WIAS Preprint No. 1979, (2014)

Guaranteed energy error estimators for a modified robust Crouzeix--Raviart Stokes element


  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian
    ORCID: 0000-0002-3390-2145

2010 Mathematics Subject Classification

  • 65N30 65N15 76D07

2008 Physics and Astronomy Classification Scheme

  • 47.11.Fg


  • mixed finite elements, a posteriori error estimation, divergence-free method, incompressible Stokes equations, Crouzeix-Raviart element




This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix--Raviart finite element method for the Stokes equations. The modification from [A. Linke 2014, On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime] is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method.

Appeared in

  • J. Sci. Comput., 64 (2015) pp. 541--558.

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