Dissertation titled: Mathematical Models, Discretizations, and Solution Strategies for Coupling of Free Fluid with Poroelastic Medium. This thesis focuses on finite element computational models for solving the coupled problem that arises from the interaction between free fluid flow and flow within a deformable poroelastic medium. We adopt the fully-dynamic Stokes or Navier-Stokes equations to model the fluid flow and the fully dynamic Biot system for the flow in the poroelastic material. The two regions are coupled through dynamic and kinematic transmission conditions at the interface, including continuity of normal velocity, balance of fluid force, conservation of momentum, and the Beavers–Joseph–Saffman slip with friction condition.
The thesis consists of three major parts. First, we develop and analyze a new Banach space formulation of the Navier-Stokes and Biot model. Under a small data condition, we establish the existence, uniqueness, and stability of both the continuous and semi-discrete continuous-in-time formulations. Additionally, we provide an error analysis for the semi-discrete continuous-in-time formulation. Finally, we present numerical experiments to verify the theoretical rates of convergence and illustrate the performance of the method for application to flow through a filter.
 Next, we introduce a new Robin-Robin partitioned method for the classical Stokes-Biot problem, which applies Robin boundary conditions on the interface defined by transmission conditions. The splitting method involves single and decoupled Stokes and Biot solves at each time step. The Robin data is represented by an auxiliary interface variable. We prove that the numerical scheme is unconditionally stable and conduct an error analysis, achieving optimal-order convergence for all variables. We further study the iterative version of the algorithm, which involves an iteration between the Stokes and Biot sub-problems at each time step. We prove that the iteration converges to a monolithic scheme with a Robin Lagrange multiplier used to impose the continuity of the velocity. We then perform a series of numerical experiments to validate the theoretical convergence rate over time and evaluate the robustness and accuracy of the numerical schemes using two benchmarks: a test with an exact solution and a simplified blood flow problem.
Lastly, we develop a Robin-Robin split scheme for a fully-mixed formulation  of the quasi-static Stokes-Biot model, ensuring local poroelastic and Stokes momentum conservation while maintaining robustness for nearly incompressible materials. This formulation enables explicit stress computation, making it well-suited for the Robin-Robin scheme. We establish stability and error analysis results for the non-iterative Robin-Robin split scheme and conclude with numerical tests, validating the theoretical findings and illustrating the behavior of the splitting method.
Committee Chair and Advisor: Dr. Ivan Yotov