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Most classical finite element schemes for the (Navier-)Stokes equations areneither pressure-robust, nor are they inf-sup stable on general anisotropictriangulations. A lack of pressure-robustness may lead to large velocityerrors, whenever the Stokes momentum balance is dominated by a strong andcomplicated pressure gradient. It is a consequence of a method, which does notexactly satisfy the divergence constraint. However, inf-sup stable schemes canoften be made pressure-robust just by a recent, modified discretization of theexterior forcing term, using $\mathbf{H}(\operatorname{div})$-conformingvelocity reconstruction operators. This approach has so far only been analyzedon shape-regular triangulations. The novelty of the present contribution isthat the reconstruction approach for the Crouzeix-Raviart method, which has astable Fortin operator on arbitrary meshes, is combined with results on theinterpolation error on anisotropic elements for reconstruction operators ofRaviart-Thomas and Brezzi-Douglas-Marini type, generalizing the method to alarge class of anisotropic triangulations. Numerical examples confirm thetheoretical results in a 2D and a 3D test case.

A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes