The Peterlin viscoelastic model describes the motion of certain incompressible polymeric fluids. It employs a nonlinear dumbbell model with a nonlinear spring force law making it more nonlinear than other viscoelastic models. In this paper, we propose and study a fully implicit stabilized Crank-Nicolson time stepping scheme for finite element spatial discretization of the non-stationary Peterlin viscoelastic fluid model with non-homogeneous boundary conditions. The proposed scheme adds a suitable stabilizing term to improve the structural and stability properties of the scheme. We prove that the scheme is almost unconditionally stable, i.e., stable when the time step is less than or equal to a constant. Further, with the help of the a priori error bounds of the Stokes and Ritz projections, optimal error estimates for the velocity, the conformation tensor and the pressure are presented in suitable norms. Numerical examples are presented that illustrate the accuracy and stability of the scheme.

Peterlin粘弹性模型描述了某些不可压缩的聚合物流体的运动。它采用具有非线性弹力定律的非线性哑铃模型，使其比其他粘弹性模型更具非线性。在本文中，我们提出并研究了具有非均匀边界条件的非平稳Peterlin粘弹性流体模型的有限元空间离散化的全隐式稳定Crank-Nicolson时间步长方案。所提出的方案增加了合适的稳定化术语以改善方案的结构和稳定性。我们证明了该方案几乎是无条件稳定的，即在时间步长小于或等于常数时是稳定的。此外，借助Stokes和Ritz投影的先验误差范围，可以对速度进行最佳误差估计，构造张量和压力以合适的范数表示。数值例子表明了该方案的准确性和稳定性。