Guaranteed error control for the pseudostress approximation of the Stokes equations

The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H (div) and the velocity in L2. Any standard mixed finite element function space can be utilized for this mixed formulation, e.g. the Raviart-Thomas discretization which is related to the Crouzeix-Raviart nonconforming finite element scheme in the lowest-order case. The effective and guaranteed a posteriori error control for this nonconforming velocity-oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf-sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy

Guaranteed error control for the pseudostress approximation of the Stokes equations

The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in $L^2$. Any standard mixed finite element function space can be utilized for this mixed formulation, e.g. the Raviart-Thomas discretization which is related to the Crouzeix-Raviart nonconforming finite element scheme in the lowest-order case. The effective and guaranteed a posteriori error control for this nonconforming velocity-oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf-sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy

Stabilized nonconforming finite element methods for data assimilation in incompressible flows

We consider a stabilized nonconforming finite element method for data assimilation in incompressible flow subject to the Stokes' equations. The method uses a primal dual structure that allows for the inclusion of nonstandard data. Error estimates are obtained that are optimal compared to the conditional stability of the ill-posed data assimilation problem