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The resolution of the incompressible Navier-Stokes equations is tricky, andit is well known that one of the major issue is to compute a divergence freevelocity. The non-conforming Crouzeix-Raviart finite element are convenientsince they induce local mass conservation. Moreover they are such that thestability constant of the Fortin operator is equal to 1. This implies that theycan easily handle anisotropic mesh [1, 2]. However spurious velocities mayappear and damage the approximation. We propose a scheme here that allows toreduce the spurious velocities. It is based on a new discretisation for thegradient of pressure based on the symmetric MPFA scheme (finite volumeMultiPoint Flux Approximation) [3, 4, 5].

Improved Crouzeix-Raviart scheme for the Stokes and Navier-Stokes problem