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

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 $L^{\infty }\left(L^2\right)$-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.