# American Institute of Mathematical Sciences

July  2012, 17(5): 1551-1573. doi: 10.3934/dcdsb.2012.17.1551

## Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows

 1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331 2 Center for Computational Geosciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China 3 Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China

Received  August 2010 Revised  December 2011 Published  March 2012

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.
Citation: Kun Wang, Yinnian He, Yanping Lin. Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1551-1573. doi: 10.3934/dcdsb.2012.17.1551
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##### References:
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