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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.
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 w
We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity, and nudging applied to the velocity only. We prove that under
We analyze Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is co
In this work, several multilevel decoupled algorithms are proposed for a mixed Navier-Stokes/Darcy model. These algorithms are based on either successively or parallelly solving two linear subdomain problems after solving a coupled nonlinear coarse g
We consider Chorin-Temam scheme (the simplest pressure-correction projection method) for the time-discretization of an unstationary Stokes problem. Inspired by the analyses of the Backward Euler scheme performed by C.Bernardi and R.Verfurth, we deriv