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A mass conserving mixed stress formulation for Stokes flow with weakly imposed stress symmetry

Jay Gopalakrishnan, Philip L. Lederer, Joachim Schöberl

TL;DR

This work develops a mass-conserving mixed finite element discretization for the steady Stokes equations that directly approximates the symmetric viscous stress $\sigma = \nu \varepsilon(u)$ in the $H(\operatorname{curl}\operatorname{div},\Omega)$ setting, while weakly enforcing symmetry via a Lagrange multiplier. The method uses $H(\operatorname{div})$-conforming velocity spaces to obtain exact mass conservation and enriches the nonconforming stress space with matrix bubbles to secure discrete inf-sup stability; a carefully designed mapping preserves normal-tangential continuity. The authors prove stability, consistency, and optimal error estimates, including pressure robustness, and introduce a local postprocessing that yields a velocity $u_h^*$ with $O(h^{k+1})$ accuracy in the energy norm while preserving divergence-free and mass-conserving properties. Numerical experiments in 2D and 3D validate the theory and demonstrate the practical benefit of postprocessing for improved velocity accuracy. This approach provides a robust, symmetry-preserving, mass-conserving framework for Stokes flow with direct stress-quantity approximation and reliable a priori guarantees.

Abstract

We introduce a new discretization of a mixed formulation of the incompressible Stokes equations that includes symmetric viscous stresses. The method is built upon a mass conserving mixed formulation that we recently studied. The improvement in this work is a new method that directly approximates the viscous fluid stress $σ$, enforcing its symmetry weakly. The finite element space in which the stress is approximated consists of matrix-valued functions having continuous "normal-tangential" components across element interfaces. Stability is achieved by adding certain matrix bubbles that were introduced earlier in the literature on finite elements for linear elasticity. Like the earlier work, the new method here approximates the fluid velocity $u$ using $H(\operatorname{div})$-conforming finite elements, thus providing exact mass conservation. Our error analysis shows optimal convergence rates for the pressure and the stress variables. An additional post processing yields an optimally convergent velocity satisfying exact mass conservation. The method is also pressure robust.

A mass conserving mixed stress formulation for Stokes flow with weakly imposed stress symmetry

TL;DR

This work develops a mass-conserving mixed finite element discretization for the steady Stokes equations that directly approximates the symmetric viscous stress in the setting, while weakly enforcing symmetry via a Lagrange multiplier. The method uses -conforming velocity spaces to obtain exact mass conservation and enriches the nonconforming stress space with matrix bubbles to secure discrete inf-sup stability; a carefully designed mapping preserves normal-tangential continuity. The authors prove stability, consistency, and optimal error estimates, including pressure robustness, and introduce a local postprocessing that yields a velocity with accuracy in the energy norm while preserving divergence-free and mass-conserving properties. Numerical experiments in 2D and 3D validate the theory and demonstrate the practical benefit of postprocessing for improved velocity accuracy. This approach provides a robust, symmetry-preserving, mass-conserving framework for Stokes flow with direct stress-quantity approximation and reliable a priori guarantees.

Abstract

We introduce a new discretization of a mixed formulation of the incompressible Stokes equations that includes symmetric viscous stresses. The method is built upon a mass conserving mixed formulation that we recently studied. The improvement in this work is a new method that directly approximates the viscous fluid stress , enforcing its symmetry weakly. The finite element space in which the stress is approximated consists of matrix-valued functions having continuous "normal-tangential" components across element interfaces. Stability is achieved by adding certain matrix bubbles that were introduced earlier in the literature on finite elements for linear elasticity. Like the earlier work, the new method here approximates the fluid velocity using -conforming finite elements, thus providing exact mass conservation. Our error analysis shows optimal convergence rates for the pressure and the stress variables. An additional post processing yields an optimally convergent velocity satisfying exact mass conservation. The method is also pressure robust.

Paper Structure

This paper contains 14 sections, 25 theorems, 123 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The new method
  4. Velocity, pressure, and vorticity spaces
  5. Stress space
  6. Equations of the method
  7. Degrees of freedom of the new stress space
  8. A priori error analysis
  9. Norms
  10. Norm equivalences
  11. Stability analysis
  12. Error estimates
  13. Postprocessing
  14. Numerical exampels

Key Result

Theorem 1

Suppose ${{\sigma}}$ is in $H^1(\mathcal{T}_h, {\mathbb{M}})$ and ${{\sigma}}_{{{n}}{{n}}}|_{\partial T} \in H^{1/2}(\partial T)$ for all elements $T \in \mathcal{T}_h$. Assume that the normal-tangential trace ${{\sigma}}_{nt}$ is continuous across element interfaces. Then $\sigma$ is in $H(\operato for all $v \in H_0(\operatorname{div}, \Omega).$

Theorems & Definitions (51)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • Remark 4
  • Theorem 5
  • proof
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 41 more