WIAS Preprint No. 2374, (2017)

Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes problem



Authors

  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian
  • Neilan, Michael
  • Neumann, Felix

2010 Mathematics Subject Classification

  • 65N12 65N30 65N15 76D07 76M10

Keywords

  • incompressible Stokes equations, mixed finite element methods, nonconforming discretizations, pressure-robustness, a-priori error estimates, Helmholtz projector

DOI

10.20347/WIAS.PREPRINT.2374

Abstract

Nearly all classical inf-sup stable mixed finite element methods for the incompressible Stokes equations are not pressure-robust, i.e., the velocity error is dependent on the pressure. However, recent results show that pressure-robustness can be recovered by a non-standard discretization of the right hand side alone. This variational crime introduces a consistency error in the method which can be estimated in a straightforward manner provided that the exact velocity solution is sufficiently smooth. The purpose of this paper is to analyze the pressure-robust scheme with low regularity. The numerical analysis applies divergence-free H¹-conforming Stokes finite element methods as a theoretical tool. As an example, pressure-robust velocity and pressure a-priori error estimates will be presented for the (first order) nonconforming Crouzeix--Raviart element. A key feature in the analysis is the dependence of the errors on the Helmholtz projector of the right hand side data, and not on the entire data term. Numerical examples illustrate the theoretical results.

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