An Augmented Fully-Mixed Formulation for the Navier-Stokes - Biot Model

We present a fully-mixed formulation and a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Navier--Stokes equations and a poroelastic medium modeled by the Biot system. We employ dual-mixed formulations in both domains and impose the transmission conditions weakly by introducing the traces of the structure velocity and the Darcy pressure on the interface as Lagrange multipliers. The fluid velocity is controlled by augmenting 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, together with stability bounds and error analysis with rates of convergence. 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.