A Nitsche method for Navier--Stokes/generalized poroelasticity interface problems

Technical Summary: A Nitsche Method for Navier–Stokes/Generalized Poroelasticity Interface Problems

Problem Statement
This work addresses the mathematical and numerical modeling of the interaction between an incompressible free-fluid and a deformable porous medium, a phenomenon known as Fluid–Poroelastic Structure Interaction (FPSI). The free-fluid region is governed by the time-dependent Navier–Stokes equations, while the poroelastic medium is modeled using a generalized, linearized poro-hyperelastic system (incorporating the Brinkman model for viscous effects within the porous matrix). The two domains are coupled through a set of physical interface conditions: continuity of normal velocity, balance of normal stresses, balance of contact forces, and the Beavers–Joseph–Saffman (BJS) slip-with-friction condition for tangential flow. The authors focus on the challenge of enforcing these interface constraints within a finite element (FE) framework, particularly when dealing with non-matching meshes at the interface.

Methodology
The paper proposes a unified, monolithic finite element discretization based on implicit time stepping (specifically the backward Euler method). To handle the fluid-structure interface without introducing additional unknowns (such as Lagrange multipliers), the authors employ a Nitsche-type formulation. This approach weakly enforces the continuity of normal velocity across the interface using a penalty parameter $\gamma$.

Key methodological components include:

• Weak Formulation: The continuous problem is cast into a variational form involving bilinear forms for the Navier–Stokes, Brinkman, and elasticity operators, alongside interface terms derived from the BJS conditions and the Nitsche penalty.
• Discretization: The spatial discretization uses standard finite element spaces (polynomials of degree $k$ and $k+1$) for velocities, pressures, and displacements. The time discretization utilizes the backward Euler scheme.
• Theoretical Analysis:
  • Well-posedness: The existence and uniqueness of the semi-discrete solution are established using the theory of Differential-Algebraic Equations (DAEs) and the Banach fixed-point theorem. This analysis relies on specific conditions regarding the penalty parameter $\gamma$ and the magnitude of the external forces.
  • Stability: A comprehensive stability analysis is performed for the fully discrete scheme, deriving energy estimates that ensure the solution remains bounded.
  • Error Estimates: A priori error estimates are derived for the fully discrete scheme, demonstrating convergence rates dependent on the mesh size $h$ and time step $\tau$. The analysis explicitly details the dependence of these estimates on the stabilization parameter $\gamma$.

Key Contributions
The authors identify their work as the first to address the monolithic discretization of Navier–Stokes/generalized poroelasticity interface problems specifically using Nitsche's method. Their primary contributions are:

1. Extension of Analysis: The work extends previous studies on Stokes–generalized poroelasticity by incorporating the nonlinear convective term of the Navier–Stokes equations and applying the Nitsche method to weakly enforce normal velocity continuity.
2. Theoretical Foundation: They establish the well-posedness of the semi-discrete problem and provide a complete convergence analysis for the fully discrete scheme, including stability and error estimates.
3. Numerical Validation: The paper presents numerical experiments to validate the theoretical convergence rates and demonstrate the method's capability to capture coupled dynamics.

Numerical Results
The authors present three sets of numerical tests:

1. Convergence Tests: Using manufactured solutions, the method is tested for spatial and temporal convergence. The results confirm the theoretical convergence rates, showing optimal convergence for most variables and sub-optimal (but acceptable) rates for relative velocity in specific configurations.
2. 2D Channel Flow: A simulation of flow in a channel with rigid obstacles situated between porous layers. The results illustrate the formation of recirculation zones, pressure drops across the interface, and solid deformation, highlighting the strong coupling between the fluid and the poroelastic medium.
3. 3D Microfluidic Simulation: A three-dimensional simulation of blood flow through a microfluidic chip containing cylindrical poroelastic obstacles. This test demonstrates the method's stability and accuracy in complex 3D geometries, capturing the deformation of the obstacles and the associated pressure and velocity fields.

Significance and Claims
The paper claims that the proposed Nitsche-based approach offers a robust and computationally efficient alternative to Lagrange multiplier methods. By avoiding the introduction of extra unknowns, the method maintains the symmetry (or skew-symmetry) of the algebraic system and simplifies implementation, particularly for non-matching meshes. The authors assert that the stability of the scheme is ensured by a penalty parameter that is independent of the mesh size, thereby avoiding the need for additional inf-sup stability conditions often required in multiplier-based approaches.

The work concludes that the method successfully captures the complex interplay between free-fluid dynamics and poroelastic responses. The authors note that while the current study focuses on the linearized poro-hyperelastic model, future work could extend this framework to fully nonlinear regimes and other transmission conditions to address a broader class of poromechanical problems.