7_7_1","doi-asserted-by":"publisher","DOI":"10.1002\/qj.2327"},{"key":"e_1_2_7_8_1","doi-asserted-by":"publisher","DOI":"10.1137\/16M108450X"},{"key":"e_1_2_7_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2015.03.014"},{"key":"e_1_2_7_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jocs.2018.12.006"},{"key":"e_1_2_7_11_1","doi-asserted-by":"crossref","unstructured":"RudiJ MalossiACI IsaacT StadlerG GurnisM StaarPWJ et al. An extreme\u2010scale implicit solver for complex pdes: highly heterogeneous flow in earth's mantle. Proceedings of the International Conference for High Performance Computing Networking Storage and Analysis SC '15; Austin TX;2015. p. 1\u201312.","DOI":"10.1145\/2807591.2807675"},{"key":"e_1_2_7_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jocs.2016.06.006"},{"key":"e_1_2_7_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.compfluid.2012.04.012"},{"key":"e_1_2_7_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/3425193"},{"key":"e_1_2_7_15_1","doi-asserted-by":"publisher","DOI":"10.1093\/gji\/ggx195"},{"volume-title":"Theory and practice of finite elements","year":"2010","author":"Ern A","key":"e_1_2_7_16_1"},{"key":"e_1_2_7_17_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3658-8"},{"key":"e_1_2_7_18_1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198528678.001.0001","volume-title":"Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics","author":"Elman H","year":"2005"},{"issue":"5","key":"e_1_2_7_19_1","first-page":"967","article-title":"A new convergence proof for the multigrid method including the v\u2010cycle","volume":"20","author":"Braess D","year":"1983","journal-title":"SIAM J Sci Comput"},{"volume-title":"Multigrid","year":"2000","author":"Trottenberg U","key":"e_1_2_7_20_1"},{"key":"e_1_2_7_21_1","doi-asserted-by":"publisher","DOI":"10.1137\/0731070"},{"key":"e_1_2_7_22_1","doi-asserted-by":"publisher","DOI":"10.1137\/0914028"},{"key":"e_1_2_7_23_1","doi-asserted-by":"publisher","DOI":"10.1137\/090778523"},{"key":"e_1_2_7_24_1","doi-asserted-by":"publisher","DOI":"10.1515\/jnma-2019-0064"},{"key":"e_1_2_7_25_1","doi-asserted-by":"publisher","DOI":"10.1145\/1089014.1089021"},{"key":"e_1_2_7_26_1","doi-asserted-by":"publisher","DOI":"10.1137\/100791634"},{"key":"e_1_2_7_27_1","unstructured":"HeisterT ClevengerTC. 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