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We introduce a new discretization of a mixed formulation of the incompressible Stokes equations that includes symmetric stresses. The method is built upon the mass conserving mixed stress (MCS) formulation that we recently introduced. 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 presenting 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. We present an error analysis showing optimal rates for the pressure and the stress variables as well as a post processing for the velocity that yields exact mass conservation and optimal convergence.

A Mass Conserving Mixed Stress Formulation for Stokes Flow with Weakly Imposed Stress Symmetry