Pressure-robustness for the axisymmetric Stokes problem by velocity reconstruction

TL;DR

This paper develops a pressure-robust finite element method for the axisymmetric Stokes problem by introducing a specialized reconstruction operator that ensures divergence-free velocity fields and improves accuracy near the axis of symmetry.

Contribution

It proposes a novel reconstruction operator using Raviart--Thomas functions that vanish on the axis, restoring pressure-robustness in axisymmetric Stokes discretizations.

Findings

The new method achieves optimal consistency error estimates.

Numerical examples show improved accuracy, especially near the axis.

The approach is feasible and enhances pressure-robustness in axisymmetric flows.

Abstract

This paper studies pressure-robustness for the axisymmetric Stokes problem. The transformation to cylindrical coordinates requires that the radially weighted velocity is divergence-free in the classical sense. Consequently, traditional divergence-free finite element methods from the Cartesian setting -- even if inf-sup stable -- are in general not divergence-free in the axisymmetric formulation. We therefore explore the approach that restores pressure-robustness via reconstruction operators for a low-order Bernardi--Raugel discretization. We show that an application of standard interpolation operators from the Cartesian setting to radially weighted test functions works in principle, but it lacks properties needed to derive optimal consistency error estimates. To address this, we introduce a reconstruction operator into a finite element space spanned by Raviart--Thomas functions that are modified such that they vanish on the rotation axis. This vanishing-on-axis property is the key to obtain optimal consistency error estimates. Numerical examples demonstrate the overall feasibility of the approach and include cases where the vanishing-on-axis property yields significantly better results.