A quasi-Lagrangian finite element method for the Navier-Stokes equations in a time-dependent domain

TL;DR

This paper introduces a stable, convergence-verified finite element method for simulating incompressible fluid flow in moving domains, validated through numerical experiments and application to cardiac flow modeling.

Contribution

It develops a quasi-Lagrangian finite element approach for Navier-Stokes in moving domains with stability and convergence analysis without CFL restrictions.

Findings

Method is unconditionally stable under mild conditions.

Convergence rates match theoretical predictions.

Successfully applied to simulate heart ventricle flow.

Abstract

The paper develops a finite element method for the Navier-Stokes equations of incompressible viscous fluid in a time-dependent domain. The method builds on a quasi-Lagrangian formulation of the problem. The paper provides stability and convergence analysis of the fully discrete (finite-difference in time and finite-element in space) method. The analysis does not assume any CFL time-step restriction, it rather needs mild conditions of the form $\Delta t\le C$, where $C$ depends only on problem data, and $h^{2m_u+2}\le c\,\Delta t$, $m_u$ is polynomial degree of velocity finite element space. Both conditions result from a numerical treatment of practically important non-homogeneous boundary conditions. The theoretically predicted convergence rate is confirmed by a set of numerical experiments. Further we apply the method to simulate a flow in a simplified model of the left ventricle of a human heart, where the ventricle wall dynamics is reconstructed from a sequence of contrast enhanced Computed Tomography images.