A mass conserving mixed stress formulation for the Stokes equations. (17th May 2019) - Gopalakrishnan, Jay; Lederer, Philip L; Schöberl, Joachim
Abstract: We propose 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 $σ $ 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 $σ $ (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.