A quasi-Lagrangian finite element method for the Navier-Stokes equations in a time-dependent domain
This work provides a theoretically rigorous numerical method for fluid-structure interaction problems in biomedical applications, though it is an incremental improvement over existing quasi-Lagrangian approaches.
The paper develops a stable and convergent finite element method for the Navier-Stokes equations in time-dependent domains, requiring only mild time-step conditions. Numerical experiments confirm the predicted convergence rates, and the method is applied to simulate blood flow in a simplified left ventricle model.
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 $Δt\le C$, where $C$ depends only on problem data, and $h^{2m_u+2}\le c\,Δt$, $m_u$ 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.