{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:31:34Z","timestamp":1787337094820,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>We consider a finite element method of higher order on a quadrilateral or triangular, hexahedral or tetrahedral mesh for solving the linearized Navier--Stokes equations (Oseen equations) by means of discontinuous elements for the pressure and suitable conforming elements for the velocity such that the global and the local inf-sup condition are satisfied. Our goal is the construction of a coupled multigrid solver for the corresponding algebraic system of equations. The idea of this solver is to switch inside of the multigrid method to a new velocity basis which leads to a reduced system with a much lower number of unknowns as well as a very small number of couplings between them. We call this new basis quasi divergence free since most of the basis functions are discretely divergence free which implies that they do not have any coupling to the pressure. The quasi divergence free basis functions can be constructed locally during the assembling process of the stiffness matrix. We create a multigrid method for solving the reduced problem efficiently. Since most of the velocity basis functions are completely decoupled from the pressure we can construct a smoother with low computational costs. The efficiency of the new multigrid method compared with other known coupled multigrid solvers is demonstrated by numerical experiments for a Stokes problem, the driven cavity problem, and a benchmark problem.<\/jats:p>","DOI":"10.1137\/04061814x","type":"journal-article","created":{"date-parts":[[2006,3,15]],"date-time":"2006-03-15T21:00:21Z","timestamp":1142456421000},"page":"141-171","source":"Crossref","is-referenced-by-count":1,"title":["A Multigrid Method for Incompressible Flow Problems Using Quasi Divergence Free Functions"],"prefix":"10.1137","volume":"28","author":[{"given":"Gunar","family":"Matthies","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Friedhelm","family":"Schieweck","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-02-01439-4"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000212"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1990-1035927-5"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100300"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2004.09.017"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61623-5"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1002\/mma.1670010103"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1002\/fld.1650010405"},{"key":"R10","unstructured":"F. Hecht, Construction of a basis at free divergence in finite element and application to the Navier\u2010Stokes equations, INRIA, Rocquencourt, 1984, 284\u201329787f:65114"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1002\/fld.377"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-18682-0"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-002-1444-2"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1002\/fld.195"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/s00791-003-0120-1"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1023\/A:1013860707381"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-002-1451-3"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827500374881"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/0914028"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"M. Sch\u00e4fer and S. Turek,\n                      The benchmark problem \u201cFlow around a cylinder\u201d\n                      , in Flow Simulation with High\u2010Performance Computers II, E. H. Hirschel, ed., Notes on Numerical Fluid Mechanics 52, Vieweg, Braunschweig, Germany, 1996, pp. 547\u2013566.","DOI":"10.1007\/978-3-322-89849-4_39"},{"key":"R21","unstructured":"F. Schieweck,\n                      A General Transfer Operator for Arbitrary Finite Element Spaces\n                      , preprint 25\/00, Otto\u2010von\u2010Guericke\u2010Universit\u00e4t Magdeburg, Fakult\u00e4t f\u00fcr Mathematik, 2000."},{"key":"R22","unstructured":"F. Schieweck,\n                      Construction of Higher Order Discretely Divergence Free Finite Elements for Incompressible Flow\n                      , preprint 34\/02, Otto\u2010von\u2010Guericke\u2010Universit\u00e4t Magdeburg, Fakult\u00e4t f\u00fcr Mathematik, 2002."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-87047-7"},{"key":"R24","first-page":"229","volume":"2","author":"Turek S.","year":"1994","journal-title":"East\u2010West J. Numer. Math."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58393-3"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(94)00056-7"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620361307"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1023\/A:1019159702198"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/04061814X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:44:28Z","timestamp":1787334268000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/04061814X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/04061814X"],"URL":"https:\/\/doi.org\/10.1137\/04061814x","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}