In this paper, we consider the complex flows when all three regimes pre-Darcy, Darcy and post-Darcy may be present in different portions of a same domain. We unify all three flow regimes under mathematics formulation. We describe the flow of a single-phase fluid in $\R^d, d\ge 2$ by a nonlinear degenerate system of density and momentum. A mixed finite element method is proposed for the approximation of the solution of the above system. The stability of the approximations are proved; the error estimates are derived for the numerical approximations for both continuous and discrete time procedures. The continuous dependence of numerical solutions on physical parameters are demonstrated. Experimental studies are presented regarding convergence rates and showing the dependence of the solution on the physical parameters.

翻译：在本文中,当所有三种制度在达西前、达西和达西后都存在于同一领域的不同部分时,我们考虑复杂的流动情况。我们根据数学公式将所有三种流动制度统一在一起。我们用非线性退化密度和动力系统来描述单阶段流体的流量,以美元=d=Ge=2美元。为接近上述系统的解决办法,提出了混合的有限要素法。近似值的稳定性得到了证明;为连续和离散时间程序的数字近似值得出了误差估计数。数字解决方案对物理参数的持续依赖得到了证明。还介绍了关于趋同率和解决方案对物理参数的依赖性的实验研究。