Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Lagrange multiplier formulation for the fully dynamic Navier-Stokes-Biot system

Xing Wang, Ivan Yotov

TL;DR

The paper develops a rigorous framework for the fully dynamic Navier–Stokes–Biot fluid–poroelastic interaction, introducing a stress/pressure Lagrange multiplier to enforce flux continuity at the fluid–poroelastic interface. It proves existence, uniqueness, and stability of a divergence-free weak formulation, then designs and analyzes a fully discrete finite element method with backward Euler time stepping and standard mixed spaces, obtaining optimal-order error estimates under a small-data condition. The analysis combines Galerkin approximations, ODE reductions, energy methods, and compactness arguments, supported by discrete inf-sup results. Numerical tests validate the theoretical convergence rates and demonstrate the method’s applicability to arterial-flow-like FPSI problems with challenging parameter regimes.

Abstract

We study a mathematical model of fluid -- poroelastic structure interaction and its numerical solution. The free fluid region is governed by the unsteady incompressible Navier-Stokes equations, while the poroelastic region is modeled by the Biot system of poroelasticity. The two systems are coupled along an interface through continuity of normal velocity and stress and the Beavers-Joseph-Saffman slip with friction condition. The variables in the weak formulation are velocity and pressure for Navier-Stokes, displacement for elasticity and velocity and pressure for Darcy flow. A Lagrange multiplier of stress/pressure type is employed to impose weakly the continuity of flux. Existence, uniqueness, and stability of a weak solution is established under a small data assumption. A fully discrete numerical method is then developed, based on backward Euler time discretization and finite element spatial approximation. We establish solvability, stability, and error estimates for the fully discrete scheme. Numerical experiments are presented to verify the theoretical results and illustrate the performance of the method for an arterial flow application.

A Lagrange multiplier formulation for the fully dynamic Navier-Stokes-Biot system

TL;DR

The paper develops a rigorous framework for the fully dynamic Navier–Stokes–Biot fluid–poroelastic interaction, introducing a stress/pressure Lagrange multiplier to enforce flux continuity at the fluid–poroelastic interface. It proves existence, uniqueness, and stability of a divergence-free weak formulation, then designs and analyzes a fully discrete finite element method with backward Euler time stepping and standard mixed spaces, obtaining optimal-order error estimates under a small-data condition. The analysis combines Galerkin approximations, ODE reductions, energy methods, and compactness arguments, supported by discrete inf-sup results. Numerical tests validate the theoretical convergence rates and demonstrate the method’s applicability to arterial-flow-like FPSI problems with challenging parameter regimes.

Abstract

We study a mathematical model of fluid -- poroelastic structure interaction and its numerical solution. The free fluid region is governed by the unsteady incompressible Navier-Stokes equations, while the poroelastic region is modeled by the Biot system of poroelasticity. The two systems are coupled along an interface through continuity of normal velocity and stress and the Beavers-Joseph-Saffman slip with friction condition. The variables in the weak formulation are velocity and pressure for Navier-Stokes, displacement for elasticity and velocity and pressure for Darcy flow. A Lagrange multiplier of stress/pressure type is employed to impose weakly the continuity of flux. Existence, uniqueness, and stability of a weak solution is established under a small data assumption. A fully discrete numerical method is then developed, based on backward Euler time discretization and finite element spatial approximation. We establish solvability, stability, and error estimates for the fully discrete scheme. Numerical experiments are presented to verify the theoretical results and illustrate the performance of the method for an arterial flow application.

Paper Structure

This paper contains 17 sections, 8 theorems, 183 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Navier-Stokes-Biot model problem
  3. Well posedness of ($\mathbf{LMWF1}$)
  4. A divergence-free Lagrange multiplier variational formulation
  5. A Galerkin approximation of $(\bold{LMWF2})$
  6. Reduction of ($\bold{GP}$) to an ODE system
  7. Existence, uniqueness, and stability of the solution of ( GP)
  8. Existence, uniqueness, and stability of the solution of ( LMWF1)
  9. Fully discrete numerical scheme
  10. Well-posedness of the solution to the fully discrete numerical scheme
  11. Error analysis
  12. Approximation error
  13. Error estimates
  14. Numerical results
  15. Example 1: convergence test
  16. ...and 2 more sections

Key Result

Lemma 3.1

Let ${\bf{f}}_{f}\in C^0(0,T;L^2(\Omega_{f}))$, ${\bf{f}}_{p}\in C^0(0,T;L^2(\Omega_{p}))$, and $q_p \in C^0(0,T; L^2(\Omega_{p}))$. Then there exists $T_1 \in (0,T]$ such that there exists a unique solution of semi1--semi3 in $[0,T_1]$ satisfying ${\bf{u}}_{f,h}(0) = 0$, ${\bf{u}}_{p,h}(0) = 0$, $p

Figures (5)

  • Figure 1: Computational domains for Example 1 (left) and Example 2 (right).
  • Figure 2: Example 2: fluid pressure (color) and velocity (arrows) at times $t=1.8$ ms, $t=3.6$ ms, and $t=5.4$ ms (from left to right). The velocity arrows are scaled proportionally to the vector magnitude.
  • Figure 3: Example 2: structure displacement in the normal direction $\boldsymbol{\eta}_p \cdot {\bf n}$ along the top interface at times $t=1.8$ ms, $t=3.6$ ms, and $t=5.4$ ms (left to right).
  • Figure 4: Example 2: normal filtration velocity ${\bf{u}}_{p}\cdot {\bf n}$ (top) and tangential filtration velocity ${\bf{u}}_{p}\cdot {\boldsymbol{\tau}}$ (bottom) along the top interface at times $t=1.8$ ms, $t=3.6$ ms, and $t=5.4$ ms (left to right).
  • Figure 5: Example 2: normal fluid velocity ${\bf{u}}_{f}\cdot {\bf n}$ (top) and tangential fluid velocity ${\bf{u}}_{f}\cdot {\boldsymbol{\tau}}$ (bottom) along the top interface at times $t=1.8$ ms, $t=3.6$ ms, and $t=5.4$ ms (left to right).

Theorems & Definitions (16)

  • Lemma 3.1
  • proof
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.2
  • proof
  • Remark 4.1
  • ...and 6 more