We introduce and analyze a partially augmented fully-mixed formulation and a mixed finite element method for the coupled problem arising in the interaction between a free fluid and a poroelastic medium. The flows in the free fluid and poroelastic regions are governed by the Navier-Stokes and Biot equations, respectively, and the transmission conditions are given by mass conservation, balance of fluid force, conservation of momentum, and the Beavers-Joseph-Saffman condition. We apply dual-mixed formulations in both domains, where the symmetry of the Navier-Stokes and poroelastic stress tensors is imposed in an ultra-weak and weak sense. In turn, since the transmission conditions are essential in the fully mixed formulation, they are imposed weakly by introducing the traces of the structure velocity and the poroelastic medium pressure on the interface as the associated Lagrange multipliers. Furthermore, since the fluid convective term requires the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. Existence and uniqueness of a solution are established for the continuous weak formulation, as well as a semidiscrete continuous-in-time formulation with non-matching grids, together with the corresponding stability bounds and error analysis with rates of convergence. Several numerical experiments are presented to verify the theoretical results and illustrate the performance of the method for applications to arterial flow and flow through a filter.

翻译：本文针对自由流体与多孔弹性介质相互作用产生的耦合问题，引入并分析了一种部分增广的全混合格式及混合有限元方法。自由流体区域与多孔弹性区域中的流动分别由Navier-Stokes方程和Biot方程控制，传递条件由质量守恒、流体力平衡、动量守恒以及Beavers-Joseph-Saffman条件给出。我们在两个区域均采用对偶混合格式，其中Navier-Stokes张量与多孔弹性应力张量的对称性分别以超弱形式和弱形式施加。由于传递条件在全混合格式中本质性地存在，我们通过引入界面上的结构速度迹线与多孔弹性介质压力迹线作为相应拉格朗日乘子，以弱形式施加这些条件。此外，由于流体对流项要求速度位于比常规更小的空间中，我们通过添加合适的Galerkin型项对变分格式进行增广。建立了连续弱格式解的存在唯一性，以及基于非匹配网格的半离散连续时间格式解的存在唯一性，并给出了相应的稳定性界与含收敛速率的误差分析。通过多个数值实验验证了理论结果，并展示了该方法在动脉血流及过滤器流动中的应用性能。