Axisymmetric Stokes flow with an H(div)-conforming discretization

Many engineering and geophysical flows have rotational symmetry and can be modeled as axisymmetric problems, reducing a three-dimensional problem to two space dimensions. This project develops an H(div)-conforming finite element discretization for the axisymmetric Stokes problem that is pointwise divergence-free in the cylindrical sense — replacing the reconstruction-based approach of the recent group paper [LLMvB_ARXIV_2026].

The axisymmetric Stokes problem

Assume cylindrical coordinates $(r, \theta, z)$ and full rotational symmetry (no $\theta$-dependence). The incompressibility constraint $\operatorname{div}\mathbf{u} = 0$ reads:

$\frac{1}{r}\frac{\partial(r u_r)}{\partial r} + \frac{\partial u_z}{\partial z} = 0,$

and the weighted formulation on the meridional half-plane $\hat\Omega = \{(r,z): r>0\}$ introduces factors of $r$ (the cylindrical volume element $r\,dr\,dz$). The weighted divergence-free condition $\operatorname{div}_r(r\mathbf{u}) = 0$ differs from the standard Cartesian divergence-free condition, so that classical divergence-free elements (Raviart–Thomas, BDM) are not divergence-free in the axisymmetric sense without modification.

The reconstruction approach (prior work)

The recent paper [LLMvB_ARXIV_2026] (Lederer, Lehrenfeld, Merdon, van Beeck) achieves pressure-robustness for a low-order Bernardi–Raugel discretization on the axisymmetric domain by introducing a reconstruction operator into a modified Raviart–Thomas space whose basis functions vanish on the rotation axis $r=0$. The vanishing-on-axis property is essential for achieving optimal consistency error estimates.

The H(div) approach (this project)

Instead of a reconstruction post-processing, this project aims to construct H(div)-conforming elements directly in the axisymmetric weighted setting. The starting point is to find a pair of finite element spaces $(V_h, Q_h)$ such that the Piola transformation from the reference element to the physical element respects the cylindrical metric, giving:

• Conformity: $V_h \subset H(\operatorname{div}_r;\hat\Omega)$, i.e. normal traces of $r\mathbf{u}_h$ are continuous across element edges in the weighted sense.
• Exact divergence-freeness: for any $\mathbf{u}_h \in V_h$ with $(\operatorname{div}_r(r\mathbf{u}_h), q_h)_r = 0$ for all $q_h \in Q_h$, there holds pointwise $\operatorname{div}_r(r\mathbf{u}_h) = 0$.
• Vanishing on axis: basis functions in $V_h$ vanish at $r=0$ (as in [LLMvB_ARXIV_2026]) to handle the coordinate singularity.
• Inf-sup stability: verify a discrete inf-sup condition $\inf_{q_h} \sup_{\mathbf{u}_h} \frac{(\operatorname{div}_r(r\mathbf{u}_h), q_h)_r}{\|\mathbf{u}_h\|_{V}\|q_h\|_Q} \geq \beta > 0$, uniformly in $h$.

Approach and expected tasks

1. Weighted function spaces and Piola transform. Derive the weighted Sobolev spaces for axisymmetric problems and determine what the Piola transformation looks like for the cylindrical case. Identify the correct reference-element shape functions.
2. Construction of the modified RT/BDM space. Adapt the standard Raviart–Thomas (or BDM) space to the weighted divergence operator. Ensure the basis functions vanish at $r=0$ and satisfy the weighted normal-trace continuity.
3. Inf-sup analysis. Prove (or test numerically) the inf-sup stability of the resulting pair $(V_h, Q_h)$ for the weighted $L^2$ pressure space.
4. Implementation in NGSolve. Implement the method using NGSolve's finite element infrastructure and validate on exact axisymmetric Stokes solutions (e.g. Poiseuille flow in a pipe, Stokes flow around a sphere).
5. Pressure-robustness tests. Verify that velocity errors are independent of the magnitude of $\nabla p$, contrasting with a standard (non-divergence-free) inf-sup stable discretization.
6. Comparison with reconstruction approach. Compare accuracy, implementation complexity, and computational cost with the reconstruction method of [LLMvB_ARXIV_2026].

References

• LLMvB_ARXIV_2026 — Lederer, Lehrenfeld, Merdon, van Beeck: Pressure-robustness for the axisymmetric Stokes problem by velocity reconstruction (2026)
• LLS_SIAM_2017 — Lehrenfeld, Schöberl: High-order exactly divergence-free HDG methods for incompressible flows
• LvBV_MC_2024 — Lehrenfeld, van Beeck, Voulis: Divergence-preserving unfitted FEM for the mixed Poisson problem (Math. Comp. 2024)