  In this paper we advance the analysis of discretizations for a
fluid-structure interaction model of the monolithic coupling between the free
flow of a viscous Newtonian fluid and a deformable porous medium separated by
an interface. A five-field mixed-primal finite element scheme is proposed
solving for Stokes velocity-pressure and Biot displacement-total pressure-fluid
pressure. Adequate inf-sup conditions are derived, and one of the distinctive
features of the formulation is that its stability is established robustly in
all material parameters. We propose robust preconditioners for this perturbed
saddle-point problem using appropriately weighted operators in fractional
Sobolev and metric spaces at the interface. The performance is corroborated by
several test cases, including the application to interfacial flow in the brain.
