{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:49:06Z","timestamp":1776840546519,"version":"3.51.2"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"267","license":[{"start":{"date-parts":[[2010,1,30]],"date-time":"2010-01-30T00:00:00Z","timestamp":1264809600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with a linear equation of state\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho equals p\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c1\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\rho =p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , based on Crouzeix-Raviart elements. The approximation of the momentum balance is obtained by usual finite element techniques. Since the pressure is piecewise constant, the discrete mass balance takes the form of a finite volume scheme, in which we introduce an upwinding of the density, together with two additional stabilization terms. We prove\n                    <italic>a priori<\/italic>\n                    estimates for the discrete solution, which yields its existence by a topological degree argument, and then the convergence of the scheme to a solution of the continuous problem.\n                  <\/p>","DOI":"10.1090\/s0025-5718-09-02216-9","type":"journal-article","created":{"date-parts":[[2009,4,27]],"date-time":"2009-04-27T13:47:33Z","timestamp":1240840053000},"page":"1333-1352","source":"Crossref","is-referenced-by-count":46,"title":["A convergent finite element-finite volume scheme for the compressible Stokes problem. Part I: The isothermal case"],"prefix":"10.1090","volume":"78","author":[{"given":"T.","family":"Gallou\u00ebt","sequence":"first","affiliation":[]},{"given":"R.","family":"Herbin","sequence":"additional","affiliation":[]},{"given":"J.-C.","family":"Latch\u00e9","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,1,30]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"867","DOI":"10.1002\/num.20126","article-title":"Analysis and computation of least-squares methods for a compressible Stokes problem","volume":"22","author":"Bochev, Pavel","year":"2006","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"key":"2","isbn-type":"print","first-page":"11","article-title":"On the stabilization of finite element approximations of the Stokes equations","author":"Brezzi, F.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/3528080841"},{"key":"3","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet, Handbook of numerical analysis volume II: Finite elements methods \u2013 Basic error estimates for elliptic problems, Handbook of Numerical Analysis, Volume II (P. Ciarlet and J.L. Lions, eds.), North Holland, 1991, pp. 17\u2013351.","DOI":"10.1016\/S1570-8659(05)80039-0"},{"key":"4","first-page":"33","article-title":"Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I","volume":"7","author":"Crouzeix, M.","year":"1973","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge"},{"key":"5","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4355-5","volume-title":"Theory and practice of finite elements","volume":"159","author":"Ern, Alexandre","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0387205748"},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1016\/S1570-8659(00)07005-8","article-title":"Finite volume methods","author":"Eymard, Robert","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0444503501"},{"key":"7","unstructured":"R. Eymard and R. Herbin, Entropy estimate for the approximation of the compressible barotropic Navier-Stokes equations using a collocated finite volume scheme, in preparation (2007)."},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1051\/m2an:2006024","article-title":"On a stabilized colocated finite volume scheme for the Stokes problem","volume":"40","author":"Eymard, Robert","year":"2006","journal-title":"M2AN Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"9","series-title":"Oxford Lecture Series in Mathematics and its Applications","isbn-type":"print","volume-title":"Dynamics of viscous compressible fluids","volume":"26","author":"Feireisl, Eduard","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0198528388"},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1051\/m2an:2008005","article-title":"An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations","volume":"42","author":"Gallou\u00ebt, Thierry","year":"2008","journal-title":"M2AN Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"11","unstructured":"L. Gastaldo, R. Herbin, and J.-C. Latch\u00e9, An entropy-preserving finite element\u2013finite volume pressure correction scheme for the drift-flux model, submitted (2008)."},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"780","DOI":"10.1137\/0733039","article-title":"A finite element method for the compressible Stokes equations","volume":"33","author":"Kellogg, R. Bruce","year":"1996","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"1093","DOI":"10.1137\/S0036142994273276","article-title":"A penalized finite-element method for a compressible Stokes system","volume":"34","author":"Kellogg, R. Bruce","year":"1997","journal-title":"SIAM J. Numer. 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