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Advances on the finite element discretization of fluid-structure interaction problems

Najwa Alshehri, Daniele Boffi, Fabio Credali, Lucia Gastaldi

TL;DR

The paper surveys an unfitted finite element method for interface and fluid-structure interaction problems based on a distributed Lagrange multiplier (FD-DLM). By extending one domain into the other and solving on fixed, independent meshes, it avoids remeshing while enforcing matching across the interface via a Lagrange multiplier. It proves well-posedness, stability and convergence, develops a posteriori error estimators with adaptive refinement, and designs robust block preconditioners and scalable solvers, including analysis of exact versus inexact assembly of the coupling term. Numerical results for elliptic interfaces and FSI demonstrate optimal convergence, stability under challenging mass-imbalance conditions, and practical performance, underscoring the method’s robustness and applicability.

Abstract

We review the main features of an unfitted finite element method for interface and fluid-structure interaction problems based on a distributed Lagrange multiplier in the spirit of the fictitious domain approach. We recall our theoretical findings concerning well-posedness, stability, and convergence of the numerical schemes, and discuss the related computational challenges. In the case of elliptic interface problems, we also present a posteriori error estimates.

Advances on the finite element discretization of fluid-structure interaction problems

TL;DR

The paper surveys an unfitted finite element method for interface and fluid-structure interaction problems based on a distributed Lagrange multiplier (FD-DLM). By extending one domain into the other and solving on fixed, independent meshes, it avoids remeshing while enforcing matching across the interface via a Lagrange multiplier. It proves well-posedness, stability and convergence, develops a posteriori error estimators with adaptive refinement, and designs robust block preconditioners and scalable solvers, including analysis of exact versus inexact assembly of the coupling term. Numerical results for elliptic interfaces and FSI demonstrate optimal convergence, stability under challenging mass-imbalance conditions, and practical performance, underscoring the method’s robustness and applicability.

Abstract

We review the main features of an unfitted finite element method for interface and fluid-structure interaction problems based on a distributed Lagrange multiplier in the spirit of the fictitious domain approach. We recall our theoretical findings concerning well-posedness, stability, and convergence of the numerical schemes, and discuss the related computational challenges. In the case of elliptic interface problems, we also present a posteriori error estimates.
Paper Structure (21 sections, 5 theorems, 58 equations, 12 figures, 3 tables)

This paper contains 21 sections, 5 theorems, 58 equations, 12 figures, 3 tables.

Table of Contents

  1. Fictitious domain formulation for elliptic interface problems
  2. Fictitious domain formulation
  3. Discretization
  4. How to deal with the coupling term
  5. Exact procedure
  6. Inexact procedure
  7. Solvers
  8. A posteriori error analysis
  9. Residuals and error indicators
  10. Adaptive mesh and refinement strategy
  11. Adaptive refinement procedure
  12. Numerical results
  13. Preconditioning strategies
  14. Application to fluid-structure interaction problems
  15. The continuous problem
  16. ...and 6 more sections

Key Result

Proposition 1.4

Let $\nu$ and $\nu_2$ be two positive constants. Given $f\in L^2(\Omega)$ and $f_2\in L^2(\Omega_2)$, Problem pro:fd_ell admits a unique solution ${(u,u_2,\lambda)\in H^1_0(\Omega)\times H^1(\Omega_2)\times\Lambda}$ satisfying

Figures (12)

  • Figure 1: Two possible domain configurations for the interface problem. Left: $\Omega_2$ is completely immersed in $\Omega$. Right: $\Omega_2$ is not immersed.
  • Figure 2: (A) The immersion of $K_2\in\mathcal{T}_h^2$ into the background mesh $\mathcal{T}_h$. The immersed element, depicted in beige, is not matching the support of the finite element functions defined on $\mathcal{T}_h$ (blue). (B) The exact computation of the interface matrix requires the implementation of a composite quadrature rule on the intersection $\mathcal{T}_h^2\cap\mathcal{T}_h$. (C) Quadrature points of an inexact rule in $K_2$.
  • Figure 3: Examples of element-based $\omega_{K}$ and edge-based ${\omega_{e}}$ neighborhoods
  • Figure 4: (a) Domain configuration of the immersed circle, initial mesh. (b) Domain configuration after adaptive refinement. (c) Profile of the computed solution.
  • Figure 5: Left: comparison of the error $e=u-u_h$ between uniform (uni) and adaptive (adp) refinement. Right: comparison between the exact error $\|e\|_{1}$ and the a posteriori indicator $\eta$.
  • ...and 7 more figures

Theorems & Definitions (15)

  • Remark 1.2
  • Proposition 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 1.8
  • Remark 1.9
  • Remark 1.10
  • Remark 1.11
  • Remark 1.12
  • Proposition 1.13
  • ...and 5 more