In this paper, we propose and analyze a mixed formulation for the Kelvin-Voigt-Brinkman-Forchheimer equations for unsteady viscoelastic flows in porous media. Besides the velocity and pressure, our approach introduces the vorticity as a further unknown. Consequently, we obtain a three-field mixed variational formulation, where the aforementioned variables are the main unknowns of the system. We establish the existence and uniqueness of a solution for the weak formulation, and derive the corresponding stability bounds, employing a fixed-point strategy, along with monotone operators theory and Schauder theorem. Afterwards, we introduce a semidiscrete continuous-in-time approximation based on stable Stokes elements for the velocity and pressure, and continuous piecewise polynomial spaces for the vorticity. Additionally, employing backward Euler time discretization, we introduce a fully discrete finite element scheme. We prove well-posedness, derive stability bounds, and establish the corresponding error estimates for both schemes. We provide several numerical results verifying the theoretical rates of convergence and illustrating the performance and flexibility of the method for a range of domain configurations and model parameters.

翻译：本文针对多孔介质中非定常黏弹性流动的Kelvin-Voigt-Brinkman-Forchheimer方程，提出并分析了一种混合变分公式。除速度和压力外，本方法还将涡量作为新的未知量引入。由此，我们得到了一个以速度、压力和涡量为主要未知量的三场混合变分公式。通过不动点策略，结合单调算子理论和Schauder定理，我们证明了该弱解形式解的存在唯一性，并推导了相应的稳定性界。随后，我们提出了一种半离散连续时间近似格式，其中速度和压力采用稳定的Stokes单元离散，涡量采用连续分段多项式空间离散。此外，通过后向欧拉时间离散，我们构建了全离散有限元格式。我们证明了两种格式的适定性，推导了稳定性界，并建立了相应的误差估计。我们提供了多组数值结果，验证了理论收敛阶，并展示了该方法在不同区域构型和模型参数下的性能与灵活性。