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Unconditional stability and convergence of fully discrete schemes for $ 2D$ viscous fluids models with mass diffusion

Authors: Francisco Guillén-González and Juan Vicente Gutiérrez-Santacreu
Journal: Math. Comp. 77 (2008), 1495-1524
MSC (2000): Primary 35Q35, 65M12, 65M60
Published electronically: February 28, 2008
MathSciNet review: 2398778
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Abstract: 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 $ H^1$-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 $ H^2$ regularity. A backward Euler in time scheme is considered decoupling the computation of the density from the velocity and pressure.

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 Gagliardo-Nirenberg interpolation inequality in $ 2D$ domains. Moreover, the discrete density is truncated in some adequate terms of the velocity-pressure problem.

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Additional Information

Francisco Guillén-González
Affiliation: Departamento E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain

Juan Vicente Gutiérrez-Santacreu
Affiliation: Departamento E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain

Keywords: Kazhikhov-Smagulov models, finite elements, stability, convergence
Received by editor(s): August 24, 2006
Received by editor(s) in revised form: July 10, 2007
Published electronically: February 28, 2008
Additional Notes: The first author was supported in part by the Spanish projects BFM2003–06446-C02-01 and PHB2005-0042-PC
The second author was supported by the Spanish projects BFM2003–06446-C02-01 and PHB2005-0042-PC
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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