Mixed Finite Element Methods for the Navier–Stokes–Biot Model

We present two mixed finite element methods for the quasistatic Navier–Stokes–Biot model. The methods are based on a fully-mixed formulation, using pseudostress, velocity, and vorticity for Navier–Stokes, stress, displacement, and rotation for elasticity, and velocity and pressure for Darcy flow. To handle the advective term, augmenting Galerkin-type terms are introduced in the Navier–Stokes model in the first method, while the second method uses a Banach space formulation. We discuss the relative advantages and disadvantages of the two methods and illustrate their behavior with numerical experiments.