A priori error estimates of fully discrete finite element Galerkin method for Kelvin-Voigt viscoelastic fluid flow model

Abstract

In this article, a finite element Galerkin method is applied to the Kelvin-Voigt viscoelastic fluid model, when its forcing function is in $L^{\infty}(\bL^2)$. Some new {\it a priori} bounds for the velocity as well as for the pressure are derived which are independent of inverse powers of the retardation time $κ$. Optimal error estimates for the velocity in $L^{\infty} (\bL^2)$ as well as in $L^{\infty}(\bH^1_0)$-norms and for the pressure in $L^{\infty}(L^2)$-norm of the semidiscrete method are discussed which hold uniformly with respect to $κ$ as $κ\rightarrow 0$ with the initial condition only in $\bH^2\cap\bH_0^1$. Further, under uniqueness condition, these estimates are shown to be uniformly in time as $t \mapsto \infty$. For the complete discretization of the semidiscrete system, a first-order accurate backward Euler method is applied and fully discrete optimal error estimates are established. Finally, numerical experiments are conducted to verify the theoretical results. The results derived in this article are sharper than those derived earlier for finite element analysis of the Kelvin-Voigt fluid model in the sense that the error estimates in this article hold true uniformly even as $κ\rightarrow 0$.