Guaranteed error control for the pseudostress approximation of the Stokes equations
- Bringmann, Philipp
- Carstensen, Carsten
- Merdon, Christian
2010 Mathematics Subject Classification
- 65N30 65N15 76D07
- nonconforming finite element method, Crouzeix-Raviart element, Stokes equations, pseudostress finite element method, adaptive finite element method, a posteriori error estimation
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.
- Numer. Methods Partial Differential Equations, 32 (2016) pp. 1411--1432.