{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:21:27Z","timestamp":1787329287087,"version":"build-2736575974"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We construct a fully discrete numerical scheme for three-dimensional incompressible fluids with mass diffusion (in density-velocity-pressure formulation), also called the Kazhikhov\u2013Smagulov model. We will prove conditional stability and convergence, by using at most $C^0$-finite elements, although the density of the limit problem will have $H^2$-regularity. The key idea of our argument is first to obtain pointwise estimates for the discrete density by imposing the constraint $\\lim_{(h,k)\\to 0}h\/k=0$ on the time and space parameters $(k,h)$. Afterwards, under the same constraint on the parameters, strong estimates for the discrete density in $l^\\infty(H^1)$ and for the discrete Laplacian of the density in $l^2(L^2)$ are obtained. From here, the compactness and convergence of the scheme can be concluded with similar arguments as we used in [Math. Comp., to appear], where a different scheme is studied for two-dimensional domains which is unconditionally stable and convergent. Moreover, we study the asymptotic behavior of the numerical scheme as the diffusion parameter $\\lambda$ goes to zero, obtaining convergence as $(k,h,\\lambda)\\to 0$ towards a weak solution of the density-dependent Navier\u2013Stokes system provided that the constraint $\\lim_{(\\lambda,h,k)\\to 0}h\/(\\lambda^2 k)=0$ on $(h,k,\\lambda)$ is satisfied.<\/jats:p>","DOI":"10.1137\/07067951x","type":"journal-article","created":{"date-parts":[[2008,5,12]],"date-time":"2008-05-12T09:52:42Z","timestamp":1210585962000},"page":"2276-2308","source":"Crossref","is-referenced-by-count":10,"title":["Conditional Stability and Convergence of a Fully Discrete Scheme for Three-Dimensional Navier\u2013Stokes Equations with Mass Diffusion"],"prefix":"10.1137","volume":"46","author":[{"given":"F.","family":"Guill\u00e9n-Gonz\u00e1lez","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. V.","family":"Guti\u00e9rrez-Santacreu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,5,9]]},"reference":[{"key":"R1","unstructured":"S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov,\n                      Boundary Value Problems in Mechanics of Nonhomogeneous Fluids\n                      , Stud. Math. Appl.  22, North-Holland, Amsterdam, 1990."},{"key":"R2","first-page":"341","volume":"10","author":"da Veiga H. Beri\u00e3o","year":"1983","journal-title":"Ann. Sc. Norm. Super. Pisa"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"S. Brenner and L. R. Scott,\n                      The Mathematical Theory of Finite Element Methods\n                      , Texts Appl. Math. 15, Springer-Verlag, Berlin, 1994.","DOI":"10.1007\/978-1-4757-4338-8"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/S1631-073X(02)02593-1"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/s00021-005-0204-4"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2005.10.005"},{"key":"R7","unstructured":"P. G. Ciarlet,\n                      The Tinite Element Method for Elliptic Problems\n                      , North-Holland, Amsterdam, 1987."},{"key":"R8","doi-asserted-by":"crossref","unstructured":"V. Girault and P. A. Raviart,\n                      Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms\n                      , Springer-Verlag, Berlin, 1986.","DOI":"10.1007\/978-3-642-61623-5"},{"key":"R9","unstructured":"F. Guill\u00e9n-Gonz\u00e1lez,\n                      Sobre un modelo asint\u00f3tico de difusi\u00f3n de masa para fluidos incompresibles, viscoso y no homog\u00e9neos\n                      , in Proceedings of the Third Catalan Days on Applied Mathematics, Lleida, Spain, 1996, pp. 103\u2013114."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2006.03.009"},{"key":"R11","unstructured":"F. Guill\u00e9n-Gonz\u00e1lez, and J. V. Guti\u00e9rrez-Santacreu,\n                      Unconditional stability and convergence of fully discrete schemes for $2D$ viscous fluids models with mass diffusion\n                      , Math. Comp., to appear."},{"key":"R12","first-page":"249","volume":"22","author":"Kazhikhov A.","year":"1977","journal-title":"Sov. Phys. Dokl.","ISSN":"https:\/\/id.crossref.org\/issn\/0038-5689","issn-type":"print"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/050629008"},{"key":"R14","first-page":"213","volume":"43","author":"Salvi R.","year":"1986","journal-title":"Port. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0032-5155","issn-type":"print"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/0519002"},{"key":"R16","first-page":"117","volume":"6","author":"Secchi P.","year":"1982","journal-title":"Boll. Unione Mat. Ital.","ISSN":"https:\/\/id.crossref.org\/issn\/0041-7084","issn-type":"print"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/BF01762360"},{"key":"R18","unstructured":"R. Temam,\n                      Navier-Stokes Equations. Theory and Numerical Analysis\n                      , North-Holland, Amsterdam, 1977."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/07067951X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:14:04Z","timestamp":1787325244000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/07067951X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":18,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/07067951X"],"URL":"https:\/\/doi.org\/10.1137\/07067951x","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}