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Pressure-robustness for the axisymmetric Stokes problem by velocity reconstruction

Philip L. Lederer, Christoph Lehrenfeld, Christian Merdon, Tim van Beeck

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.

Pressure-robustness for the axisymmetric Stokes problem by velocity reconstruction

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.
Paper Structure (17 sections, 6 theorems, 83 equations, 9 figures)

This paper contains 17 sections, 6 theorems, 83 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Model problem and preliminaries
  3. Axisymmetric geometry
  4. Strong formulation and differential operators
  5. Weak formulation and weighted Sobolev spaces
  6. Helmholtz projection
  7. Pressure-robustness for the axisymmetric Stokes problem by reconstruction operators
  8. A modified finite element method
  9. A priori error estimates
  10. Classical $H(\mathrm{div}$)-conforming reconstruction operators
  11. A modified reconstruction operator
  12. Modified $L^2_{-1}(\Omega) \cap H(\mathrm{div}, \Omega)$-conforming interpolation
  13. Improved consistency error estimate
  14. Numerical examples
  15. Example 1: linear stagnation flow
  16. ...and 2 more sections

Key Result

Lemma 3.1

If the Stokes problem inherits $\bm{H}^{1+s}_1(\Omega) \times H^s_1(\Omega)$ elliptic regularity for some $s \in (0, 1]$ (from the equivalent three-dimensional problem) such that $\nu \|\bm{u}\|_{\bm{H}^{1+s}_1} + \|p \|_{H^s_1} \lesssim \| \bf \|_{L^2_1}$, there holds

Figures (9)

  • Figure 1: Illustrating pressure-robustness for the Taylor--Hood and Scott--Vogelius pairs in Cartesian (left) and axisymmetric (right) settings for right-hand sides $\bm{f}=-\nu\Delta \bm{u}+\nabla p$. A pressure-robust method yields $\nu$-independent velocity errors, whereas non-robust methods show locking w.r.t. $\nu$. Taylor--Hood is not pressure-robust and Scott--Vogelius is only pressure-robust in the Cartesian setting. In this example, $\bm{f}$ is computed from $\bm{u} = \operatorname{curl} (x^2 (x-1)^2 y^2 (y-1)^2)$ and $p = x^5 + y^5 - \tfrac{1}{3}$ in the Cartesian case, and as in Section \ref{['sec:example2']} in the axisymmetric case.
  • Figure 1: Geometric setup: the two-dimensional meridional domain $\Omega$ (right) and its revolution about the rotation axis $\Gamma_{\! \text{rot}}$ generate the three-dimensional domain $\widehat{\Omega}$ (left). The boundary $\Gamma$ denotes the non-axis portion of $\partial\Omega$.
  • Figure 1: We distinguish between type 2 ($\text{T}2$) with two vertices on the rotation axis $\Gamma_{\! \text{rot}}$ and type 1 ($\text{T}1$) triangles with one vertex on the rotation axis $\Gamma_{\! \text{rot}}$. Both types of triangles have two edges belonging to $\mathcal{E}_R$ on which we define a modified basis function $\psi_E^{\mathrm{R}}$ in \ref{['eqn:modified_RT0_function']}.
  • Figure 1: Error convergence histories in Example 1 for $\nu = 1$.
  • Figure 2: Top row: Standard Raviart--Thomas basis functions on a simple two element mesh ($\Gamma_{\! \text{rot}}$ corresponds to the left edge). Bottom row: Modified Raviart--Thomas basis functions. Edges corresponding to the shape functions are highlighted. The color encodes the normalized magnitude.
  • ...and 4 more figures

Theorems & Definitions (18)

  • Remark 2.1: Transformation to the three-dimensional setting
  • Lemma 3.1
  • Proof 1
  • Theorem 3.2: A priori error estimate
  • Proof 2
  • Remark 3.3: Lack of pressure-robustness for $\Pi = \mathbb{I}$
  • Remark 3.4: Condition for pressure-robustness
  • Remark 4.1: Regularity issue
  • Theorem 4.2
  • Proof 3
  • ...and 8 more