WIAS Preprint No. 2106, (2015)

Guaranteed error control for the pseudostress approximation of the Stokes equations



Authors

  • Bringmann, Philipp
  • Carstensen, Carsten
  • Merdon, Christian

2010 Mathematics Subject Classification

  • 65N30 65N15 76D07

Keywords

  • nonconforming finite element method, Crouzeix-Raviart element, Stokes equations, pseudostress finite element method, adaptive finite element method, a posteriori error estimation

Abstract

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.

Appeared in

  • Numer. Methods Partial Differential Equations, 32 (2016) pp. 1411--1432.

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