A gauge-Uzawa finite element method for the time-dependent Viscoelastic Oldroyd flows

In this paper, we present the gauge-Uzawa finite element method for the time dependent Viscoelastic Oldroyd flow. This method is a fully discrete projection method, whereas most projection methods have been studied without space discretization and the ensuing analysis may not extend to this case. We also give the stability analysis and the optimal error analysis. The analysis is based on an iterated time-discrete system, with which the error function is split into a temporal error and a spatial error. In order to show the effect of the method, we present some numerical results. The numerical results confirm our analysis and show clearly the stability and optimal convergence of the gauge-Uzawa finite element method for the Viscoelastic Oldroyd flow.