Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Doi
Abstract
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