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.

在本文中，我们考虑具有最小等序对的Kelvin-Voigt粘弹性流体模型的Galerkin有限元方法（FEM）。为了克服所谓的inf-sup条件的限制，引入了基于两个局部高斯积分的差异的压力投影方法。在对初始数据和强迫函数进行适当假设的基础上，首先给出了空间离散方案数值解的稳定性和收敛性结果。通过构造对偶线性化的Kelvin-Voigt模型，建立了各种规范中数值解的稳定性和最优误差估计。其次，引入了完全离散的稳定有限元，采用后向欧拉方案来处理时间导数项，隐式方案用于处理线性项，半隐式方案用于处理非线性项，并给出了无条件稳定性和收敛性。最后，给出了一些数值例子，以验证所开发的理论分析并显示所考虑数值方案的性能。