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### Open Access Journals

NACO

In this note, the continuity results of weak vector solutions and global vector solutions to a parametric generalized Ky Fan inequality are established by using a new scalarization method. Our results improve the corresponding ones of Li and Fang (J. Optim. Theory Appl. 147: 507-515, 2010).

DCDS-B

In this article, we study the long time numerical stability and asymptotic behavior for the viscoelastic Oldroyd fluid motion equations. Firstly, with the Euler semi-implicit scheme for the temporal discretization, we deduce the global $H^2-$stability result for the fully discrete finite element solution. Secondly, based on the uniform stability of the numerical solution, we investigate the discrete asymptotic behavior and claim that the viscoelastic Oldroyd problem converges to the stationary Navier-Stokes flows if the body force $f(x,t)$ approaches to a steady-state $f_\infty(x)$ as $t\rightarrow\infty$. Finally, some numerical experiments are given to verify the theoretical predictions.

keywords:
Oldroyd model
,
Viscoelastic flows
,
asymptotic analysis.
,
$H^2-$stability
,
long time behavior

DCDS-B

In this article, a fully discrete finite element method is
considered for the viscoelastic fluid motion equations arising in
the two-dimensional Oldroyd model. A finite element method is
proposed for the spatial discretization and the time discretization
is based on the backward Euler scheme. Moreover, the stability and
optimal error estimates in the $L^2$- and $H^1$-norms for the
velocity and $L^2$-norm for the pressure are derived for all time
$t>0.$ Finally, some numerical experiments are shown to verify the
theoretical predictions.

DCDS

In this paper, the asymptotic analysis of the two-dimensional viscoelastic Oldroyd flows is presented. With the physical constant $\rho/\delta$ approaches zero, where $\rho$ is the viscoelastic coefficient and $1/\delta$ the relaxation time, the viscoelastic Oldroyd fluid motion equations converge to the viscous model known as the famous Navier-Stokes equations. Both the continuous and discrete uniform-in-time asymptotic errors are provided. Finally, the theoretical predictions are confirmed by some numerical experiments.

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