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An Eulerian finite element method for the linearized Navier--Stokes problem in an evolving domain

Michael Neilan, Maxim Olshanskii

TL;DR

This paper provides a rigorous error analysis for an Eulerian CutFEM discretization of the linearized Navier–Stokes (Oseen) problem on evolving domains with known boundary motion. By combining a backward Euler/BDF-type time stepping with Nitsche-based unfitted finite elements and ghost-penalty stabilization, the authors establish optimal-order convergence for velocity in the energy norm and for pressure in a scaled $L^2(H^1)$-type norm under the mesh-time-step constraint $h^2 \lesssim \Delta t \lesssim h$. A key methodological contribution is the use of a divergence-free extension and a nonstandard pressure norm to overcome the lack of a weakly divergence-free discrete time derivative, enabling stable and accurate fully discrete solutions even with evolving geometries. The analysis applies to general inf-sup stable unfitted element pairs and is complemented by concrete examples (Mini, Taylor–Hood, and $P_3$-$P_0$) that satisfy the required stability assumptions, supporting practical adoption in simulations of evolving-domain flows.

Abstract

The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier--Stokes problem in a time-dependent domain. In this study, the domain's evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard Backward Differentiation Formula (BDF)-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche's method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity--pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled $L^2(H^1)$-type norm for the pressure component.

An Eulerian finite element method for the linearized Navier--Stokes problem in an evolving domain

TL;DR

This paper provides a rigorous error analysis for an Eulerian CutFEM discretization of the linearized Navier–Stokes (Oseen) problem on evolving domains with known boundary motion. By combining a backward Euler/BDF-type time stepping with Nitsche-based unfitted finite elements and ghost-penalty stabilization, the authors establish optimal-order convergence for velocity in the energy norm and for pressure in a scaled -type norm under the mesh-time-step constraint . A key methodological contribution is the use of a divergence-free extension and a nonstandard pressure norm to overcome the lack of a weakly divergence-free discrete time derivative, enabling stable and accurate fully discrete solutions even with evolving geometries. The analysis applies to general inf-sup stable unfitted element pairs and is complemented by concrete examples (Mini, Taylor–Hood, and -) that satisfy the required stability assumptions, supporting practical adoption in simulations of evolving-domain flows.

Abstract

The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier--Stokes problem in a time-dependent domain. In this study, the domain's evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard Backward Differentiation Formula (BDF)-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche's method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity--pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled -type norm for the pressure component.
Paper Structure (18 sections, 8 theorems, 103 equations, 1 figure)

This paper contains 18 sections, 8 theorems, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Extensions
  4. The Fully Discrete Finite Element Method
  5. Approximate geometries
  6. Triangulations
  7. Finite Element Spaces and Assumptions
  8. The CutFEM Discretization
  9. Stability and Convergence Analysis
  10. Preliminary Results
  11. Stability analysis
  12. Consistency
  13. Error Estimates
  14. Examples of finite element pairs satisfying Assumption \ref{['ass:Pair']}
  15. The Mini element
  16. ...and 3 more sections

Key Result

Lemma 4.1

For $h$ sufficiently small, there holds for all $\mathbf v\in \mathbf V_h^{n-1}$, for a constant $c_1>0$ independent of $h$, $\Delta t$ and how the boundary cuts through the triangulation.

Figures (1)

  • Figure 1: Left: A depiction of the mesh in two dimensions, with the interior triangulation $\mathcal{T}_{h,i}^n$ in red and $\mathcal{T}_{h,e}^n\backslash \mathcal{T}_{h,i}^n$ in blue. Right: The triangulation $\mathcal{T}_{\Gamma_h}^n$ (blue) around a $\delta_h$-neighborhood tubular region of $\Gamma_h$.

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 11 more