Mathematical Models, Discretizations, and Solution Strategies for Coupling of Free Fluid With Poroelastic Medium

Abstract

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 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.

First, we develop and analyze a novel Banach space formulation of the Navier–Stokes–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 study the iterative version of the algorithm, showing that it converges to a monolithic scheme with a Robin Lagrange multiplier to enforce the continuity of the velocity. Numerical experiments are presented to illustrate the theoretical results.

Lastly, we develop a Robin-Robin split scheme for a fully-mixed formulation of the quasistatic Stokes-Biot model, ensuring local poroelastic and Stokes momentum conservation while maintaining robustness for nearly incompressible materials. 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.