{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:12:58Z","timestamp":1776787978155,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"263","license":[{"start":{"date-parts":[[2009,2,28]],"date-time":"2009-02-28T00:00:00Z","timestamp":1235779200000},"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>\n                    In this work we develop fully discrete (in time and space) numerical schemes for two-dimensional incompressible fluids with mass diffusion, also so-called Kazhikhov-Smagulov models. We propose at most\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -conformed finite elements (only globally continuous functions) to approximate all unknowns (velocity, pressure and density), although the limit density (solution of continuous problem) will have\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    regularity. A backward Euler in time scheme is considered decoupling the computation of the density from the velocity and pressure.\n                  <\/p>\n                  <p>\n                    Unconditional stability of the schemes and convergence towards the (unique) global in time weak solution of the models is proved. Since a discrete maximum principle cannot be ensured, we must use a different interpolation inequality to obtain the strong estimates for the discrete density, from the used one in the continuous case. This inequality is a discrete version of the\n                    <italic>Gagliardo-Nirenberg<\/italic>\n                    interpolation inequality in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 upper D\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>D<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2D<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    domains. Moreover, the discrete density is truncated in some adequate terms of the velocity-pressure problem.\n                  <\/p>","DOI":"10.1090\/s0025-5718-08-02099-1","type":"journal-article","created":{"date-parts":[[2008,4,22]],"date-time":"2008-04-22T17:24:54Z","timestamp":1208885094000},"page":"1495-1524","source":"Crossref","is-referenced-by-count":14,"title":["Unconditional stability and convergence of fully discrete schemes for 2\ud835\udc37 viscous fluids models with mass diffusion"],"prefix":"10.1090","volume":"77","author":[{"given":"Francisco","family":"Guill\u00e9n-Gonz\u00e1lez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juan","family":"Guti\u00e9rrez-Santacreu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2008,2,28]]},"reference":[{"key":"1","series-title":"Studies in Mathematics and its Applications","isbn-type":"print","volume-title":"Boundary value problems in mechanics of nonhomogeneous fluids","volume":"22","author":"Antontsev, S. N.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0444883827"},{"issue":"2","key":"2","first-page":"341","article-title":"Diffusion on viscous fluids. Existence and asymptotic properties of solutions","volume":"10","author":"Beir\u00e3o da Veiga, H.","year":"1983","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)","ISSN":"https:\/\/id.crossref.org\/issn\/0391-173X","issn-type":"print"},{"issue":"11","key":"3","doi-asserted-by":"publisher","first-page":"973","DOI":"10.1016\/S1631-073X(02)02593-1","article-title":"De nouveaux syst\u00e8mes de type Kazhikhov-Smagulov: mod\u00e8les de propagation de polluants et de combustion \u00e0 faible nombre de Mach","volume":"335","author":"Bresch, Didier","year":"2002","journal-title":"C. R. Math. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/1631-073X","issn-type":"print"},{"key":"4","unstructured":"D. Bresch, E. H. Essoufi, M. Sy. Effects of density dependent viscosities on multiphasic incompressible fluid models, J. Math. Fluid Mech., DOI 10.1007\/s00021-005-0204-4."},{"key":"5","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":"6","doi-asserted-by":"crossref","unstructured":"J. \u00c9tienne, E. J. Hopfinger, P. Saramito. Numerical simulations of high density ratio lock-exchange flows. Phys. Fluids 17, 036601 (2005).","DOI":"10.1063\/1.1849800"},{"issue":"12","key":"7","doi-asserted-by":"publisher","first-page":"769","DOI":"10.1016\/j.crma.2005.10.005","article-title":"Estimations d\u2019erreur a priori de la m\u00e9thode de Lagrange-Galerkin pour les syst\u00e8mes de type Kazhikhov-Smagulov","volume":"341","author":"\u00c9tienne, Jocelyn","year":"2005","journal-title":"C. R. Math. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/1631-073X","issn-type":"print"},{"key":"8","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61623-5","volume-title":"Finite element methods for Navier-Stokes equations","volume":"5","author":"Girault, Vivette","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/3540157964"},{"key":"9","unstructured":"V. Girault, F. Guill\u00e9n-Gonz\u00e1lez. Mixed formulation, approximation and decoupling algorithm for a nematic liquid crystals model.  In preparation."},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"468","DOI":"10.1016\/j.jmaa.2006.03.009","article-title":"Approximation by an iterative method for regular solutions for incompressible fluids with mass diffusion","volume":"326","author":"Guill\u00e9n-Gonz\u00e1lez, F.","year":"2007","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"key":"11","unstructured":"F. Guill\u00e9n-Gonz\u00e1lez, M. Sy. An iterative method for mass diffusion model with density dependent viscosity. Submitted."},{"key":"12","unstructured":"A. Kazhikhov, Sh. Smagulov. The correctness of boundary value problems in a diffusion model of an inhomogeneous fluid. Sov. Phys. Dokl., 22, (1977), No. 1, 249\u2013252."},{"issue":"2","key":"13","first-page":"213","article-title":"On the existence of weak solutions of boundary value problems in a diffusion model for an inhomogeneous liquid in regions with moving boundaries","volume":"43","author":"Salvi, Rodolfo","year":"1985","journal-title":"Portugal. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0032-5155","issn-type":"print"},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1137\/0519002","article-title":"On the motion of viscous fluids in the presence of diffusion","volume":"19","author":"Secchi, Paolo","year":"1988","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"3","key":"15","first-page":"1117","article-title":"On the initial value problem for the equations of motion of viscous incompressible fluids in the presence of diffusion","volume":"1","author":"Secchi, Paolo","year":"1982","journal-title":"Boll. Un. Mat. Ital. B (6)"},{"key":"16","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/BF01762360","article-title":"Compact sets in the space \ud835\udc3f^{\ud835\udc5d}(0,\ud835\udc47;\ud835\udc35)","volume":"146","author":"Simon, Jacques","year":"1987","journal-title":"Ann. Mat. Pura Appl. (4)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-4622","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\/2008-77-263\/S0025-5718-08-02099-1\/S0025-5718-08-02099-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2008-77-263\/S0025-5718-08-02099-1\/S0025-5718-08-02099-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:22:00Z","timestamp":1776784920000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2008-77-263\/S0025-5718-08-02099-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,2,28]]},"references-count":17,"journal-issue":{"issue":"263","published-print":{"date-parts":[[2008,7]]}},"alternative-id":["S0025-5718-08-02099-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-08-02099-1","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":[[2008,2,28]]}}}