WIAS Preprint No. 2140, (2015)

Optimal L2 velocity error estimate for a modified pressure-robust Crouzeix--Raviart Stokes element



Authors

  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian
  • Wollner, Winnifried

2010 Mathematics Subject Classification

  • 76M10 76D07

Keywords

  • incompressible Navier--Stokes equations, mixed finite elements, a priori error analysis, duality argument

Abstract

Recently, a novel approach for the robust discretization of the incompressible Stokes equations was proposed that slightly modifies the nonconforming Crouzeix--Raviart element such that its velocity error becomes pressure-independent. The modification results in an O(h) consistency error that allows straightforward proofs for the optimal convergence of the discrete energy norm of the velocity and of the L2 norm of the pressure. However, though the optimal convergence of the velocity in the L2 norm was observed numerically, it appeared to be nontrivial to prove. In this contribution, this gap is closed. Moreover, the dependence of the error estimates on the discrete inf-sup constant is traced in detail, which shows that classical error estimates are extremely pessimistic on domains with large aspect ratios. Numerical experiments in 2D and 3D illustrate the theoretical findings.

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