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Pressure robust finite element discretizations of the nonlinear Stokes equations

Lars Diening, Adrian Hirn, Christian Kreuzer, Pietro Zanotti

TL;DR

This work addresses the nonlinear Stokes problem with $\varphi$- or $(r,\varepsilon)$-structure by developing two first-order, nonconforming CR discretizations that are monotone and stabilization-free, ensuring unique solvability. A key feature is a smoothing operator $E$ that maps CR test functions to exactly divergence-free fields, enabling pressure-robust velocity errors measured in the natural distance $\boldsymbol{F}$ that are independent of the pressure. The authors prove a priori error bounds showing first-order convergence in $\boldsymbol{F}(\boldsymbol{\mathcal{D}}\boldsymbol{u})$ and derive pressure-error estimates that depend on the velocity error, with improved rates under additional regularity; these results are robust as $\varepsilon \to 0$. Numerical experiments on $r$-Stokes problems confirm the theoretical rates and demonstrate clear pressure robustness, even for low regularity data and in the presence of discontinuous pressures. The methods offer stable, locally implementable schemes suitable for nonlinear and non-Newtonian flow models in which pressure decoupling is crucial for reliable simulations.

Abstract

We present first-order nonconforming Crouzeix-Raviart discretizations for the nonlinear generalized Stokes equations with $(r,ε)$-structure. Thereby the velocity-errors are independent of the pressure-error; i.e., the method is pressure robust. This improves suboptimal rates previously experienced for non pressure robust methods.

Pressure robust finite element discretizations of the nonlinear Stokes equations

TL;DR

This work addresses the nonlinear Stokes problem with - or -structure by developing two first-order, nonconforming CR discretizations that are monotone and stabilization-free, ensuring unique solvability. A key feature is a smoothing operator that maps CR test functions to exactly divergence-free fields, enabling pressure-robust velocity errors measured in the natural distance that are independent of the pressure. The authors prove a priori error bounds showing first-order convergence in and derive pressure-error estimates that depend on the velocity error, with improved rates under additional regularity; these results are robust as . Numerical experiments on -Stokes problems confirm the theoretical rates and demonstrate clear pressure robustness, even for low regularity data and in the presence of discontinuous pressures. The methods offer stable, locally implementable schemes suitable for nonlinear and non-Newtonian flow models in which pressure decoupling is crucial for reliable simulations.

Abstract

We present first-order nonconforming Crouzeix-Raviart discretizations for the nonlinear generalized Stokes equations with -structure. Thereby the velocity-errors are independent of the pressure-error; i.e., the method is pressure robust. This improves suboptimal rates previously experienced for non pressure robust methods.

Paper Structure

This paper contains 13 sections, 17 theorems, 121 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation and function spaces
  4. N-functions
  5. Properties of the nonlinear operator
  6. Variational formulation of the nonlinear Stokes equations
  7. Mesh and finite element spaces
  8. Smoothing operator
  9. A pressure robust Crouzeix-Raviart method for $\boldsymbol{\mathcal{D}}=\nabla$.
  10. A pressure robust Crouzeix-Raviart method for $\boldsymbol{\mathcal{D}}=\boldsymbol{\mathrm{D}}$
  11. Estimates of the pressure-error
  12. Numerical experiments
  13. Conclusions

Key Result

Proposition 2.1

Let $\psi$ be a uniformly convex N-function with indices $r^\pm$. Then for a bounded John domain $\omega\subset\mathbb{R}^d$ and all $\boldsymbol{w}\in\boldsymbol{W}^{1,1}(\omega)$ we have The hidden constant only depends on the constants $r^\pm$ of uniform convexity of $\psi$ and the John constant of $\omega$. Note that if $\boldsymbol{w} \in \boldsymbol{W}^{1,1}_0(\omega)$, then $\langle{\nabla

Figures (4)

  • Figure 1: Test case 1. Experimental order of convergence $\mathrm{EOC}_k$ of the velocity-error versus the number $k$ of mesh refinements.
  • Figure 2: Test case 1. Experimental order of convergence $\mathrm{EOC}_k$ of the pressure-error versus the number $k$ of mesh refinements.
  • Figure 3: Test case 2. Experimental order of convergence $\mathrm{EOC}_k$ of the velocity-error versus the number $k$ of mesh refinements.
  • Figure 4: Test case 2. Experimental order of convergence $\mathrm{EOC}_k$ of the pressure-error versus the number $k$ of mesh refinements.

Theorems & Definitions (39)

  • Proposition 2.1: DieningRuzickaSchumacher:2010
  • Remark 2.2
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • Remark 2.7: Quasi-norm and natural distance
  • Lemma 2.8: DieKapSchw:2012
  • Lemma 2.9
  • proof
  • Corollary 2.10
  • ...and 29 more