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

Abstract

The resolution of the incompressible Navier-Stokes equations is tricky, and it is well known that one of the major issue is to approach the space: H1(Ω)∩ H(div 0; Ω) := {v ∈ H1(Ω) : div v = 0}. H 1 (Ω)∩ H ( div 0 ;Ω):= { v ∈ H 1 (Ω): div v = 0 }. $$ \mathbf{H}^1(\Omega) \cap \mathbf{H}(\operatorname{div} 0 ; \Omega):=\left\{\mathbf{v} \in \mathbf{H}^1(\Omega): \operatorname{div} \mathbf{v}=0\right\}. $$ The non-conforming Crouzeix-Raviart finite element are convenient since they induce local mass conservation. Moreover they are such that the stability constant of the Fortin operator is equal to 1. This implies that they can easily handle anisotropic mesh. However spurious velocities may appear and damage the approximation. We propose a scheme here that allows one to reduce the spurious velocities. It is based on a new discretisation for the gradient of pressure based on the symmetric MPFA scheme (finite volume MultiPoint Flux Approximation).