We study a mathematical model of fluid -- poroelastic structure interaction and its numerical solution. The free fluid region is governed by the unsteady incompressible Navier-Stokes equations, while the poroelastic region is modeled by the Biot system of poroelasticity. The two systems are coupled along an interface through continuity of normal velocity and stress and the Beavers-Joseph-Saffman slip with friction condition. The variables in the weak formulation are velocity and pressure for Navier-Stokes, displacement for elasticity and velocity and pressure for Darcy flow. A Lagrange multiplier of stress/pressure type is employed to impose weakly the continuity of flux. Existence, uniqueness, and stability of a weak solution is established under a small data assumption. A fully discrete numerical method is then developed, based on backward Euler time discretization and finite element spatial approximation. We establish solvability, stability, and error estimates for the fully discrete scheme. Numerical experiments are presented to verify the theoretical results and illustrate the performance of the method for an arterial flow application.

翻译：本文研究流体-多孔弹性结构相互作用的数学模型及其数值求解方法。自由流体区域由非定常不可压缩Navier-Stokes方程控制，而多孔弹性区域采用Biot多孔弹性系统进行建模。两个系统通过法向速度与应力的连续性条件以及带摩擦的Beavers-Joseph-Saffman滑移条件沿界面耦合。弱形式中的变量包括Navier-Stokes的流速与压力、弹性位移以及达西流的流速与压力。采用应力/压力型拉格朗日乘子对通量连续性进行弱约束。在小数据假设下，证明了弱解的存在性、唯一性与稳定性。随后基于后向欧拉时间离散与有限元空间近似，建立了全离散数值方法。针对全离散格式，证明了可解性、稳定性并给出了误差估计。通过数值实验验证理论结果，并以动脉血流应用为例展示了该方法的性能。