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A Revisiting of the Pressure Elimination for a Fluid-Structure PDE Interaction and its Implications

George Avalos, Yuhao Mu

TL;DR

The paper addresses the challenge of eliminating the pressure in a linear Stokes–elasticity fluid-structure interaction on bounded Lipschitz domains. It develops a new pressure-elimination method based on de Rham theory that circumvents the need for elliptic regularity, yielding an explicit generator $\mathcal{A}$ on the energy space $\mathbf{H}$ and proving analytic well-posedness via a contraction $C_0$-semigroup. Key contributions include the explicit form and domain of $\mathcal{A}$, pressure recovery formulas, and FEM convergence results on polygonal (non-smooth) domains, enabling reliable numerical analysis on general geometries. This approach broadens the applicability of FSI analysis to non-smooth domains and supports accurate, convergent finite element methods for polygonal geometries.

Abstract

In this paper we establish, for the first time, a new technique for eliminating and recovering the pressure for a fluid-structure interaction model that is valid in general bounded Lipschitz domains, without additional geometric conditions such as convexity of angles. The specific fluid-structure interaction (FSI) that we consider is a well-known model of coupled Stokes flow with linear elasticity, which constitutes a coupled parabolic-hyperbolic system. The coupling between these two distinct PDE dynamics occurs across a boundary interface, with each of the components evolving on its own distinct geometry, with the boundaries concerned being Lipschitz. For simplicity, we consider the linear version of this FSI system with Stokes flow. Our new pressure elimination technique admits of an explicit $C_{0}$-semigroup generator representation $\mathcal{A} : D(\mathcal{A}) \subset \mathbf{H} \to \mathbf{H}$, where $\mathbf{H}$ is the associated energy space of fluid-structure initial data. This leads to an analytic proof for the first time of the well-posedness of the continuous PDE in such general geometries. We illustrate some automatic consequences of our results to other fields, such as numerical approximations where it provides FEM convergence estimates over polygonal domains.

A Revisiting of the Pressure Elimination for a Fluid-Structure PDE Interaction and its Implications

TL;DR

The paper addresses the challenge of eliminating the pressure in a linear Stokes–elasticity fluid-structure interaction on bounded Lipschitz domains. It develops a new pressure-elimination method based on de Rham theory that circumvents the need for elliptic regularity, yielding an explicit generator on the energy space and proving analytic well-posedness via a contraction -semigroup. Key contributions include the explicit form and domain of , pressure recovery formulas, and FEM convergence results on polygonal (non-smooth) domains, enabling reliable numerical analysis on general geometries. This approach broadens the applicability of FSI analysis to non-smooth domains and supports accurate, convergent finite element methods for polygonal geometries.

Abstract

In this paper we establish, for the first time, a new technique for eliminating and recovering the pressure for a fluid-structure interaction model that is valid in general bounded Lipschitz domains, without additional geometric conditions such as convexity of angles. The specific fluid-structure interaction (FSI) that we consider is a well-known model of coupled Stokes flow with linear elasticity, which constitutes a coupled parabolic-hyperbolic system. The coupling between these two distinct PDE dynamics occurs across a boundary interface, with each of the components evolving on its own distinct geometry, with the boundaries concerned being Lipschitz. For simplicity, we consider the linear version of this FSI system with Stokes flow. Our new pressure elimination technique admits of an explicit -semigroup generator representation , where is the associated energy space of fluid-structure initial data. This leads to an analytic proof for the first time of the well-posedness of the continuous PDE in such general geometries. We illustrate some automatic consequences of our results to other fields, such as numerical approximations where it provides FEM convergence estimates over polygonal domains.

Paper Structure

This paper contains 17 sections, 6 theorems, 93 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Elimination of the pressure
  3. Past techniques to eliminate the pressure
  4. New method of pressure elimination
  5. Consequences of new pressure elimination method
  6. The explicit form of the fluid-structure generator
  7. Domain of A
  8. Semigroup well-posedness
  9. Convergence estimates for approximating FEM
  10. Semi-discrete scheme.
  11. Numerical Results
  12. Numerical approximation framework: static PDE
  13. A numerical example
  14. Appendix: Proof of Semigroup Well-posedness
  15. Proof of dissipativity
  16. ...and 2 more sections

Key Result

Lemma 1

Suppose $\mathbf{u}$ and $\mathbf{u}^{*}$ are in $\mathbf{H}^{1}(\Omega_f)$, $\pi$ is in $L^{2}(\Omega_f)$, and $\mathop{\mathrm{div}}\nolimits(\varepsilon(\mathbf{u})) - \nabla \pi = - \mathbf{u}^{*} + \lambda \mathbf{u} \in L^{2}(\Omega_f)$. Then we have that $\varepsilon(\mathbf{u}) \cdot \nu - \

Figures (5)

  • Figure 1: The fluid-structure geometry.
  • Figure 2: The fluid-structure geometry.
  • Figure 3: Mesh configuration for the coarsest mesh.
  • Figure 4: Error for both components of $\mathbf{u}_h$ compared to $\mathbf{u}$, for first mesh.
  • Figure 5: Error for both components of $u_h$ compared to $\mathbf{u}$, for fourth mesh.

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 2
  • Corollary 3
  • Theorem 4: Discrete uniform inf-sup inequality, see avalos2016infsup Theorem 3.1
  • Corollary 5
  • Theorem 6: Babuška-Brezzi, see e.g. p.116 of kesavan1989topics