Stability and convergence analysis of stabilized finite element method for the Kelvin-Voigt viscoelastic fluid flow model

Abstract

In this paper, we consider the Galerkin finite element method (FEM) for the Kelvin-Voigt viscoelastic fluid flow model with the lowest equal-order pairs. In order to overcome the restriction of the so-called inf-sup conditions, a pressure projection method based on the differences of two local Gauss integrations is introduced. Under some suitable assumptions on the initial data and forcing function, we firstly present some stability and convergence results of numerical solutions in spatial discrete scheme. By constructing the dual linearized Kelvin-Voigt model, stability and optimal error estimates of numerical solutions in various norms are established. Secondly, a fully discrete stabilized FEM is introduced, the backward Euler scheme is adopted to treat the time derivative terms, the implicit scheme is used to deal with the linear terms and semi-implicit scheme is applied to treat the nonlinear term, unconditional stability and convergence results are also presented. Finally, some numerical examples are presented to verify the developed theoretical analysis and show the performances of the considered numerical schemes.