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J. Gopalakrishnan, P. Lederer, J. Schöberl:
"A mass conserving mixed stress formulation for the Stokes equations";
arXiv.org e-Print archive, 1806 (2018), 07173; 32 pages.

We propose a new discretization of a mixed stress formulation of the Stokes equations. The velocity u is approximated with H(div)-conforming finite elements providing exact mass conservation. While many standard methods use H1-conforming spaces for the discrete velocity, H(div)-conformity fits the considered variational formulation in this work. A new stress-like variable σ equalling the gradient of the velocity is set within a new function space H(curldiv). New matrix-valued finite elements having continuous "normal-tangential" components are constructed to approximate functions in H(curldiv). An error analysis concludes with optimal rates of convergence for errors in u (measured in a discrete H1-norm), errors in σ (measured in L2) and the pressure p (also measured in L2). 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.