Long Time Stabilities And Error Estimates Of The Finite Element Method For The Viscoelastic Oldroyd Fluid Motion Equations

Viscoelastic Oldroyd ?uid motion problem is one of the classical Non-Newtonian ?uidmodels. This problem is used to describe polymeric ?uids, biological ?uids, suspensionsand so on. It can be considered as a perturbed Navier-Stokes problem by a integralterm. The equations not only inherit lots of characteristics of the Navier-Stokes equa-tions, but also contain the memory term, which designs the viscoelastic property. Com-pared with the Navier-Stokes equations, this model is more complicated. The researchon this problem is very limited, especially in mathematics. For the numerical approxima-tion of the problem, the investigation in the literatures all focuses on the semi-discreteschemes(spatial semi-discrete schemes and time semi-discrete schemes), so there are alot of work to do on this topic. Moreover, there is no numerical experiment in the litera-tures to verified the numerical analysis. On the other hand, it is also very interesting toinvestigate the long time behavior of the numerical scheme for the problem, which cangive a better test on the algorithm's performance. For the Navier-Stokes equations, mostof the existent analysis is done on the finite time interval. There is little research onthe long time behavior of the approximation scheme. To prove the long time stabilitiesand error estimates of the numerical algorithm for the viscoelastic Oldroyd problem isalso an interesting field. Furthermore, for the viscoelastic Oldroyd equations, to inves-tigate the asymptotic behavior of the numerical scheme and the relationship with theNavier-Stokes equations are attractive, too. In the literature, the author only analyzedthe asymptotic behavior of the continuous model. What will happen to the problem'sasymptotic behavior when the system is approximated by the numerical scheme? Thereis almost no report on this topic.For the research of this paper, firstly, in Section 3, we extend the numerical analysisof the viscoelastic Oldroyd ?uid motion equations to the fully discrete finite element scheme. By dealing with the time interval by part and using the energy technique,we prove the uniform stabilities and error estimates. At the end of this section, weperform some numerical experiments for the model. They not only verify the numericalanalysis, but also show the di?erence between it and the Navier-Stokes equations. Next,in Section 4, under the assumptions of a small viscoelastic coe?cient or relax time, weconsider the asymptotic behavior of the problem. The results show that the viscoelasticOldroyd model converges to the Navier-Stokes equations in these cases. We also derivethe uniform optimal asymptotic error estimate, which shows exactly the relationshipof the two di?erent models. Moreover, based on the long time stability of the finiteelement solution deduced in Section 3, we investigate the asymptotic behavior of thefinite element approximation for the problem and obtain similar error estimates as thatin the continuous case. Then, in Section 5 of this paper, we analyze the asymptoticbehavior of the problem if the body force approaches zero as the time approaches infinity.Under some weaker assumptions than that in other literatures, we derive the asymptoticerror estimate with respect to the body force, which can show better the procedure thatthe ?uid tends to the steady state. Similarly, we derive the asymptotic error estimateof the discrete scheme. As a classical decouple method, in Section 6, we investigate thepenalty method of the viscoelastic Oldroyd equations. By using some techniques, wederive the uniform optimal error estimate with respect to the time. Then, we extend theanalysis to the finite element discrete scheme. Finally, we give a plan of the further workin Section 7.