We study a fully discrete finite element approximation of a model for unsteady flows of rate-type viscoelastic fluids with stress diffusion in two and three dimensions. The model consists of the incompressible Navier--Stokes equation for the velocity, coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus model for the left Cauchy--Green tensor. The discretization of the model is chosen such that an energy inequality is preserved at the fully discrete level. Thus, unconditional solvability and stability for the discrete system are guaranteed and the discrete Cauchy--Green tensor is positive definite. Moreover, subsequences of discrete solutions converge to a global-in-time weak solution, as the discretization parameters tend to zero. In the end, we present numerical convergence tests.

翻译：本文研究了一种二维和三维非定常流动中具有应力扩散的率型黏弹性流体模型的完全离散有限元逼近。该模型由描述速度的不可压缩Navier-Stokes方程，与结合Oldroyd-B模型和Giesekus模型的左Cauchy-Green张量扩散变体耦合而成。所选择的模型离散化方案使得能量不等式在完全离散层面得以保持。因此，离散系统的无条件可解性与稳定性得到保证，且离散Cauchy-Green张量保持正定性。此外，当离散化参数趋于零时，离散解的子序列收敛于全局时间弱解。最后，我们给出了数值收敛性测试结果。