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Stability and error estimates of a linear and partitioned finite element method approximating nonlinear fluid-structure interactions

Bangwei She, Tian Tian, Karel Tuma

TL;DR

The paper develops and analyzes a linear partitioned finite element method for incompressible fluid–thin shell interactions within an ALE framework, using a reference-domain weak formulation and compatible discrete spaces. It proves unconditional energy stability and linear convergence in time and space for the proposed scheme, without standard simplifying assumptions about deformation or convection. Numerical experiments corroborate the theoretical rates and illustrate robustness, efficiency, and a fair comparison with monolithic schemes. The results advance partitioned FSI methods by providing rigorous stability and error estimates alongside practical validation, with potential benefits particularly in higher-dimensional problems where subproblem decomposition can reduce computational cost.

Abstract

We propose and analyze a linear and partitioned finite element method for fluid-shell interactions under the arbitrary Lagrangian-Eulerian (ALE) framework. We adopt the P1-bubble/P1/P1 elements for the fluid velocity, pressure, and structure velocity, respectively. We show the stability and error estimates of the scheme without assuming infinitesimal structural deformation nor neglecting fluid convection effects. The theoretical convergence rate is further corroborated by numerical experiments.

Stability and error estimates of a linear and partitioned finite element method approximating nonlinear fluid-structure interactions

TL;DR

The paper develops and analyzes a linear partitioned finite element method for incompressible fluid–thin shell interactions within an ALE framework, using a reference-domain weak formulation and compatible discrete spaces. It proves unconditional energy stability and linear convergence in time and space for the proposed scheme, without standard simplifying assumptions about deformation or convection. Numerical experiments corroborate the theoretical rates and illustrate robustness, efficiency, and a fair comparison with monolithic schemes. The results advance partitioned FSI methods by providing rigorous stability and error estimates alongside practical validation, with potential benefits particularly in higher-dimensional problems where subproblem decomposition can reduce computational cost.

Abstract

We propose and analyze a linear and partitioned finite element method for fluid-shell interactions under the arbitrary Lagrangian-Eulerian (ALE) framework. We adopt the P1-bubble/P1/P1 elements for the fluid velocity, pressure, and structure velocity, respectively. We show the stability and error estimates of the scheme without assuming infinitesimal structural deformation nor neglecting fluid convection effects. The theoretical convergence rate is further corroborated by numerical experiments.
Paper Structure (20 sections, 7 theorems, 84 equations, 3 figures, 2 tables)

This paper contains 20 sections, 7 theorems, 84 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem description and numerical method
  3. Problem formulation
  4. Function spaces
  5. Finite element space
  6. The numerical method
  7. Stability
  8. Error estimates
  9. Numerical experiments
  10. Experiment 1: Large deformation driven by external force
  11. Experiment 2: Convergence rates
  12. Experiment 3: Comparison with a monolithic scheme from SST
  13. Conclusion
  14. Appendix: Useful equalities and estimates
  15. Proof of the error equation \ref{['EEQ']}
  16. ...and 5 more sections

Key Result

Theorem 2.2

(SST) Let $\Omega_{\eta_h}\subset \mathbb{R}^2$ be a subgraph, the grids $\mathcal{T}_h$ and $\Sigma_h$ respectively defined on the domains $\widehat{\Omega}$ and $\Sigma$ be shape regular and quasi-uniform. Besides, let $\Gamma_S=\{(x_1,\eta_h(x_1))| x_1\in\Sigma\}$ satisfying $\min_\Sigma \eta_h \ satisfying for $\gamma <\infty$ that where the bounds depend linearly on $\frac{1}{\delta}, L, \fr

Figures (3)

  • Figure 1: Snapshots of the simulation at different time instants. The color scale depicts pressure, arrows show the direction of the velocity field.
  • Figure 2: Mesh (left) and timestep (right) convergence for $\|e_\mathbf{u}\|_{L^\infty(L^2)}$, $\|e_{\xi}\|_{L^\infty(L^2)}$, $\|e_{\eta}\|_{L^\infty(L^2)}$, $\|\nabla e_{\eta}\|_{L^\infty(L^2)}$, $\|e_{\zeta}\|_{L^\infty(L^2)}$ and $\|\nabla e_\mathbf{u}\|_{L^2(L^2)}$. For a better comparison, the plots of the errors are shifted to start from the same point.
  • Figure 3: Timestep convergence comparison monolithic vs. splitting scheme for $\|\nabla e_\mathbf{u}\|_{L^2(L^2)}$ and $\|\nabla e_{\eta}\|_{L^\infty(L^2)}$.

Theorems & Definitions (13)

  • Definition 2.1: Weak formulation on the reference domain ${\widehat{\Omega}}$
  • Theorem 2.2
  • Remark 2.3
  • Theorem 3.1: Energy estimates
  • proof
  • Corollary 3.2
  • Theorem 4.1: Convergence rate
  • proof
  • Lemma A.1
  • Lemma A.2
  • ...and 3 more