Moving Stokes with pointwise divergence-free H(div)-FEM

This project fuses two lines of work from the group: unfitted Eulerian methods for moving domains and pointwise divergence-free H(div)-conforming FEM for Stokes flows. The goal is to obtain a method for the unsteady incompressible Stokes problem on a moving domain that is exactly divergence-free — a property known to greatly improve pressure-robustness and long-time stability.

Type: Bachelor's / Master's thesis or scientific computing practical | Prerequisites: Numerics of PDEs, some familiarity with FEM and saddle-point problems | Contact: Christoph Lehrenfeld

The two building blocks

1. Unfitted Eulerian method for moving domains

In the Eulerian approach (see [IMAJNA_vWRL_2021] and [LO_ESAIM_2019]), a background mesh $\mathcal{T}_h$ on a fixed computational domain $\Omega$ is used. The physical domain $\Omega(t) \subset \Omega$ is described implicitly by a level-set function $\phi(t)$, and at each discrete time $t_n$ the active mesh is $\mathcal{T}_h^n = \{T \in \mathcal{T}_h : T \cap \Omega(t_n) \neq \emptyset\}$. A BDF-$k$ time discretization reads:

$\frac{1}{\Delta t}\sum_{j=0}^{k} \alpha_j \mathbf{u}_h^{n-j} + \nabla p_h^n - \mu\Delta\mathbf{u}_h^n = \mathbf{f}^n$ in $\Omega(t_n)$.

Ghost-penalty stabilization $j_h$ extends the solution stably off the physical domain so that degrees of freedom from previous time steps remain well-defined as $\Omega(t)$ moves.

2. Pointwise divergence-free H(div)-conforming FEM

H(div)-conforming finite element spaces (e.g. Raviart–Thomas or BDM) with appropriate interior penalty or HDG coupling yield pointwise divergence-free velocity solutions for Stokes and Navier–Stokes flows. This provides pressure-robustness: velocity errors are independent of the pressure gradient and of $\mu^{-1}$. In the unfitted setting, divergence-preserving methods require a careful reformulation of the divergence constraint (see [LvBV_MC_2024] for the mixed Poisson analogue).

The project: combining both

The existing analysis in [IMAJNA_vWRL_2021] uses inf-sup stable Taylor–Hood elements, which are not pointwise divergence-free. The project replaces the spatial discretization by an H(div)-conforming unfitted method and studies:

Formulation: an appropriate unfitted H(div) space on the active mesh $\mathcal{T}_h^n$, combined with an $L^2$ pressure space, satisfying an inf-sup condition robust in the cut configuration.
Divergence constraint on the active mesh: analogously to [LvBV_MC_2024], the constraint $\operatorname{div}\,\mathbf{u}_h = 0$ is imposed on $\mathcal{T}_h^n$ (not just on $\Omega(t_n)$) to avoid ill-conditioning from small cuts.
Ghost-penalty for H(div) spaces: the standard facet ghost-penalty must be adapted to preserve the normal-trace structure of H(div) functions.
Error analysis: a priori error estimates showing optimal convergence and pressure-robustness of the combined method.

Expected outcome and scope

Depending on the scope (practical vs. thesis), the project focuses on:

Practical: implement the method in NGSolve / ngsxfem for a 2D benchmark (e.g. Stokes flow around a moving disk), compare with Taylor–Hood reference solution, verify pressure-robustness.
Bachelor's thesis: formulation and numerical study, including ghost-penalty for H(div), convergence rates.
Master's thesis: full method including error analysis; prove stability and optimal convergence rates, incorporating geometry approximation errors from the level-set.

References

IMAJNA_vWRL_2021 — von Wahl, Richter, Lehrenfeld: Unfitted Eulerian FEM for the time-dependent Stokes problem on moving domains (IMA J. Numer. Anal. 2021)
LO_ESAIM_2019 — Lehrenfeld, Olshanskii: Eulerian FEM for PDEs in time-dependent domains (ESAIM 2019)
LvBV_MC_2024 — Lehrenfeld, van Beeck, Voulis: Divergence-preserving unfitted FEM for the mixed Poisson problem (Math. Comp. 2024)
LLS_SIAM_2017 — Lehrenfeld, Schöberl: High-order exactly divergence-free HDG methods for incompressible flows