Geodesic
Improved Crouzeix-Raviart scheme for the Stokes and Navier-Stokes problem
Eric Chénier1 ; Erell Jamelot2 ; Christophe Le Potier2 ; Andrew Peitavy2
1Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, UMR 8208, MSME, F-77454 Marne-la-Vallée, France
2Université Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des Fluides, 91191 Gif-sur-Yvette, France
ESAIM. Proceedings, Tome 76 (2024), pp. 20-34
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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).
DOI : 10.1051/proc/202476020

Eric Chénier  1   ; Erell Jamelot  2   ; Christophe Le Potier  2   ; Andrew Peitavy  2

1 Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, UMR 8208, MSME, F-77454 Marne-la-Vallée, France
2 Université Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des Fluides, 91191 Gif-sur-Yvette, France 
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     author = {Eric Ch\'enier and Erell Jamelot and Christophe Le Potier and Andrew Peitavy},
     title = {Improved {Crouzeix-Raviart} scheme for the {Stokes} and {Navier-Stokes} problem},
     journal = {ESAIM. Proceedings},
     pages = {20--34},
     year = {2024},
     volume = {76},
     doi = {10.1051/proc/202476020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/202476020/}
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Eric Chénier; Erell Jamelot; Christophe Le Potier; Andrew Peitavy. Improved Crouzeix-Raviart scheme for the Stokes and Navier-Stokes problem. ESAIM. Proceedings, Tome 76 (2024), pp. 20-34. doi: 10.1051/proc/202476020

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