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

## TL;DR

This paper presents a novel finite element method for Stokes flow that weakly enforces stress symmetry, ensuring mass conservation, stability, and optimal convergence rates for velocity, pressure, and stress variables.

## Contribution

It introduces a new discretization that directly approximates symmetric viscous stresses with weak enforcement, improving stability and accuracy over previous methods.

## Key findings

- Achieves optimal convergence rates for pressure and stress.
- Ensures exact mass conservation with $H(div)$-conforming velocity.
- Method is pressure robust and stable.

## 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 $\sigma$, 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.