{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:44:08Z","timestamp":1776725048592,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"230","license":[{"start":{"date-parts":[[2000,5,17]],"date-time":"2000-05-17T00:00:00Z","timestamp":958521600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this paper, we consider iterative algorithms of Uzawa type for solving linear nonsymmetric saddle point problems. Specifically, we consider systems, written as usual in block form, where the upper left block is an invertible linear operator with positive definite symmetric part. Such saddle point problems arise, for example, in certain finite element and finite difference discretizations of Navier\u2013Stokes equations, Oseen equations, and mixed finite element discretization of second order convection-diffusion problems. We consider two algorithms, each of which utilizes a preconditioner for the operator in the upper left block. Convergence results for the algorithms are established in appropriate norms. The convergence of one of the algorithms is shown assuming only that the preconditioner is spectrally equivalent to the inverse of the symmetric part of the operator. The other algorithm is shown to converge provided that the preconditioner is a sufficiently accurate approximation of the inverse of the upper left block. Applications to the solution of steady-state Navier\u2013Stokes equations are discussed, and, finally, the results of numerical experiments involving the algorithms are presented.<\/p>","DOI":"10.1090\/s0025-5718-99-01152-7","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"667-689","source":"Crossref","is-referenced-by-count":78,"title":["Uzawa type algorithms for nonsymmetric saddle point problems"],"prefix":"10.1090","volume":"69","author":[{"given":"James","family":"Bramble","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joseph","family":"Pasciak","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Apostol","family":"Vassilev","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,5,17]]},"reference":[{"key":"1","series-title":"Stanford Mathematical Studies in the Social Sciences, II","volume-title":"Studies in linear and non-linear programming","author":"Arrow, Kenneth J.","year":"1958"},{"key":"2","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1007\/BF03046994","article-title":"On non-linear partial differential equations of the hyperbolic type","volume":"9","author":"Sundaram, S. Minakshi","year":"1939","journal-title":"Proc. Indian Acad. Sci., Sect. A."},{"issue":"1-2","key":"3","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1016\/S0898-1221(96)00216-7","article-title":"Iterative techniques for time dependent Stokes problems","volume":"33","author":"Bramble, J. H.","year":"1997","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"1072","DOI":"10.1137\/S0036142994273343","article-title":"Analysis of the inexact Uzawa algorithm for saddle point problems","volume":"34","author":"Bramble, James H.","year":"1997","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"5","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and hybrid finite element methods","volume":"15","author":"Brezzi, Franco","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387975829"},{"key":"6","series-title":"Studies in Mathematics and its Applications, Vol. 4","isbn-type":"print","volume-title":"The finite element method for elliptic problems","author":"Ciarlet, Philippe G.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0444850287"},{"key":"7","unstructured":"H. Elman. Preconditioning for the steady-state Navier-Stokes equations with low viscosity. Technical Report CS-TR-3712, Department of Computer Science, University of Maryland, College Park, MD 20742, 1996."},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1137\/0917004","article-title":"Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations","volume":"17","author":"Elman, Howard","year":"1996","journal-title":"SIAM J. Sci. 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New error estimates for approximate solutions of convection-diffusion problems by mixed and discontinuous Galerkin methods. SIAM J. Numer. Anal. Submitted."},{"key":"15","unstructured":"M.F. Murphy and A.J. Wathen. On preconditioning for the Oseen equations. Technical Report AM 95-07, Department of Mathematics, University of Bristol, 1995."},{"issue":"3","key":"16","doi-asserted-by":"publisher","first-page":"856","DOI":"10.1137\/0907058","article-title":"GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems","volume":"7","author":"Saad, Youcef","year":"1986","journal-title":"SIAM J. Sci. Statist. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-5204","issn-type":"print"},{"key":"17","series-title":"Studies in Mathematics and its Applications, Vol. 2","isbn-type":"print","volume-title":"Navier-Stokes equations. Theory and numerical analysis","author":"Temam, Roger","year":"1977","ISBN":"https:\/\/id.crossref.org\/isbn\/0720428408"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2000-69-230\/S0025-5718-99-01152-7\/S0025-5718-99-01152-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-230\/S0025-5718-99-01152-7\/S0025-5718-99-01152-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:16:43Z","timestamp":1776723403000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-230\/S0025-5718-99-01152-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,5,17]]},"references-count":17,"journal-issue":{"issue":"230","published-print":{"date-parts":[[2000,4]]}},"alternative-id":["S0025-5718-99-01152-7"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01152-7","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1999,5,17]]}}}