Backward Euler Schemes for the Kelvin-Voigt Viscoelastic Fluid Flow Model

Abstract

In this paper, we discuss the backward Euler method along with its linearized version for the Kelvin-Voigt viscoelastic fluid flow model with non zero forcing function, which is either independent of time or in $\rm{L}^∞(\rm{L^2})$. After deriving some bounds for the semidiscrete scheme, a priori estimates in Dirichlet norm for the fully discrete scheme are obtained, which are valid uniformly in time using a combination of discrete Gronwall's lemma and Stolz-Cesaro's classical result for sequences. Moreover, an existence of a discrete global attractor for the discrete problem is established. Further, optimal a priori error estimates are derived, whose bounds may depend exponentially in time. Under uniqueness condition, these estimates are shown to be uniform in time. Even when $\rm{f}$ = 0, the present result improves upon earlier result of Bajpai et al. (IJNAM,10 (2013), pp.481-507) in the sense that error bounds in this article depend on $1 / \sqrt{\kappa}$ as against $1 / \kappa^r$, $r \geq 1$. Finally, numerical experiments are conducted which confirm our theoretical findings.