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∞(L2). Some new 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. The second order error estimate in L∞(L2)-norm, the first order error estimate in L∞(H01)-norm for the velocity and the first order error estimate in L∞(L2)-norm for the pressure for the semidiscrete method are derived which hold uniformly with respect to κ as κ→0 with the initial condition only in H2∩H01. Further under the smallness assumption on the data, these error estimates are shown to be uniform in time as t↦∞. For the complete discretization of the semidiscrete system, a first-order accurate backward Euler method is applied and fully discrete 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→0.