A mass conserving mixed stress formulation for the Stokes equations

TL;DR

This paper introduces a novel mass conserving mixed stress finite element method for the Stokes equations, achieving optimal convergence rates and pressure robustness with computational costs comparable to existing methods.

Contribution

It develops a new discretization using $H( ext{div})$-conforming elements and a novel stress space in $H( ext{curl} ext{div})$, ensuring exact mass conservation and pressure robustness.

Findings

Achieves optimal convergence rates for velocity, stress, and pressure.

Ensures exact mass conservation and pressure robustness.

Maintains computational cost comparable to existing methods.

Abstract

We propose a new discretization of a mixed stress formulation of the Stokes equations. The velocity $u$ is approximated with $H(\operatorname{div})$-conforming finite elements providing exact mass conservation. While many standard methods use $H^1$-conforming spaces for the discrete velocity, $H(\operatorname{div})$-conformity fits the considered variational formulation in this work. A new stress-like variable $\sigma$ equalling the gradient of the velocity is set within a new function space $H(\operatorname{curl} \operatorname{div})$. New matrix-valued finite elements having continuous "normal-tangential" components are constructed to approximate functions in $H(\operatorname{curl} \operatorname{div})$. An error analysis concludes with optimal rates of convergence for errors in $u$ (measured in a discrete $H^1$-norm), errors in $\sigma$ (measured in $L^2$) and the pressure $p$ (also measured in $L^2$). The exact mass conservation property is directly related to another structure-preservation property called pressure robustness, as shown by pressure-independent velocity error estimates. The computational cost measured in terms of interface degrees of freedom is comparable to old and new Stokes discretizations.