{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T23:02:03Z","timestamp":1777676523445,"version":"3.51.4"},"reference-count":26,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2013,3,5]],"date-time":"2013-03-05T00:00:00Z","timestamp":1362441600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["The International Journal of High Performance Computing Applications"],"published-print":{"date-parts":[[2014,2]]},"abstract":"<jats:p>The Stokes equations are frequently used to simulate geodynamic processes, including mantle convection, lithospheric dynamics, lava flow, and among others. In this study, the multigrid (MG) method is adopted to solve Stokes and continuity equations with strongly temperature-dependent viscosity. By taking advantage of the enhanced computing power of graphics processing units (GPUs) and the new version of Matlab 2010b, MG codes are optimized through Compute Unified Device Architecture (CUDA). Herein, we illustrate the approach that implements a GPU-based MG solver with Red\u2013Black Gauss\u2013Seidel (RBGS) smoother for the three-dimensional Stokes and continuity equations, in a hope that it helps solve the synthetic incompressible sinking problem in a cubic domain with strongly variable viscosity, and finally analyze our solver\u2019s efficiency on a GPU.<\/jats:p>","DOI":"10.1177\/1094342013478640","type":"journal-article","created":{"date-parts":[[2013,3,6]],"date-time":"2013-03-06T21:29:48Z","timestamp":1362605388000},"page":"50-60","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":4,"title":["Implementation of a multigrid solver on a GPU for Stokes equations with strongly variable viscosity based on Matlab and CUDA"],"prefix":"10.1177","volume":"28","author":[{"given":"Liang","family":"Zheng","sequence":"first","affiliation":[{"name":"Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, China"},{"name":"College of Earth Science, University of Chinese Academy of Sciences, China"},{"name":"Minnesota Supercomputing Institute, University of Minnesota, MN, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huai","family":"Zhang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, China"},{"name":"College of Earth Science, University of Chinese Academy of Sciences, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Taras","family":"Gerya","sequence":"additional","affiliation":[{"name":"Institute of Geophysics, ETH-Zurich, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matthew","family":"Knepley","sequence":"additional","affiliation":[{"name":"Computational Institute, University of Chicago, Chicago, IL, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David A","family":"Yuen","sequence":"additional","affiliation":[{"name":"Minnesota Supercomputing Institute, University of Minnesota, MN, USA"},{"name":"School of Environmental Sciences, China University of Geosciences, Wuhan, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yaolin","family":"Shi","sequence":"additional","affiliation":[{"name":"Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, China"},{"name":"College of Earth Science, University of Chinese Academy of Sciences, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2013,3,5]]},"reference":[{"key":"bibr1-1094342013478640","doi-asserted-by":"publisher","DOI":"10.1046\/j.1365-246x.1999.00833.x"},{"key":"bibr2-1094342013478640","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(66)90118-2"},{"key":"bibr3-1094342013478640","unstructured":"Bell N, Garland M (2012) Cusp: Generic parallel algorithms for sparse matrix and graph computations. http:\/\/cusp-library.googlecode.com."},{"key":"bibr4-1094342013478640","unstructured":"Cohen JM, Molemaker MJ (2009) A fast double precision CFD code using CUDA. http:\/\/www.jcohen.name\/papers\/Cohen_Fast_2009_final.pdf."},{"key":"bibr5-1094342013478640","doi-asserted-by":"publisher","DOI":"10.1016\/j.pepi.2008.06.023"},{"key":"bibr6-1094342013478640","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198528678.001.0001","volume-title":"Finite elements and fast iteratice solvers with applications to incompressible fluid dynamics","author":"Elman H","year":"2005"},{"key":"bibr7-1094342013478640","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(64)90253-8"},{"key":"bibr8-1094342013478640","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2011.09.007"},{"key":"bibr9-1094342013478640","volume-title":"Introduction to numerical Geodynamic modelling","author":"Gerya T","year":"2010"},{"key":"bibr10-1094342013478640","unstructured":"Hackbusch W (1977) On the Convergence of a Multi-grid Iteration Applied to Finite Element Equations. 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