Guaranteed velocity error control for the pseudostress approximation of the Stokes equations

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.

Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations

“…Table 1 shows the different inf-sup stable velocity pressure pairs that we consider for the primal formulation (5). Further we give the abbreviation that we use, the expected convergence rate of the error r and the used reconstruction operator in (5) that ensures pressure-robustness.…”

“…In this paper we turn our interest now to guaranteed error control for the velocity and thereby refine existing approaches in [17,34,5,27,32]. In principle, the unified approach from e.g.…”

Projection stabilized nonconforming finite element methods for the stokes problem

“…Carstensen et al proposed and analyzed an a posteriori error estimator for discontinuous Galerkin methods by using a stress‐velocity‐pressure formulation for the Stokes equations. Recently, Bringmann et al presented a survey where they compared different strategies for guaranteed velocity error control for the pseudostress approximation of the Stokes equations using the lowest‐order nonconforming Crouzeix‐Raviart finite element method.…”