{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:31:11Z","timestamp":1787337071032,"version":"build-2736575974"},"reference-count":25,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2007,1]]},"abstract":"<jats:p>The purpose of this paper is to develop and analyze a multigrid solver for the finite element discretization of the pseudostress system associated with the differential operator $\\mathcal{A}-\\gamma\\,\\mathbf{grad}\\,\\mathbf{div}$ over $2\\times 2$ matrix-valued functions. This system is derived from the pseudostress-velocity formulation [Z. Cai, B. Lee, and P. Wang, SIAM J. Numer. Anal., 42 (2004), pp. 843\u2013859] of two-dimensional Stokes problems through the penalty method or natural time discretization for the respective steady- or unsteady-state problems. Here $\\gamma&gt;0$ is a constant associated with either the penalty parameter or the time-step size, and $\\mathcal{A}$ is a singular map. In this paper, we develop a multigrid solver for the discrete problem using both the Raviart\u2013Thomas (RT) and the Brezzi\u2013Douglas\u2013Marini (BDM) finite elements. We show that the multigrid convergence rate is $O(\\frac{1+\\gamma^{-2}}{1+\\gamma^{-2} + m})$, where m is the number of smoothings. This convergence rate is independent of the mesh size and the number of levels used in multigrid. Moreover, numerical results presented in this paper show that the multigrid convergence rate for the BDM elements does not depend on the parameter $\\gamma$.<\/jats:p>","DOI":"10.1137\/060661429","type":"journal-article","created":{"date-parts":[[2007,10,8]],"date-time":"2007-10-08T18:22:50Z","timestamp":1191867770000},"page":"2078-2095","source":"Crossref","is-referenced-by-count":30,"title":["A Multigrid Method for the Pseudostress Formulation of Stokes Problems"],"prefix":"10.1137","volume":"29","author":[{"given":"Zhiqiang","family":"Cai","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yanqiu","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,9,28]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-97-00826-0"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/PL00005386"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01379659"},{"key":"R4","first-page":"169","volume":"15","author":"Arnold D. N.","year":"1988","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100348"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0257(98)00122-0"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(93)90205-C"},{"key":"R8","unstructured":"J. H. Bramble,\n                      Multigrid Methods\n                      , Pitman Res. Notes Math. Ser. 294, Longman Scientific & Technical, Harlow, UK, 1993."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389710"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"F. Brezzi and M. Fortin,\n                      Mixed and Hybrid Finite Element Methods\n                      , Springer-Verlag, New York, 1991.","DOI":"10.1007\/978-1-4612-3172-1"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142903422673"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/S003614290139696X"},{"key":"R13","unstructured":"Z. Cai, C. Tong, P. Vassilevski, and C. Wang,\n                      Mixed Finite Element Methods for Incompressible Flow: Stationary Stokes Equations\n                      , manuscript, 2005."},{"key":"R14","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , North\u2013Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827597324846"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"V. Girault and P. A. Raviart,\n                      Finite Element Methods for Navier\u2013Stokes Equations\n                      , Springer Ser. Comput. Math. 5, Springer-Verlag, New York, 1986.","DOI":"10.1007\/978-3-642-61623-5"},{"key":"R17","unstructured":"P. Grisvard,\n                      Singularities in Boundary Value Problems\n                      , Research in Applied Mathematics 22, Springer-Verlag, Berlin, 1992."},{"key":"R18","doi-asserted-by":"crossref","unstructured":"W. Hackbusch,\n                      Multigrid Methods and Applications\n                      , Springer-Verlag, Berlin, 1985.","DOI":"10.1007\/978-3-662-02427-0"},{"key":"R19","first-page":"133","volume":"6","author":"Hiptmair R.","year":"1997","journal-title":"Electron. Trans. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1097-4067","issn-type":"print"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/BF01403910"},{"key":"R21","first-page":"449","volume":"9","author":"Keunings R.","year":"2001","journal-title":"Comp. Fluid Dyn. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0918-6654","issn-type":"print"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"J. Mandel, S. McCormick, and R. Bank,\n                      Variational multigrid theory\n                      , in Multigrid Methods, S. F. McCormick, ed., SIAM, Philadelphia, 1987, pp. 131\u2013177.","DOI":"10.1137\/1.9781611971057.ch5"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"P. A. Raviart and J. M. Thomas,\n                      A mixed finite element method for 2nd order elliptic problems\n                      , in Mathematical Aspects of Finite Element Methods, Lecture Notes in Math. 606, I. Galligani and E. Magenes, eds., Springer-Verlag, 1977, pp. 292\u2013315.","DOI":"10.1007\/BFb0064470"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385872"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/1034116"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060661429","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:43:21Z","timestamp":1787334201000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060661429"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1]]},"references-count":25,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/060661429"],"URL":"https:\/\/doi.org\/10.1137\/060661429","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1]]}}}