Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dynamic Ritz Projection for Finite Element Methods in Fluid-Structure Interaction

Erik Burman, Buyang Li, Rong Tang

TL;DR

This paper addresses optimal-order convergence of finite element discretizations for a fluid-structure interaction (FSI) model with a thick structure, where standard Ritz projections are incompatible with the dynamic interface condition $u=\partial_t\eta$ on $\Gamma$ and cannot yield optimal-order convergence in $L^\infty(0,T;L^2)$. It introduces a dynamic Ritz projection $(R_h \eta, R_h u, R_h p)$ that enforces the interface condition dynamically and proves existence, uniqueness, and approximation properties in $L^\infty(0,T;H^1)$ and $L^\infty(0,T;L^2)$, enabling optimal-order convergence in $L^\infty(0,T;L^2)$. Using these results, it proves optimal-order convergence of FEM solutions in $L^\infty(0,T;L^2)$, with error bounds of order $O(\tau^2 + h^{k+1})$ for the velocity and structure displacement, via a space-time duality framework and defect analysis. Numerical experiments corroborate the theory and illustrate the method's applicability to monolithic Crank–Nicolson schemes and partitioned approaches under the stated regularity assumptions.

Abstract

Regardless of the development of various finite element methods for fluid-structure interaction (FSI) problems, optimal-order convergence of finite element discretizations of the FSI problems in the $L^\infty(0,T;L^2)$ norm has not been proved due to the incompatibility between standard Ritz projections and the interface conditions in the FSI problems. To address this issue, we define a dynamic Ritz projection (which satisfies a dynamic interface condition) associated to the FSI problem and study its approximation properties in the $L^\infty(0,T;H^1)$ and $L^\infty(0,T;L^2)$ norms. Existence and uniqueness of the dynamic Ritz projection of the solution, as well as estimates of the error between the solution and its dynamic Ritz projection, are established. By utilizing the established results, we prove optimal-order convergence of finite element methods for the FSI problem in the $L^\infty(0,T;L^2)$ norm.

Dynamic Ritz Projection for Finite Element Methods in Fluid-Structure Interaction

TL;DR

This paper addresses optimal-order convergence of finite element discretizations for a fluid-structure interaction (FSI) model with a thick structure, where standard Ritz projections are incompatible with the dynamic interface condition $u=\partial_t\eta$ on $\Gamma$ and cannot yield optimal-order convergence in $L^\infty(0,T;L^2)$. It introduces a dynamic Ritz projection $(R_h \eta, R_h u, R_h p)$ that enforces the interface condition dynamically and proves existence, uniqueness, and approximation properties in $L^\infty(0,T;H^1)$ and $L^\infty(0,T;L^2)$, enabling optimal-order convergence in $L^\infty(0,T;L^2)$. Using these results, it proves optimal-order convergence of FEM solutions in $L^\infty(0,T;L^2)$, with error bounds of order $O(\tau^2 + h^{k+1})$ for the velocity and structure displacement, via a space-time duality framework and defect analysis. Numerical experiments corroborate the theory and illustrate the method's applicability to monolithic Crank–Nicolson schemes and partitioned approaches under the stated regularity assumptions.

Abstract

Regardless of the development of various finite element methods for fluid-structure interaction (FSI) problems, optimal-order convergence of finite element discretizations of the FSI problems in the norm has not been proved due to the incompatibility between standard Ritz projections and the interface conditions in the FSI problems. To address this issue, we define a dynamic Ritz projection (which satisfies a dynamic interface condition) associated to the FSI problem and study its approximation properties in the and norms. Existence and uniqueness of the dynamic Ritz projection of the solution, as well as estimates of the error between the solution and its dynamic Ritz projection, are established. By utilizing the established results, we prove optimal-order convergence of finite element methods for the FSI problem in the norm.

Paper Structure

This paper contains 21 sections, 3 theorems, 120 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Basic settings: Triangulation and regularity assumptions
  4. FEM for the FSI problem
  5. Dynamic Ritz projection and error estimates
  6. Proof of Theorem \ref{['THM-Ritz']}
  7. Existence and uniqueness of the dynamic Ritz projection
  8. Estimates of $\|\nabla (R_h \eta - \eta)\|_{L^\infty(0,T;L^2(\Omega_2))}$ and $\|R_h u - u\|_{L^2(0,T;H^1(\Omega_1))}$
  9. Estimate of $\|R_h u - u\|_{L^\infty(0,T;H^1(\Omega_1))}$
  10. Estimates of $\|\nabla \partial _t ( R_h \eta - \eta)\|_{L^\infty(0,T;L^2(\Omega_2))}$ and $\|\partial _t ( R_h u - u)\|_{L^\infty(0,T;H^1(\Omega_1))}$
  11. Estimates of $\|\nabla \partial _{t}^j ( R_h \eta - \eta)\|_{L^\infty(0,T;L^2(\Omega_2))}$ and $\| \partial _{t}^j( R_h u - u)\|_{L^\infty(0,T;H^1(\Omega_1))}$ for $j\geq 1$
  12. Estimates of $\|(R_h \eta - \eta)(0)\|_{L^2(\Omega_2)}$ and $\|(R_h u - u)(0)\|_{L^2(\Omega_1)}$
  13. Space-time dual problem of the dynamic Ritz projection
  14. Estimates of $\|\partial_t ( R_h \eta - \eta)\|_{L^\infty(0,T;L^2(\Omega_2))}$ and $\|\partial_t (R_h u - u)\|_{L^\infty(0,T;L^2(\Omega_1))}$
  15. Estimates of $\|\partial_{tt} ( R_h \eta - \eta)\|_{L^\infty(0,T;L^2(\Omega_2))}$ and $\|\partial_{tt} (R_h u - u)\|_{L^\infty(0,T;L^2(\Omega_1))}$
  16. ...and 6 more sections

Key Result

Theorem 2.2

Under Assumption AS0--AS2, problem weak_Ritz determines a unique solution $(R_h \eta, R_h u, R_h p) \in X_h(\Omega_{2}) \times X_h(\Omega_{1})\times M_h(\Omega_{1})$, called the dynamic Ritz projection of the solution. Moreover, this dynamic Ritz projection has the following approximation properties

Figures (7)

  • Figure 5.1: Fluid velocity magnitude (Example \ref{['section:bloodflow']}).
  • Figure 5.2: Spatial discretization errors (Example \ref{['section:bloodflow']}).
  • Figure 5.3: Time discretization errors (Example \ref{['section:bloodflow']}).
  • Figure 5.4: Fluid velocity magnitude (Example \ref{['section:bloodflow2']}).
  • Figure 5.5: Spatial errors and convergence rates (Example \ref{['section:bloodflow2']}).
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 2.2: Dynamic Ritz projection
  • Theorem 2.3: Optimal-order convergence
  • Lemma 4.1
  • proof
  • Example 5.1: Fluid flow in a channel, under periodic boundary condition
  • Example 5.2: Fluid flow in a channel, under the traction boundary condition
  • Example 5.3: The heat-wave system