{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:09:39Z","timestamp":1776791379279,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"270","license":[{"start":{"date-parts":[[2010,12,8]],"date-time":"2010-12-08T00:00:00Z","timestamp":1291766400000},"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 an equation of state of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p equals rho Superscript gamma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03c1\n                                \n                              <\/mml:mi>\n                              <mml:mi>\n                                \u03b3\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p=\\rho ^\\gamma<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    stands for the pressure and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"rho\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c1\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\rho<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for the density). This scheme is based on Crouzeix-Raviart approximation spaces. The discretization of the momentum balance is obtained by the usual finite element technique. The discrete mass balance is obtained by a finite volume scheme, with an upwinding of the density, and two additional stabilization terms. We prove\n                    <italic>a priori<\/italic>\n                    estimates for the discrete solution, which yield its existence. Then the convergence of the scheme to a solution of the continuous problem is established. The passage to the limit in the equation of state requires the a.e. convergence of the density. It is obtained by adapting at the discrete level the \u201ceffective viscous pressure lemma\u201d of the theory of compressible Navier-Stokes equations.\n                  <\/p>","DOI":"10.1090\/s0025-5718-09-02310-2","type":"journal-article","created":{"date-parts":[[2010,2,4]],"date-time":"2010-02-04T11:19:48Z","timestamp":1265282388000},"page":"649-675","source":"Crossref","is-referenced-by-count":30,"title":["A convergent finite element-finite volume scheme for the compressible Stokes problem. Part II: the isentropic case"],"prefix":"10.1090","volume":"79","author":[{"given":"R.","family":"Eymard","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"T.","family":"Gallou\u00ebt","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R.","family":"Herbin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"Latch\u00e9","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2009,12,8]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1142\/S0218202503002544","article-title":"A proof of the inf-sup condition for the Stokes equations on Lipschitz domains","volume":"13","author":"Bramble, James H.","year":"2003","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"key":"2","series-title":"Handbook of Numerical Analysis, II","isbn-type":"print","volume-title":"Handbook of numerical analysis. Vol. II","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0444703659"},{"key":"3","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":"4","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":"5","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, 2008."},{"issue":"2","key":"6","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":"7","doi-asserted-by":"crossref","unstructured":"T. Gallou\u00ebt, R. Herbin, and J.-C. Latch\u00e9. A convergent finite element\u2013finite volume scheme for the compressible Stokes problem. Part I - The isothermal case. to appear in Mathematics of Computation, 2009.","DOI":"10.1090\/S0025-5718-09-02216-9"},{"key":"8","doi-asserted-by":"crossref","unstructured":"L. Gastaldo, R. Herbin, and J.-C. Latch\u00e9. A discretization of phase mass balance in fractional step algorithms for the drift-flux model. to appear in IMA Journal of Numerical Analysis, 2009.","DOI":"10.1093\/imanum\/drp006"},{"key":"9","doi-asserted-by":"crossref","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, 2009.","DOI":"10.1051\/m2an\/2010002"},{"key":"10","doi-asserted-by":"crossref","unstructured":"M. Jobelin, B. Piar, P. Angot, and J.-C. Latch\u00e9. Une m\u00e9thode de p\u00e9nalit\u00e9-projection pour les \u00e9coulements dilatables. Revue Europ\u00e9ene de m\u00e9canique num\u00e9rique, 17:453\u2013480, 2008.","DOI":"10.3166\/remn.17.153-480"},{"key":"11","series-title":"Oxford Lecture Series in Mathematics and its Applications","isbn-type":"print","volume-title":"Mathematical topics in fluid mechanics. Vol. 2","volume":"10","author":"Lions, Pierre-Louis","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0198514883"},{"key":"12","series-title":"Oxford Lecture Series in Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198530848.001.0001","volume-title":"Introduction to the mathematical theory of compressible flow","volume":"27","author":"Novotn\u00fd, A.","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0198530846"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02310-2\/S0025-5718-09-02310-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02310-2\/S0025-5718-09-02310-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:22:48Z","timestamp":1776788568000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02310-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,12,8]]},"references-count":12,"journal-issue":{"issue":"270","published-print":{"date-parts":[[2010,4]]}},"alternative-id":["S0025-5718-09-02310-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02310-2","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":[[2009,12,8]]}}}