Stability of instantaneous pressures in an Eulerian finite element method for moving boundary flow problems

TL;DR

The paper analyzes spurious pressure oscillations in sharp-interface unfitted Eulerian FEMs for the Navier–Stokes equations on moving domains and identifies divergence mismatch as a key instability source. It proposes a globally redefined ghost-penalty stabilization to achieve unconditional stability of the instantaneous pressure in $L^\infty$-temporal norms, while preserving optimal convergence in velocity and pressure in standard norms. Numerical experiments show that global ghost-penalty stabilization dramatically reduces pressure oscillations, especially when combined with grad-div stabilization and higher-order elements, and that oscillations are not primarily caused by geometry approximation. The results imply increased robustness and accuracy of moving-boundary simulations with unfitted FEMs and suggest extensions to nonlinear FSI settings, with code publicly available for replication.

Abstract

This paper focuses on identifying the cause and proposing a remedy for the problem of spurious pressure oscillations in a sharp-interface immersed boundary finite element method for incompressible flow problems in moving domains. The numerical method belongs to the class of Eulerian unfitted finite element methods. It employs a cutFEM discretization in space and a standard BDF time-stepping scheme, enabled by a discrete extension of the solution from the physical domain into the ambient space using ghost-penalty stabilization. To investigate the origin of spurious temporal pressure oscillations, we revisit a finite element stability analysis for the steady domain case and extend it to derive a stability estimate for the pressure in the $L^\infty(L^2)$-norm that is uniform with respect to discretization parameters. By identifying where the arguments fail in the context of a moving domain, we propose a variant of the method that ensures unconditional stability of the instantaneous pressure. As a result, the modified method eliminates spurious pressure oscillations. We also present extensive numerical studies aimed at illustrating our findings and exploring the effects of fluid viscosity, geometry approximation, mass conservation, discretization and stabilization parameters, and the choice of finite element spaces on the occurrence and magnitude of spurious temporal pressure oscillations. The results of the experiments demonstrate a significant improvement in the robustness and accuracy of the proposed method compared to existing approaches.

Figures (14)

• Figure 1: Sketch of different elements and facets used for the unfitted Eulerian time-stepping scheme.
• Figure 2: \ref{['sec.numerics:subsec.convergence']} -- Convergence behavior of the error of the velocity and pressure under uniform mesh refinement and combined mesh and time-step refinement for the isoparametric Taylor-Hood elements with the sufficient and global ghost-penalty facet choices.
• Figure 3: \ref{['sec.numerics:subsec.convergence']} -- Behavior of pressure, velocity, and discrete velocity time derivative norms under uniform time-step refinement for the isoparametric Taylor-Hood elements with the narrow-band ghost-penalty stabilization.
• Figure 4: \ref{['sec.numerics:subsec.convergence']} -- Behavior of the pressure, velocity, and discrete velocity time derivative under uniform time-step refinement for the isoparametric Taylor-Hood elements with the global ghost-penalty stabilization.
• Figure 5: \ref{['sec.numerics:subsec.ex2']} -- $p - p_{mean}$ integrated over the moving boundary for the isoparametric Taylor-Hood method using narrow-band ghost-penalty stabilization and parameters $\nu=0.01$, $h=0.1$, $k=2$, $\gamma_{\text{gp}}^{v}=\gamma_{\text{gp}}^{p}=0.1$ and $\gamma_{\text{gd}}=0$.

Theorems & Definitions (1)

• Remark 1: Computational effort